Download Joint Probability Density Function Closure of Turbulent

Mathias Leander Hack
Joint Probability Density Function (PDF) Closure of Turbulent Premixed Flames
The accurate and reliable prediction of turbulent premixed flames is a crucial task and
even after 50 years of intense research this problem could not be solved. One reason is
found in the sophisticated turbulence-chemistry-interaction occurring on a wide range of
length and time scales. With the help of direct numerical simulations (DNS) and
experimental investigations of such flames, the general understanding of the underlying
processes within a premixed turbulent flame could be ameliorated tremendously, but
remains far from being complete. Moreover, the computational effort makes it infeasible to
predict turbulent flames without any modeling assumptions. An optimal balance between
computational efficiency and level of closure has to be found. By that reason a variety of
approaches dealing with the description of phenomena within turbulent flames have been
proposed. The Reynolds-averaged Navier-Stokes (RANS) equations can be solved very
efficiently, but the resolution remains at a level of mean flow and thermodynamic
quantities, such that predictions strongly depend on the disposed models. In large eddy
simulations (LES) the large turbulent scales are resolved, such that only the so-called
subgrid scale (SGS) phenomena have to be modeled. This approach allows to study
instantaneous flame dynamics at the price of increased simulation times. A sophisticated
alternative are transported probability density function (PDF) methods. While providing the
full statistical information of flow and thermodynamic quantities at a certain location, the
numerical effort remains between those of RANS and LES simulations. Moreover,
turbulent convection and mean source terms appear in closed form in the PDF transport
equation. Due to the high dimensionality of the corresponding sample space, the PDF
transport equation is solved by a Monte-Carlo particle method, representing the PDF by an
ensemble of computational particles. For these particles, an equipollent set of stochastic
differential equations (SDE) is solved. In addition, a finite-volume method provides the
mean flow velocity by solving the RANS equations, where the RANS equations get closed
by the joint statistics of the PDF method. Even though consistency is ensured at the level
of governing equations, it is not straightforward to achieve convergence and consistency
between the two methods. Therefore, various correction schemes have been proposed to
achieve consistency and convergence. The first part of this work deals with the hybrid
solution strategy. In addition, a study on the topic of particle number control mechanisms is
presented.
In the second part of the thesis, a novel model for turbulent premixed combustion is
presented for the corrugated flamelet regime. It is based on a transported joint PDF of
velocities, turbulence frequency and scalars. A binary progress variable indicates the
arrival of embedded quasi laminar flames within a turbulent flame brush at the particle
location. In addition, a flame residence time is introduced to resolve the embedded quasi
laminar flame structure. Under the assumption of undisrupted embedded flame structures
and of constant laminar flame speed (i.e. unaffected by strain effects), the particle
composition can be retrieved from precomputed one-dimensional laminar flame tables
knowing its flame residence time. The ''ignition'' of reactive unburnt fluid elements by the
propagating embedded flame is described by the ''ignition'' probability P, which describes
the rate at which unburnt particles get consumed by the flame. First, an empirical ansatz
for P is proposed; second with the help of a flag indicating whether the flame residence
time lies within a specified range, the ignition probability is calculated based on an
estimate of the mean flame surface density. Latter gets transported by the PDF method,
but to account for flame stretching, curvature effects, collapse and cusp formation, a
mixing model for the residence time is employed. The same mixing model also accounts
for molecular mixing of the products with a co-flow. Numerical simulations of three piloted
premixed Bunsen flames show excellent agreement with the experimental measurements
and demonstrate the applicability of the proposed PDF model.
Joint Probability Density
Function (PDF) Closure of
Turbulent Premixed Flames
Mathias Leander Hack
Dissertation ETH No. 19927
Mean temperature field of a piloted Bunsen flame; the red color represents
hot burnt gas and the blue color cold gas. This figure is generated from a
simulation of the Aachen flame F1 which is presented in this work.
An online version of this thesis is available at the ETH e-collection:
http://e-collection.ethbib.ethz.ch/
DISS. ETH NO. 19927
Joint Probability Density Function (PDF)
Closure of Turbulent Premixed Flames
A dissertation submitted to
ETH ZURICH
for the degree of
Doctor of Sciences
presented by
MATHIAS LEANDER HACK
Dipl. Rech. Wiss. ETH
born on August 24, 1978
citizen of Wolfenschiessen NW (Switzerland)
accepted on the recommendation of
Prof. Dr. Patrick Jenny, examiner
Prof. Dr.-Ing. Johannes Janicka, co-examiner
2011
I
Abstract
The accurate and reliable prediction of turbulent premixed flames is a crucial task and even after 50 years of intense research this problem could not
be solved. One reason is found in the sophisticated turbulence-chemistryinteraction occurring on a wide range of length and time scales. With the
help of direct numerical simulations (DNS) and experimental investigations
of such flames, the general understanding of the underlying processes within
a premixed turbulent flame could be ameliorated tremendously, but remains
far from being complete. Moreover, the computational effort makes it infeasible to predict turbulent flames without any modeling assumptions. An
optimal balance between computational efficiency and level of closure has to
be found. By that reason a variety of approaches dealing with the description
of phenomena within turbulent flames have been proposed.
The Reynolds-averaged Navier-Stokes (RANS) equations can be solved
very efficiently, but the resolution remains at a level of mean flow and thermodynamic quantities, such that predictions strongly depend on the disposed
models. In large eddy simulations (LES) the large turbulent scales are resolved, such that only the so-called subgrid scale (SGS) phenomena have to
be modeled. This approach allows to study instantaneous flame dynamics
at the price of increased simulation times. A sophisticated alternative are
transported probability density function (PDF) methods. While providing
the full statistical information of flow and thermodynamic quantities at a
certain location, the numerical effort remains between those of RANS and
LES simulations. Moreover, turbulent convection and mean source terms
appear in closed form in the PDF transport equation.
Due to the high dimensionality of the corresponding sample space, the
PDF transport equation is solved by a Monte-Carlo particle method, representing the PDF by an ensemble of computational particles. For these particles, an equipollent set of stochastic differential equations (SDE) is solved.
In addition, a finite-volume method provides the mean flow velocity by solving the RANS equations, where the RANS equations get closed by the joint
statistics of the PDF method. Even though consistency is ensured at the level
of governing equations, it is not straightforward to achieve convergence and
consistency between the two methods. Therefore, various correction schemes
have been proposed to achieve consistency and convergence. The first part
of this work deals with the hybrid solution strategy. In addition, a study on
the topic of particle number control mechanisms is presented.
In the second part of the thesis, a novel model for turbulent premixed combustion is presented for the corrugated flamelet regime. It is based on a
II
transported joint PDF of velocities, turbulence frequency and scalars. A
binary progress variable indicates the arrival of embedded quasi laminar
flames within a turbulent flame brush at the particle location. In addition,
a flame residence time is introduced to resolve the embedded quasi laminar
flame structure. Under the assumption of undisrupted embedded flame structures and of constant laminar flame speed (i.e. unaffected by strain effects),
the particle composition can be retrieved from precomputed one-dimensional
laminar flame tables knowing its flame residence time. The ”ignition” of reactive unburnt fluid elements by the propagating embedded flame is described
by the ”ignition” probability P , which describes the rate at which unburnt
particles get consumed by the flame. First, an empirical ansatz for P is proposed; second with the help of a flag indicating whether the flame residence
time lies within a specified range, the ignition probability is calculated based
on an estimate of the mean flame surface density. Latter gets transported by
the PDF method, but to account for flame stretching, curvature effects, collapse and cusp formation, a mixing model for the residence time is employed.
The same mixing model also accounts for molecular mixing of the products
with a co-flow. Numerical simulations of three piloted premixed Bunsen
flames show excellent agreement with the experimental measurements and
demonstrate the applicability of the proposed PDF model.
III
Zusammenfassung
Die zuverlässige und exakte Vorhersage von turbulenten Vormischflammen
ist eine wichtige Aufgabe und auch nach 50 Jahren der intensiven Forschung
ist dieses Problem noch nicht vollständig gelöst. Der Hauptgrund hierfür sind
die komplexen Wechselwirkungen zwischen Turbulenz und Chemie über eine
grosse Bandbreite von Längen- und Zeitskalen. Mittels Direkten Numerischen
Simulationen (DNS) und experimentellen Untersuchungen konnte das allgemeine Verständnis der zugrundeliegenden Phänomene innerhalb turbulenter
Vormischflammen stark verbessert werden; allerdings verhindert der enorme
Rechenaufwand Simulationen turbulenter Flammen ohne Modellierungsannahmen. Deswegen wurden in der Vergangenheit zahlreiche Ansätze zur Beschreibung turbulenter Flammen vorgeschlagen, um ein Optimum zwischen
numerischem Aufwand und Schliessungsgrad zu finden.
Die Reynolds-gemittelten Navier-Stokes (RANS) Gleichungen kön-nen
sehr effizient gelöst werden, wobei die Simulationsresultate stark von den
verwendeten Modellen abhängen, da nur die mittleren Strömungsfelder und
thermodynamischen Grössen aufgelöst werden. In Grobstruktursimulationen
(LES), in welchen die grossen turbulenten Skalen aufgelöst sind, müssen lediglich Modelle für Subskalenphänomene verwendet werden. Dies ermöglich
ein Studium der instantanen Flammendynamik zu Lasten eines erhöhten
Simulationsaufwandes. Eine attraktive Alternative hierzu bilden die transportierten Verbundswahrscheinlichtkeitsdichtefunktionen (JPDF) Methoden.
Während diese Methoden die kompletten Ein-Punkt-Statistiken von strömungsrelevanten Grössen zur Lösung heranziehen, liegt deren numerischer
Aufwand zwischen dem von RANS und LES. Zudem treten in der PDFTransportgleichung turbulente Konvektion und mittlere Quellterme in geschlossener Form auf.
Aufgrund ihres hochdimensionalen Ereignisraumes wird die PDF-Transportgleichung mittels Monte-Carlo Partikelmethode gelöst, wobei die PDF
durch ein Kollektiv imaginärer Partikel repräsentiert wird. Für diese Partikel wird ein äquivalentes System von stochastischen Differenzialgleichungen
gelöst. Zudem berechnet ein Finite-Volumen-Löser die mittlere Strömungsgeschwindigkeit anhand der RANS Gleichungen, wobei die ungeschlossenen
Terme mittels der Verbundsstatistik berechnet werden. Konvergenz und Konsistenz zwischen den beiden Methoden sind numerisch nicht gewährleistet,
obwohl diese beiden Methoden, auf dem Niveau der mathematischen Formulierung, konsistent sind. Aus diesem Grund wurden in der Vergangenheit
diverse Korrekturschemata vorgeschlagen. Im ersten Teil dieser Arbeit wird
eine Hybridlösungsstrategie betrachtet und eine Studie zum Thema der Partikelanzahlkontrolle pro Zelle präsentiert.
IV
Im zweiten Teil der Arbeit wird ein neuartiges Modell für turbulente Vormischflammen im ”Corrugated Flamelet”-Regime, basierend auf einer transportierten JPDF von Geschwindigkeiten, turbulenter Frequenz und Skalaren,
vorgestellt. Eine binäre Fortschrittsvariable kennzeichnet die Ankunft von
eingebetteten quasi-laminaren Flammen innerhalb einer turbulenten Flamme am Ort des Partikels. Um die eingebettete quasi-eindimensionale Flammenstruktur aufzulösen, wird eine Flammenaufenthaltszeit eingeführt. Zudem wird angenommen, dass die eingebetteten Flammenstrukturen zusammenhängend und die laminaren Flammengeschwindigkeiten konstant bleiben
(d.h nicht durch Verzerrungseffekte beeinflusst werden). Daher kann die Partikelkomposition aus vorab berechneten eindimensionalen laminaren Flammentabellen ausgelesen werden, sofern die Flammenaufenthaltszeit bekannt
ist. Die ”Zündung” von reaktiven unverbrannten Fluidelementen durch propagierende eingebettete Flammen wird durch die Verbrennungswahrscheinlichkeit P beschrieben. Einerseits wird ein empirischer Ansatz für P vorgeschlagen, andererseits wird P , mit der Hilfe eines Partikelindikators für
einen bestimmten Bereich der Flammenaufenthaltszeit, basierend auf einer
Approximation der mittleren Flammenoberflächendichte berechnet. Letztere
wird durch die PDF-Methode transportiert. Um aber Phänomene wie Flammenstreckung, Krümmungseffekte, zusammenlaufende Flammenfronten und
von der Flamme abgelöste brennende Fluidvolumen zu beschreiben, wird ein
Mischungsmodell auf die Flammenaufenthaltszeit angewendet. Das gleiche
Mischungsmodell trägt auch dem molekularen Mischen von heissen Produkten mit der umgebenden Luft Rechnung. Numerische Simulationen von drei
pilotierten vorgemischen Bunsenflammen zeigen eine exzellente Übereinstimmung mit den experimentellen Messungen und demonstrieren die Anwendbarkeit des vorgeschlagenen PDF-Modelles.
V
Acknowledgments
I am deeply grateful to Prof. Patrick Jenny giving me the opportunity to
join this multifarious and very interesting project and being my PhD supervisor at the Institute of Fluid Dynamics. His advises and many intense and
fruitful discussions during the past five years were important ingredients for
the progress of this project and a huge motivation for myself.
I would like to thank Prof. Johannes Janicka for agreeing to become my
co-referent and for the valuable feedback regarding my thesis.
I am also thankful to my comrades at the Institute of Fluid Dynamics at
the ETH Zurich for the great spirit among us, and especially to the member
of the ”combustion group” for many many small but crucial talks on both
scientific and personal matter. I am also indebted to our secretaries Ms.
Bianca Maspero and Ms. Sonja Atkinson for their administrative guidance
and to our system administrator Mr. Hans Peter Caprez for his IT-support.
My parents have always believed in me and enforced me to become who
I am today, wherefore I am very grateful to. Moreover, I am much obliged
to my father spending many hours as lector for my thesis.
My greatest gratefulness belongs to my wife Olivia. She has supported me
during my whole doctoral studies, and with her, I have always felt secure,
even in stressful times.
This work was financially supported by the ETH Zurich.
VI
to Olivia
Contents
1 Preface
1
I
7
Probability Density Function (PDF) Methods
2 Computational Approaches
2.1 Direct Numerical Simulations . . . . . . . .
2.2 Reynolds-averaged Navier-Stokes Equations
2.3 Large Eddy Simulations . . . . . . . . . . .
2.4 Probability Density Function Methods . . .
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9
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3 Joint PDF Framework
3.1 Foundation of Probability Density Functions . . . . .
3.1.1 Definition and Properties . . . . . . . . . . . .
3.1.2 Dirac Delta Function and Heavyside Function
3.1.3 Joint Probability Density Function . . . . . .
3.2 Eulerian Joint PDF . . . . . . . . . . . . . . . . . . .
3.2.1 Different Types of JPDFs . . . . . . . . . . .
3.2.2 JPDF Transport Equation . . . . . . . . . . .
3.3 Lagrangian Joint PDF . . . . . . . . . . . . . . . . .
3.3.1 Lagrangian System . . . . . . . . . . . . . . .
3.3.2 JPDF Transport Equation . . . . . . . . . . .
3.3.3 Relation to Eulerian Joint PDF . . . . . . . .
3.4 Stochastic Systems . . . . . . . . . . . . . . . . . . .
3.4.1 Markov Process . . . . . . . . . . . . . . . . .
3.4.2 Differential Chapman-Kolmogorov Equation .
3.4.3 Jump Process . . . . . . . . . . . . . . . . . .
3.4.4 Diffusion Process . . . . . . . . . . . . . . . .
3.4.5 Stochastic Differential Equations (SDE) . . .
3.5 Modeled Joint PDF . . . . . . . . . . . . . . . . . . .
3.5.1 Example Model PDF . . . . . . . . . . . . . .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . .
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17
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4 Hybrid Solution Strategy
4.1 Hybrid Method . . . . .
4.1.1 Set of Equations
4.1.2 Particle Method .
4.2 Particle Number Control
4.2.1 Motivation . . . .
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4.2.2
4.2.3
4.2.4
5 1D
5.1
5.2
5.3
5.4
II
Clone/Cluster - A Review . . . . . . . . . . . . . . . . 40
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Summary . . . . . . . . . . . . . . . . . . . . . . . . . 46
Setup
Idea . . . . . . . . .
Governing Equations
Convergence Study .
Summary . . . . . .
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Modeling of Turbulent Premixed Combustion
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6 Turbulent Premixed Combustion
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6.1 Laminar Premixed Flames . . . . . . . . . . . . . . . . . . . . 65
6.2 Regimes of Turbulent Premixed Combustion . . . . . . . . . . 67
7 Existing Modeling Approaches
7.1 BML Approach . . . . . . . . . . . . . .
7.1.1 Flame Crossing Frequencies . . .
7.1.2 Flame Surface Density Approach
7.2 G-Equation Model . . . . . . . . . . . .
7.3 PDF Approach . . . . . . . . . . . . . .
7.4 Reduced Chemical Mechanisms . . . . .
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8 Novel JPDF Combustion Model
8.1 Motivation . . . . . . . . . . . . . . . . . .
8.2 JPDF Method . . . . . . . . . . . . . . . .
8.3 Combustion Model . . . . . . . . . . . . .
8.4 Ignition Probability . . . . . . . . . . . . .
8.5 Tabulation . . . . . . . . . . . . . . . . . .
8.6 Mixing Model . . . . . . . . . . . . . . . .
8.7 Results . . . . . . . . . . . . . . . . . . . .
8.7.1 Quasi 1D Swirl Burner Simulation .
8.7.2 Piloted Bunsen Burner . . . . . . .
8.7.3 Sensitivity and Convergence Study
8.8 Summary . . . . . . . . . . . . . . . . . .
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9 Combustion Model Extension
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9.1 Combustion Modeling Review . . . . . . . . . . . . . . . . . . 108
9.2 Ignition Probability . . . . . . . . . . . . . . . . . . . . . . . . 109
9.3 Analogy to the G-Equation Approach . . . . . . . . . . . . . . 111
9.4
9.5
9.6
Molecular Mixing . . . . . . . . .
Results . . . . . . . . . . . . . . .
9.5.1 Non-Reactive Results . . .
9.5.2 Reactive Results . . . . .
9.5.3 Mechanical-to-Scalar-Time
9.5.4 Numerical Analysis . . . .
Summary . . . . . . . . . . . . .
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Scale Ratio
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112
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10 Conclusions
131
References
135
Curriculum Vitae
145
List of Figures
1
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11
Sketch of the geometry of the generic piloted jet flame. . . . .
Bias convergence study: Mean downstream velocity against
1/NANPC at four locations; reference (◦), statistical elimination
(∗), random elimination (+) and quant elimination (). . . . .
Bias convergence study: Favre averaged turbulent kinetic energy against 1/NANPC at four locations; reference (◦), statistical elimination (∗), random elimination (+) and quant elimination (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bias convergence study: Mean temperature against 1/NANPC
at four locations; reference (◦), statistical elimination (∗), random elimination (+) and quant elimination (). . . . . . . . .
Schematic illustration of the simplified quasi one-dimensional
simulation setup. . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic illustration of the simplified quasi one-dimensional
simulation of a weak swirl burner. The shaded area is the
computational domain, the dashed lines are iso-temperature
levels and the arrows represent mean flow field stream lines. .
Schematic illustration of the simplified configuration to simulate planar turbulent flames, where the turbulent flame speed
is computed. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evolution of the turbulent flame speed sT for different turbulent intensities over a period of 2 million time steps: urms =
0.2m/s (solid line), urms = 0.8m/s (dashed line), urms =
1.4m/s (dotted line) and urms = 2.0m/s (dashed-dotted line).
Simulation result for an rms velocity fluctuation of 2.0m/s:
f1 (solid line), mean
fixed downstream momentum profile hρi U
f1 (dashed line), normalized dendownstream velocity profile U
00 00
sity 10.0 hρi (dotted line) and Reynolds stresses ug
1 u1 (dasheddotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Evolution of the location of the flame front xT over a period of
2 million time steps: xT,init = 1.2m (solid line), xT,init = 0.9m
(dashed line), xT,init = 0.6m (dotted line) and xT,init = 0.3m
(dashed-dotted line). . . . . . . . . . . . . . . . . . . . . . . .
Evolution of the turbulent flame speed sT over a period of
2 million time steps: xT,init = 1.2 (solid line), xT,init = 0.9
(dashed line), xT,init = 0.6 (dotted line) and xT,init = 0.3
(dashed-dotted line). . . . . . . . . . . . . . . . . . . . . . . .
43
45
46
47
50
51
52
55
56
57
57
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Evolution of the turbulent flame speed sT over a period of 1
million time steps: Nx = 100 (solid line with circles), Nx = 200
(dashed line), Nx = 400 (dotted line), Nx = 800 (dasheddotted line) and Nx = 1600 (solid line). . . . . . . . . . . . . .
Evolution of the turbulent flame speed sT over a period of 0.5
million time steps: µmem = 0.99 (solid line), µmem = 0.999
(dashed line), µmem = 0.9999 (dotted line) and µmem = 0.9999
(dashed-dotted line). . . . . . . . . . . . . . . . . . . . . . . .
Evolution of the turbulent flame speed sT over a period of 0.5
million time steps: µmem = 0.999 and NN P C = 10 (solid line)
and µmem = 0.99 and NN P C = 100 (dashed line). . . . . . . . .
Sketch of a steady laminar premixed flame containing the preheat zone (I), the inner or reaction zone (II) and the oxidation
layer (III). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Regimes of premixed turbulent combustion by [64]. . . . . . .
Sketch of a possible bimodal mass weighted PDF h̃ of the
progress variable c. . . . . . . . . . . . . . . . . . . . . . . . .
Sketch of the laminar 1D flame profile showing T ∗ , c∗ and τ ∗ . .
Sketch of α(ω) used for the simulations in [25]. . . . . . . . . .
Sketch of the normalized temperature T̂ as a function of the
mixture fraction Z and of the flame residence time τ . . . . . .
Axial profiles of the normalized mean temperature for all six
flow rates FR: experiments (dashed line) and simulations (solid
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Validation of the turbulent flame speed sT as a function of
the turbulence intensity ν 0 = (2k/3)0.5 , both scaled with the
laminar flame speed sL = 0.2: theoretical model [64] with
lt /lF = 43 (solid line), experiments (dashed line with crosses)
and numerical simulations (dashed-dotted line with circles). .
Sketch of the Aachen flame with the unignited jet (J), the hot
pilot (P ) and the co-flow (C). . . . . . . . . . . . . . . . . . .
Radial profiles of the normalized mean downstream velocity Û
at several downstream locations in the piloted Bunsen flame:
experiments (circles) and numerical simulations (solid lines). .
Radial profiles of the normalized turbulent kinetic energy k̂
at several downstream locations in the piloted Bunsen flame:
experiments (circles) and numerical simulations (solid lines). .
Radial profiles of the normalized mean temperature at several
downstream locations of the piloted Bunsen flame: experiments (circles) and numerical simulations (solid lines). . . . .
58
59
59
66
70
86
86
89
90
93
94
95
96
96
97
27
28
29
30
31
32
33
34
35
36
Radial profiles of the normalized rms temperature at several
downstream locations of the piloted Bunsen flame: experiments (circles) and numerical simulations (solid lines). . . .
Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the
piloted Bunsen flame: experiments (circles), reference simulations (solid lines) and 80 × 80 grid with 20 particles per cell
(dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . .
Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the
piloted Bunsen flame: experiments (circles), reference simulations (solid lines), CF = 0.8 (dashed lines) and ωmax =
700.0s−1 (dotted lines). . . . . . . . . . . . . . . . . . . . . .
Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the
piloted Bunsen flame: experiments (circles), reference simulations (solid lines), zf lamable = 0.5 (dashed lines) and zf lamable =
0.9 (dotted lines). . . . . . . . . . . . . . . . . . . . . . . . .
Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the
piloted Bunsen flame: experiments (circles), reference simulations (solid lines) and the manifold constructed out of 7 onedimensional laminar profiles between a mixture fraction of 0.7
and 1.0 (dashed lines). . . . . . . . . . . . . . . . . . . . . .
Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the
piloted Bunsen flame: experiments (circles), reference simulations (solid lines) and Cφ = 4.0 (dashed lines). . . . . . . . .
Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the
piloted Bunsen flame: experiments (circles), reference simulae1,co−f low = 1.0m/s (dashed lines). . .
tions (solid lines) and U
Sketch of an instantaneous flame surface with the volumes Ωu
(left) and Ωd (shaded). . . . . . . . . . . . . . . . . . . . .
Radial profiles of the normalized mean downstream velocity
Û1 for the three cold cases at several downstream locations:
experiments (circles) and numerical simulation (solid lines). .
Radial profiles of the normalized turbulent kinetic energy k̂
for the three cold cases at several downstream locations: experiments (circles) and numerical simulation (solid lines). . .
. 97
. 98
. 99
. 100
. 101
. 102
. 103
. 109
. 115
. 116
37
Radial profiles of the normalized mean downstream velocity
Û for all three flames at several downstream locations: experiments (circles) and numerical simulation (solid lines). . . . . . 117
38
Radial profiles of the normalized turbulent kinetic energy k̂ for
all three flames at several downstream locations: experiments
(circles) and numerical simulation (solid lines). . . . . . . . . . 118
39
e
Radial profiles of the normalized mean temperature T̂ for all
three flames at several downstream locations: experiments
(circles) and numerical simulation (solid lines). . . . . . . . . . 119
40
Radial profiles of the normalized rms-temperature T̂ rms for
all three flames at several downstream locations: experiments
(circles) and numerical simulation (solid lines). . . . . . . . . . 120
41
The manifold T ∗ −Z ∗ −τ ∗ represented by the particle ensemble
in the simulation of flame F3. . . . . . . . . . . . . . . . . . . 121
e
Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream
locations: experiments (circles), numerical simulations: Cφ =
2.0 (solid lines), Cφ = 4.0 (dashed lines), Cφ = 6.0 (dasheddotted lines) and Cφ = 8.0 (dotted lines). . . . . . . . . . . . . 122
42
43
e
Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream
locations: experiments (circles), numerical simulations: Nx =
50, Ny = 50, Np = 20 (solid lines) and Nx = 80, Ny = 80,
Np = 40 (dashed lines). . . . . . . . . . . . . . . . . . . . . . . 123
44
MC
Radial profiles of ρF Vρ−ρ
(left), ue001 (center) and ue002 (right) at
FV
two downstream locations in the reactive piloted Bunsen flame. 124
45
Sketch of particle trajectories (solid lines) within the numerical
grid (dashed lines). The bullets show the location of the flame
surface on the trajectories and the hatched area indicates the
locations where d∗ equals 1 for a particle. . . . . . . . . . . . 125
46
e
Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream
locations. d∗ is equal 1, if its normalized temperature lies
within [0.2, 0.8] (solid lines), [0.3, 0.7] (dashed lines) and [0.4, 0.6]
(dotted lines); experiments (circles). . . . . . . . . . . . . . . . 126
47
e
Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream
locations. d∗ is equal 1, if its normalized temperature lies
within [0.1, 0.9] (solid lines), [0.1, 0.7] (dashed lines) and [0.1, 0.5]
(dotted lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
List of Tables
1
2
3
Model constants for equations (86), (87) and (89). . . . . . . 37
Locations, at which the quantities are measured for the convergence study. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Averaged number of particles per cell (NANPC ) for all simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1
Preface
Worldwide, more than 80% of the total global primary energy supply is based
on fossil fuels [1]. During the past 35 years, the global primary energy supply
has been doubled up to approximately 12 · 109 tonnes of oil equivalent. The
percental fraction of energy based on burning fossil fuels has been decreased
from 86.6% down to 81.3%, whereas the total amount has been almost doubled. Therefore, it is ceaseless that the conversion of fossil energy sources is
further optimized in terms of efficiency and reduction of pollutant formations.
To achieve such improvements of the combustion process, experimental and
numerical techniques are used, where the accurate prediction of the underlying physical processes is essential. Most combustion applications operate
within turbulent flow fields. Examples are gas turbines, internal combustion
engines and spark-ignition engines. Turbulence itself is a huge research field
and far from fully understood. It is combined with highly non-linear combustion processes, such that the fundamental physical processes of turbulent
reactive flows are rather complex, especially the interaction of chemistry and
turbulence is very crucial. From a numerical point of view, no satisfying
general approach to predict turbulent reactive flows exists so far.
Turbulent reactive flows are usually distinct into premixed and non-premixed
flames. In a non-premixed or diffusion flame, fuel and oxidizer are induced
separately into the combustion chamber. At the interface of the two streams,
molecular diffusion is enhanced by the turbulence such that the gas mixture
becomes flammable. Thus, the location of the (diffusion) flame is determined
by the mixture of fuel and oxidizer, which is controlled by the turbulence and
the diffusion. These kinds of flames do not propagate; the diffusion transfers the necessary supply of fuel and oxidizer towards the flame, whereas the
burnt products are transported away from the flame. Therefore, one talks of
a reactive-diffusive flame structure. On the other hand, a premixed flame is
not bounded by the mixture at a certain location. Fuel and oxidizer streams
are mixed before entering the combustion chamber and thus the gas mixture
is ignitable everywhere. For an equivalence ratio in the flammable range,
2
1 PREFACE
i.e. between 0.5 and 1.5 approximately, the flame feeds into the fresh unburnt gas. This flame propagation is driven by the thermal diffusion into
the fresh gas, enhanced by the turbulence. Moreover, the flame gets wrinkled, stretched and strained by the turbulence such that the surface of the
flame increases, whereas locally the flame front still propagates at the laminar flame speed. Extinction of premixed flames may occur, when the flame
structure is disrupted and broken by eddies, entering into the inner reaction
zone of the flame. Compared to diffusion flames, no soot formation occurs
in premixed flames, but premixed combustion processes are less secure than
non-premixed ones, since once mixed, the gas is ignitable and less controllable. Therefore, it is said that premixed combustion is conditionally stable.
A critical key parameter is the crossover temperature, which is defined as the
temperature Tc at which the competing chain branching and chain breaking
reactions are in equilibrium. If the temperature of a system is lower than Tc ,
chain breaking reactions dominate chain branching ones, but a strong heat
source (spark) may initiate the combustion process in the whole combustion
chamber. By that reason, the premixing occurs under low temperature conditions. In the presented work, only premixed flames are considered.
This work mainly deals with the interaction of turbulence and chemistry
within premixed turbulent flames, where the accurate treatment of these
phenomena is essential for reliable predictions. This interaction is complex
and not fully understood, since a variety of superposed processes at different scales has to be considered. At interfaces of eddies, strain and shear
are amplified, which results in steepening of concentration gradients. These
strengthened gradients intensify the molecular and thermal diffusion such
that the fresh unignited gas is consumed faster by the propagating flame.
But combustion impacts the turbulence too; the Reynolds number decreases
through the flame, leading to a ”laminarization” of the flow. On the other
hand, the generated heat release may induce flow instabilities, that further
enhance the turbulent fluctuations. Another example of processes depending
on various time scales, is the formation of nitrogen oxides. NOx production
itself is a slow process compared to primary reaction time scales within a
premixed flame, but the formation depends on certain radicals occuring on
very small time scales.
Besides experimental techniques, the accomplishment of numerical simulations provides a powerful tool to investigate the underlying physical processes
within turbulent flames. To accurately predict reactive flows with the help of
numerical calculations, two necessary prerequisites have to be considered: (1)
the understanding of the underlying physical processes is essential to reliably
3
formulate models and (2) the numerical treatment of the governing equations has to be correct and efficient. Part I of this work mainly deals with
the second point, i.e. the governing equations and the numerical solution
strategies, where the focus lies on the usage of Monte-Carlo probability density function (PDF) methods applied to turbulent reactive flows. In Part II,
the main focus of attention is on the modeling of premixed flame-turbulence
interaction.
In a direct numerical simulation (DNS), even the smallest scales are resolved.
Such simulations are limited by the Reynolds-number barrier to simplified
flow configurations, and thus are inapplicable to industrial applications. By
that reason, simulations are performed based on transport equations for mean
quantities. An early model is the eddy break-up (EBU) model proposed by
Spalding in 1971 [86], expressing the mean reaction rate of a product proportional to the variance of the product. A series of similar closures was provided
in the following decades solving transport equations for mean flow quantities and their variance. Nevertheless, the increase of available computational
power gave raise for more sophisticated numerical techniques. In 1985, Pope
[72] presented the underlying mathematical framework for Monte-Carlo particle PDF methods applied to reactive flows, where the joint statistics of
flow properties as well as thermochemical quantities are computed. Issues
like counter-gradient diffusion could be circumvented by this approach, since
turbulent convection appears in closed form. With the introduction of hybrid methods, e.g. hybrid RANS/PDF methods, the numerical cost could be
drastically reduced. In spite of the exact mathematical formulation of the
governing PDF transport equation, it is not straightforward to achieve consistency within such hybrid methods. In large eddy simulations (LES), the
large energy-containing scales are resolved and only the small scales (so-called
subgrid scales (SGS)) have to be modeled. The advantages and drawbacks
of the mentioned approaches are briefly reviewed in chapter 2.
Because the presented combustion modeling approaches are based on a Lagrangian joint PDF consideration, the mathematical framework of PDF methods is presented in chapter 3. The joint PDF of velocity and composition is
introduced and a solution strategy is described. Furthermore, the relation
between Eulerian and Lagrangian PDF approaches is shown. The strategy
to solve PDF transport equations based on stochastic processes described by
the differential Chapman-Kolmogorov equation is reviewed.
In chapter 4, the PDF solution strategy is presented. A hybrid finitevolume/Monte-Carlo particle method is used to solve the PDF transport
4
1 PREFACE
equation in an efficient way. An algorithmic issue arises due to the employment of stochastic particles, namely the number control of computational
particles within a control volume. Such control mechanisms are required by
two reasons: to increase the numerical efficiency and to reduce the statistical
error. While producing particles is straightforward, particle elimination is
more complex and introduces a bias error for a low number of particles. A
quantitative study on the influence of different particle elimination schemes
for statistically stationary problems is presented with the help of a generic
jet flame. A new clustering approach is proposed, but is not able to improve
the best performing statistical elimination procedure. In chapter 5, a stand
alone stochastic particle method is proposed to perform simplified quasi onedimensional simulations of weak swirl burners.
Part II of this work deals with turbulent premixed combustion modeling. In
chapter 6, the chemical processes within a laminar premixed flame are briefly
described; for a detailed introduction into chemical kinetics it is referred to
standard textbooks on that topic, e.g. the book “Principles of Combustion“
by Kuo [42]. Since turbulence-chemistry-interactions occur on a wide range of
scales, the different regimes of premixed turbulent combustion are illustrated,
and several phenomena within turbulent flames are described. Chapter 7 reviews existing modeling approaches for turbulent premixed flames.
The next two chapters are dedicated to a novel model approach for turbulent premixed flames in the corrugated flamelet regime. The presented
models in chapters 8 and 9 are based on a transported joint probability density function (PDF) of velocities, turbulence frequency and scalars, where
the PDF is represented by an ensemble of computational (stochastic) particles. The progress variable from the BML model becomes a computational
particle property describing the arrival of the flame front at the particle location. By introducing a flame residence time, together with precomputed
one-dimensional laminar flame tables, the embedded quasi laminar flame
structure can be resolved, similar as in flamelet approaches for premixed
flames. The remaining challenge is to accurately describe the probability P
for an individual particle to ”ignite” during a time step. Note that P reflects
the coupled fine-scale convection-diffusion-reaction dynamics in the flame.
An empirical approach is proposed in chapter 8, where this ansatz is closed
by the usage of simplified one-dimensional flame simulations of a low swirl
burner. Moreover, in chapter 9, the ignition probability is calculated based
on an estimate of the mean flame surface density. Latter gets transported
by the PDF method, but to account for flame stretching, curvature effects,
collapse and cusp formation, a mixing model for the residence time is em-
5
ployed. The same mixing model also accounts for molecular mixing of the
products with a co-flow. Numerical validation studies of piloted premixed
Bunsen flames reveal the applicability of the proposed model for the prediction of premixed turbulent flames in the corrugated flamelet regime, and to
a small extent also in the thin reaction zone regime.
Finally, in chapter 10 the work is summarized and conclusions are given.
Part I
Probability Density Function
(PDF) Methods
2
Computational Approaches for Turbulent
Combustion
The aim of this chapter is to briefly summarize the different numerical approaches, that are employed to predict turbulent flames. The consideration is
at the bottom of the system described by equations (1) - (3), i.e. from top to
bottom, the continuity equation, the momentum conservation equations and
the scalar transport equations for the composition vector including species
mass fractions and enthalpy. This set of equations fully describes turbulent
reactive flows. The continuity equation reads
∂ρUj
∂ρ
+
= 0 ,
∂t
∂xj
(1)
the momentum evolution equations read
ρ
DUi
∂ρUi
∂ρUj Ui
∂p
∂τij
=
+
= −
+
Dt
∂t
∂xj
∂xi
∂xj
(2)
and the scalar transport equations are
∂ρφα
∂ρUj φα
Dφα
∂
=
+
ρ
=
Dt
∂t
∂xj
∂xj
∂φα
Γ(α) ρ
∂xj
+ ρSα
.
(3)
The density is denoted by ρ, the velocity component in the xi -direction by
Ui , the pressure by p and the viscous shear stress tensor by τij . Equation (3)
is solved for all Ns scalars φα . The molecular diffusion term (first term on
the right-hand-side of equation (3)) contains the diffusion coefficient Γ(α) of
the scalar with index α. The brackets around α denote that the Einstein
summation convention is circumvented and Sα is the source term of scalar
α. In addition, the viscous shear stress tensor τij is defined as
∂Ui ∂Uj
2 ∂Uk
+
− µ
δij ,
(4)
τij = µ
∂xj
∂xi
3 ∂xk
where δij is the Kronecker delta function.
10
2.1
2 COMPUTATIONAL APPROACHES
Direct Numerical Simulations
In direct numerical simulations (DNS), equations (1)-(3)) are directly calculated at a level, where all length and time scales are resolved. The advantage
of this method is that no models are required for the turbulence-chemistry
interaction. But it is only feasible with the penalty of tremendous numerical
cost, such that it is inapplicable for industrial applications. The numerical cost is approximately proportional to Re3 , where Re is the Reynolds
number. This criterion is called the Reynolds barrier and limits DNS to simple, academic flow configurations with low Reynolds numbers. Nevertheless,
DNS is an important tool for the understanding and the model development.
In geometrically simplified flows, propagating statistically planar premixed
flames within a homogeneous turbulent flow field can be studied, where high
fidelity insight into a turbulent flame is achieved. An example is found in
the investigation of the interplay between a single vortex and a propagating
laminar flame. Such simulations are often performed without any modeling
of the turbulence, but the chemistry is treated by reduced (single or two step)
mechanisms, since chemical scales may become even smaller than turbulent
ones.
2.2
Reynolds-averaged Navier-Stokes Equations
With the help of Reynolds decomposition, a scalar quantity Q can be split
into Reynolds-averaged value hQi and fluctuating part Q0 , and the quantity
Q reads
Q = hQi + Q0 .
(5)
In a similar way, Q may be decomposed into its density (or Favre) weighted
e and the corresponding fluctuation Q00
average Q
e + Q00
Q = Q
,
(6)
where the Favre average is defined as
e = hρQi
Q
hρi
.
(7)
This averaging technique becomes important in variable density flows. Employing the variable decomposition to equations (1)-(3) and averaging the
decomposed equations result in transport equations for the averaged density,
averaged momentum and averaged scalars. The averaged continuity equation
reads
fj
∂ hρi
∂ hρi U
+
= 0 ,
(8)
∂t
∂xj
2.2 Reynolds-averaged Navier-Stokes Equations
11
the evolution of the mean momenta are governed by
00 00
fj Uei
∂ hρi ug
∂ hρi Uei
∂ hρi U
∂ hpi
∂ hτij i
i uj
+
= −
+
−
∂t
∂xj
∂xi
∂xj
∂xj
(9)
and the mean scalar transport equations are
!
00 u00
∂ hρi φg
α j
.
∂xj
(10)
There is the same number of equations as for DNS, but the number of unknowns increases by the 6 Reynolds-stresses (last terms on the r.h.s. of
eq. (9)), by the Ns turbulent scalar fluxes (last terms on the r.h.s. of eq. (10))
and the Ns mean source terms (2nd r.h.s. term of eq. (10)) resulting in a
closure problem. Solving for the mean flow quantities instead of the instantaneous ones drastically reduces the computational cost and coarse grids can
be employed since only the averaged quantities have to be resolved. On
the other hand, predictions are only made about mean values, whereas only
averaged quantities are available to model the unclosed terms. Thus, the
obtained results strongly depend on the required models.
The closure of the Reynolds-stresses can be obtained by two major strategies. The first one is based on the Boussinesq eddy-viscosity hypothesis,
where the six unknown Reynolds stresses are modeled by a single variable,
the eddy-viscosity νt . The relation is given by
!
fj
ei
1 g
∂
U
∂
U
00 00
−
+
u00 u00 δij ,
(11)
− ug
i u j = νt
∂xj
∂xi
3 k k
fα
fα
fj φ
∂ hρi φ
∂ hρi U
∂
+
=
∂t
∂xj
∂xj
g
∂φα
hρi Γ(α)
∂xj
fα −
+ hρi S
where the eddy-viscosity νt is determined with the help of algebraic expressions (like Prandtl’s mixing length approach) or by solving additional transport equation, where the most popular among these approaches for νt is
the two equation model considering the turbulent kinetic energy k and the
dissipation rate ε of turbulent kinetic energy (so called k-ε model). The
second possibility are the Reynolds-stress models, where one transport equation (containing unknown correlation like e.g. triple correlations u00ig
u00j u00k or
00 00
double correlations ug
i p ) is solved for each Reynolds stress. Mostly, for the
turbulent scalar fluxes a gradient-diffusion ansatz is employed, where it is
known that for premixed combustion this gradient assumption may become
wrong due to counter-gradient diffusion. Additionally, since the source term
Sα usually is a highly non-linear function of φ1 , . . . , φNs , the estimation of
fα is a major challenge. Nevertheless, RANS models
the mean source term S
are still the main choice for industrial applications.
12
2.3
2 COMPUTATIONAL APPROACHES
Large Eddy Simulations
Large eddy simulations (LES) resolve the instantaneous large-scale structures
of turbulent flows, whereas the small scale structures have to be modeled.
In LES, filtered quantities are considered and the filtered quantity Q(x) is
defined as
Z
Q(x) =
Q(y)F (x − y)dy ,
(12)
Ω
where F is the LES filter. Examples of LES filters in physical space are
Gaussian or top hat filters, where the top hat filter function reads
(1/∆)3 , if |xi | ≤ ∆/2 ∀i = 1, 2, 3
F (x) =
.
(13)
0
, else
Note that ∆ is the filter width. Again, mass-weighted filtered quantities can
be calculated by
Z
d
ρQ(x) = ρQ(x) =
ρQ(x0 )F (x − x0 )dx0 .
(14)
Ω
Different than the averaging properties in the RANS context, for the LES
filters, in general, it does not hold that filtering of a filtered quantity is equal
to the filtered quantity itself. Moreover, filtering of the unresolved part
Q0 = Q − Q is not equal 0, which is also true for density weighted filtered
quantities. The following properties have to be kept in mind:
d
d =
d , Q0 (x) 6= 0 , Qd
00 (x) 6= 0 .
Q(x) 6= Q(x) , Q(x)
6 Q(x)
(15)
Applying the filtering procedure to equations (1)-(3) leads to the filtered
equations
cj
∂ρU
∂ρ
+
= 0 ,
(16)
∂t
∂xj
cj Ubi
cb
∂ρUbi
∂ρU
∂p
∂τ ij
∂ρ(Ud
j Ui − Uj Ui )
+
= −
+
−
∂t
∂xj
∂xi
∂xj
∂xj
(17)
and
cα
cα
cj φ
∂ρφ
∂ρU
∂
+
=
∂t
∂xj
∂xj
d
∂φα
ρΓ(α)
∂xj
!
d
cc
cα − ∂ρ(Uj φα − Uj φα )
+ ρS
∂xj
. (18)
The last terms on the r.h.s. of equation (17) and (18) represent the unresolved small scales of turbulence and chemistry by the LES. The unresolved
cb
residual stresses τijSGS = (Ud
j Ui − Uj Ui ) and unresolved residual scalar fluxes
2.4 Probability Density Function Methods
13
SGS
cc
τjα
= (Ud
j φα − Uj φα ) are closed with so-called subgrid scale (SGS) models.
A model for the residual stress tensor similar to the Boussinesq approximation was proposed in 1963 by Smagorinsky, which is still popular due to its
simplicity. Besides the fact that instantaneous flames can be predicted, a
second convenience of LES is that the large energy-containing scales are already captured and, therefore, only the small scales have to be modeled. To
model these unfiltered structures is less crucial, since small scale turbulence
has more uniform characteristics as proposed by Kolmogorov’s first similarity hypothesis, stating that the small scale turbulence only depends on the
dissipation rate ε and the viscosity ν. Nevertheless, compared to RANS, a
higher grid resolution is required, which increases the numerical effort. A
review on LES of turbulent reactive flows is given in [34].
2.4
Probability Density Function Methods
A very general statistical description of turbulent reactive flows is given by
probability density function (PDF) approaches. Usually in PDF methods,
the statistical representation is given at one physical location at one specific
time, thus one talks of one-point one-time PDFs. Several types of PDF approaches exist.
In presumed PDF methods [50, 7], transport equations for different moments
are solved, usually mean value and variance, and the shape of the PDF is
predefined as a function of these moments. Thus, these methods only work
for scenarios where the distribution function is approximately known, but
the computational cost remains relatively low, e.g. like that of RANS.
The joint distribution of a set of scalars (e.g. compositions) is considered in
composition PDF methods [81, 46, 90, 44, 98, 41, 53, 89]. There, the flow
field is solved within the LES or RANS context described previously, but
for the scalar PDF a transport equation has to be solved. Micromixing is
not closed within this approach and has to be modeled with so-called mixing
models, but the joint statistics of the scalars allow to close the average of the
non-linear source term without bounding assumptions.
A more general description is given by the joint velocity-composition PDF
approach, where the one-point one-time JPDF f = fU,Φ (V, Ψ; x, t) is considered. Such approaches were proposed and successfully applied to turbulent
flame simulations [14, 57, 92, 101, 37, 36, 82]. The evolution equation for f
14
2 COMPUTATIONAL APPROACHES
can be derived exactly from equations (2) - (3) and reads
∂ρf
∂t
∂ρf
∂xj
∂ hpi ∂f
∂ hτij i ∂f
+
∂xi ∂Vi
∂xj ∂Vi
∂
Sα ρf
∂Ψα
0
∂τij0 ∂p
∂
V, Ψ f
−
∂Vj
∂xj
∂xi ∂
∂
∂φα ρΓ(α)
V, Ψ f
.
∂Ψα
∂xj
∂xj + Vj
−
+
=
−
(19)
The pressure p = hpi + p0 and the viscous stress tensor τij = hτij i + τij0 are decomposed into mean and fluctuation; h·|·i denotes a conditional expectation,
where the conditioning is upon V, Ψ. Here, it is assumed that the density
only depends on the scalars Ψ, i.e. ρ = ρ(Ψ). The evolution of the JPDF is
described in terms of changes in the velocity-composition sample space V-Ψ.
The mean pressure and mean viscous stress tensor gradients (3rd and 4th
term in eq. (19)) appear in closed form. The last term on the left-hand-side of
eq. (19) contains the influence of the chemical source terms on the evolution
of f and is closed, but its evolution is highly non-linear and stiff. The two
terms on the right-hand-side are unclosed conditional expectations. The first
one can be modeled e.g. with the simplified Langevin model (SLM), but also
more general descriptions are available, e.g. the generalized Langevin model
[28]. For the last term, similar as in composition PDF methods, molecular
mixing models are required. Note that such one-point one-time PDFs do
not provide information about multipoint statistics. Thus, the integral time
scale has to be modeled in addition to achieve a closed system of equations
to describe turbulent reactive flows. By that reason, joint PDF methods of
velocity, composition and turbulence frequency were proposed.
It is not feasible to solve PDF transport equations like equation (19) directly
by conventional finite-volume or finite-element methods. The reason is the
high dimensionality of the sample space (3 + 3 + Ns ), in which the PDF f
evolves. Therefore, very often, Monte-Carlo particle methods are employed
to solve PDF equations, since the computational cost increases linearly with
the dimensionality of the sample space. A set of stochastic differential equations is solved for each computational particle. Further improvement was
achieved by so-called hybrid PDF/RANS methods [37, 36, 57], where the
mean velocities are obtained from the RANS equations, such that the number of particles to achieve smooth mean velocities could be reduced. More
2.4 Probability Density Function Methods
15
recently, the PDF approach has also been applied to LES, where the unresolved subgrid scale models are based on filtered density functions (FDF). A
review on the achievements on this field is presented in [29].
3
Joint Probability Density Function (PDF)
Framework
The Navier-Stokes equations (NSE) together with proper boundary conditions exactly describe turbulent flows, whereas the resulting solutions strongly
depend on the boundary conditions. Small variations e.g. at the inflow
boundaries may lead to completely different flows. A similar observation is
achieved when looking at experimental measurements of turbulent flows. On
one hand, one is not capable to measure in a certain flow all length scales
occurring in this flow. On the other hand, one is not interested in predicting
flows of very specific boundary conditions, since usually it is not possible
to enforce these conditions in an experimental apparatus. Therefore, one is
interested in a more general description - less dependent on small variations
of the boundary conditions. By that reason, the flow variables are considered
to be random variables.
For the joint PDF methods described in this chapter, low Mach numbers
are assumed. Usually the momentum equations and the thermochemical
equations are coupled through density and pressure. Assuming low Mach
numbers, this coupling is given by the density only. Furthermore, for a given
reference pressure p0 , the density ρ and the scalar source term Sα depend only
on the set of scalars Φ, i.e. ρ = ρ(Φ) and Sα = Sα (Φ). Thus, a complete
description of the thermochemical state of the gas (e.g. density or source
terms) is provided by the set of scalars Φ.
The properties of PDFs are briefly revised in section 3.1. In the next two sections, the mathematical framework of both transported Eulerian (section 3.2)
and Lagrangian (section 3.3) PDF methods are presented and their relation
is described. Then, in section 3.4, systems of stochastic differential equations
(SDE) are presented and related to the evolution equation of the corresponding probability density function (Chapman-Kolmogorov equation). Furthermore, the employment of Monte-Carlo particle methods to solve a system of
18
3 JOINT PDF FRAMEWORK
stochastic differential equations is presented. In section 3.5, models based
on SDEs are presented to close the conditional expectations in the exact,
but unclosed, transport equation of an Eulerian PDF. Finally, this chapter
is summarized in section 3.6.
3.1
3.1.1
Foundation of Probability Density Functions
Definition and Properties
Let ϕ be a random variable (e.q. concentration or normalized temperature)
and ξ its sample space variable. The probability that an event {ϕ ≤ ψ}
occurs is denoted by Prob(ϕ ≤ ψ), where each event describes a region in the
sample space. The cumulative distribution function Fϕ (ξ) is defined through
Fϕ (ξ) := Prob(ϕ ≤ ξ) and describes the probability that the random variable
ϕ is smaller than a certain value ξ. The probability density function (PDF)
fϕ (ξ) of the scalar ϕ can now be defined as the derivative of the cumulative
distribution function, i.e.
fϕ (ξ) :=
∂Fϕ (ξ)
∂ξ
.
(20)
Consequently, one obtains the following relation by integration with respect
to the sample space variable ξ
Z ξ2
Prob(ξ1 ≤ ϕ ≤ ξ2 ) = Fϕ (ξ2 ) − Fϕ (ξ1 ) =
fϕ (ξ)dξ .
(21)
ξ1
Or in other words, fϕ (ξ)dξ is the probability that ϕ has a value between ξ
and ξ + dξ. The properties of a PDF fϕ (ξ) are
• fϕ (ξ) ≥ 0
• fϕ (ξ → ±∞) = 0
Z +∞
fϕ (ξ)dξ = 1 ,
•
(22)
−∞
i.e. the PDF has to be non-negative with a compact support and the integral
of the PDF has to equal 1. The expectation hϕi is defined as
Z +∞
hϕi =
ξfϕ (ξ)dξ ,
(23)
−∞
where the integration is done with respect to the whole sample space. Let
Q(ϕ) be a function of the random variable ϕ (i.e. Q is a random variable
3.1 Foundation of Probability Density Functions
itself), then the expectation of Q reads
Z +∞
Q(ξ)fϕ (ξ)dξ
hQ(ϕ)i =
19
.
(24)
−∞
Since the random variable ϕ is fully described by its PDF fϕ , knowing fϕ
allows to calculate each statistical moment of ϕ, i.e. the moment of n-th
order is calculated by
Z +∞
n
(ξ − hϕi)n fϕ (ξ)dξ .
(25)
h(ϕ − hϕi) i =
−∞
3.1.2
Dirac Delta Function and Heavyside Function
In the mathematical description of multivariate PDF methods, the Dirac
delta function δ(ξ) and its cumulative distribution function, the Heavyside
function H(ξ), play an important role. Therefore, these two functions are
described next.
The Dirac delta function δ(ξ) is zero everywhere except for ξ = 0, where
it has an infinite value, and it holds that
Z +∞
δ(ξ)dξ = 1
.
(26)
−∞
Since this is not a proper definition, a more mathematical description is
presented. For any continuous and integrable function g(ξ), it holds that
Z +∞
g(ξ)δ(ξ)dξ = g(0) .
(27)
−∞
Moreover, δ(ξ) fulfills the properties of a PDF (eq. (22)) and thus is a PDF
itself. Its corresponding cumulative distribution function is the Heavyside
function H(ξ) and is defined as
0 ξ≤0
H(ξ) =
.
(28)
1 ξ>0
Alternatively, the Heavyside function can be defined with the help of its
integral property
Z +∞
Z +∞
g(ξ)H(ξ)dξ =
g(ξ)dξ ,
(29)
−∞
0
which holds for any continuous and integrable function g(ξ).
20
3 JOINT PDF FRAMEWORK
When dealing with the derivation of PDF methods, the fine-grained PDF
plays an important role. The expectation of H(ξ − ϕ) becomes hH(ξ − ϕ)i =
1 · P (ϕ < ξ) + 0 · (1 − P (ϕ < ξ)) = Fϕ (ξ). Differentiating with respect to
ξ leads to hδ(ξ − ϕ)i = fϕ (ξ), which is a fundamental property of the Dirac
delta function. Therefore, the fine-grained PDF fϕ0 (ξ) is defined as
fϕ0 (ξ) = δ(ξ − ϕ) ,
(30)
where f 0 (ξ) is the PDF of one single realization of the flow and its expectation
0 ϕ
fϕ (ξ) equals the PDF fϕ (ξ).
3.1.3
Joint Probability Density Function
Let us consider the multivariate or joint probability density function (JPDF)
fU,φ (V, ψ) of the velocity U with sample space variable V and of the scalar
φ with corresponding sample space variable ψ. The probability that {ψ ≤
φ ≤ ψ + dψ} and {V ≤ U ≤ V + dV } can be determined as a function of
fU,φ (V, ψ) and reads
Prob(V ≤ U ≤ V + dV, ψ ≤ φ ≤ ψ + dψ) = fU,φ (V, ψ)dV dψ
.
(31)
Similar to the univariate PDF, the properties of the JPDF fU,φ (V, ψ) are
• fU,φ (V, ψ) ≥ 0
• fU,φ (±∞, ψ) = fU,φ (V, ±∞) = 0
Z +∞ Z +∞
fU,φ (V, ψ)dV dψ = 1 .
•
−∞
(32)
−∞
Furthermore, the marginal PDFs fφ (ψ) and fU (V ) are derivable from the
multivariate PDF fU,φ (V, ψ). The integration of fU,φ (V, ψ) with respect to V
leads to the marginal PDF fφ (ψ) and fU (V ) is obtained by integration with
respect to ψ:
Z +∞
fφ (ψ) =
fU,φ (V, ψ)dV
−∞
Z +∞
fU (V ) =
fU,φ (V, ψ)dψ .
(33)
−∞
Note that the two marginal PDFs can be determined from the joint PDF, but
the knowledge of the two marginal PDFs does not suffice to calculate their
joint PDF. Using Bayes’ theorem, the conditional PDF fU |φ (V |ψ) (i.e. the
probability of {V ≤ U ≤ V +dV } conditioned on the state {ψ ≤ φ ≤ ψ+dψ})
3.2 Eulerian Joint PDF
21
can be calculated as the ratio of the joint PDF fU,φ (V, ψ) and the marginal
PDF fφ (ψ):
fU,φ (V, ψ)
.
(34)
fU |φ (V |ψ) =
fφ (ψ)
The fine-grained PDF, presented earlier, can also be defined for a multivariate
0
PDF and it reads fU,φ
(V, ψ) = δ(V − U ) · δ(ψ − φ). Consequently, the
0
expectation fU,φ (V, ψ) is identical to fU,φ (V, ψ).
3.2
3.2.1
Eulerian Joint PDF
Different Types of JPDFs
In turbulent flows, the velocities Ui (i = 1, 2, 3) and scalars φα (α = 1, . . . , Ns )
at a given location are treated as random variables. Let Vi (i = 1, 2, 3)
and ψα (α = 1, ..., Ns ) be their corresponding sample space variables; V =
(V1 , V2 , V3 ) is the sample space of the random vector U = (U1 , U2 , U3 ) (velocity space) and Ψ = (ψ1 , . . . , ψNs ) is the sample space of the random vector
Φ = (φ1 , . . . , φNs ) (scalar space). A certain (U0 , Φ0 ) corresponds to one
point in the sample space, defined by V = U0 and Ψ = Φ0 . Let f =
fU,Φ (V, Ψ; x, t) be the one-point one-time JPDF of velocities and scalars.
The semicolon indicates that f is the JPDF at a given location x and at a
given time t, but f contains no two point information in physical space or
time.
Let Q be a function of U and Φ, then its expectation is determined by
Z Z
hQi (x, t) =
Q(V, Ψ)fU,Φ (V, Ψ; x, t)dVdΨ .
(35)
Moreover, for variable-density flows, e.g. reactive flows, it is useful to consider
density-weighted means. Thus, the density (or mass) weighted expectation
of Q is defined as
e t) = hρQi
(36)
Q(x,
hρi
e The massand its corresponding fluctuation is denoted as Q00 = Q − Q.
e can also be determined as a function of the massweighted mean value Q
e
weighted joint PDF f = feU,Φ (V, Ψ; x, t), which is defined by fe = (ρf )/ hρi.
But for inhomogeneous variable-density flows, the natural dependent variable
is the mass density function (MDF) F = FU,Φ (V, Ψ, x; t). The relation
between these three types of PDFs is given in the next equation
FU,Φ (V, Ψ, x; t) = hρi feU,Φ (V, Ψ; x, t)
= ρ(Ψ)fU,Φ (V, Ψ; x, t) .
(37)
22
3 JOINT PDF FRAMEWORK
Since the MDF is the natural dependent variable for reactive flow, its normalization property reads
Z Z
FU,Φ (V, Ψ, x; t)dVdΨ = hρi .
(38)
This property states that F is the expected mass density in the V − Ψ − xspace at a given time t. Note that F evolves according to a different sample
space than the corresponding PDF, which has a velocity-scalar sample space
and is defined at a given location x and time t.
3.2.2
JPDF Transport Equation
A transport equation for the JPDF f of velocities and scalars can be derived
from the transport equations for the velocities (eq. (2)) and scalars (eq. (3)).
These equations can be rewritten as
ρ
∂τij
∂p
DUi
+
= ρÂi (x, t)
= −
Dt
∂xi
∂xj
(39)
and
Dφα
1 ∂
=
Dt
ρ ∂xj
∂φα
Γ(α) ρ
∂xj
+ Sα = B̂α (x, t) ,
(40)
where on the right hand sides the variables Âi and B̂α are introduced. There
exist different ways to derive a JPDF transport equation. One method starts
from the consideration of a PDF as the expectation of a Dirac delta function (fine-grained PDF). An
alternative
way is described in [72], where two
DQ
independent expressions for ρ Dt are equated, where Q can be almost any
function. The resulting transport equation for f reads
∂f
∂
DUi ∂f
+ ρ(Ψ)Vj
= −
V, Ψ ρ(Ψ)f
ρ(Ψ)
∂t
∂xj
∂Vj
Dt ∂
Dφα −
V, Ψ ρ(Ψ)f . (41)
∂Ψα
Dt Equivalently, the transport equation for the MDF F reads
∂Vj F
∂
DUi ∂F
+
= −
V, Ψ F
∂t
∂xj
∂Vj
Dt ∂
Dφα −
V, Ψ F
.
∂Ψα
Dt (42)
3.3 Lagrangian Joint PDF
23
The conditional expectations can be written based on the governing equations (39)) and (40). For f , the resulting exact transport equation is given
in eq. (19) and for the MDF F it reads
∂Vj F
1
∂ hpi ∂ hτij i ∂F
∂F
+
=
−
∂t
∂xj
ρ(Ψ) ∂xi
∂xj
∂Vi
0
∂
1 ∂p
1 ∂τij0 +
−
V, Ψ F
∂Vj
ρ(Ψ) ∂xj
ρ(Ψ) ∂xi 1 ∂
∂φα ∂
Γ(α) ρ
−
V, Ψ F
∂Ψα
ρ(Ψ) ∂xj
∂xj ∂
Sα F
.
(43)
−
∂Ψα
Note that this equation is exact, but the second and third terms on the right
hand side are unclosed and require modeling.
3.3
3.3.1
Lagrangian Joint PDF
Lagrangian System
Let x+ (x0 , t) define the physical location of a fluid particle, having its origin at x0 + = x+ (x0 , t0 ). Furthermore, the velocity of the fluid particle is
determined as U+ (x0 , t) = U(x+ (x0 , t), t) and the scalar of the particle is
Φ+ (x0 , t) = Φ(x+ (x0 , t), t). For a given initial location x0 , the state of a
fluid particle is fully defined by x+ , U+ , Φ+ and the rate of change of these
properties are
∂ +
x (x0 , t) =
∂t i
∂ +
U (x0 , t) =
∂t i
∂ +
φ (x0 , t) =
∂t α
Ui+ (x0 , t)
(44)
DUi
Dt x=x+
bi (x+ (x0 , t), t)
= A
(45)
Dφα
Dt x=x+
bα (x+ (x0 , t), t) ,
= B
(46)
bi and ρB
bi are defined in equations (39) and (40). If the flow is
where ρA
turbulent at (x, t) = (x0 , t0 ), then the initial state x0 + , U0 + , Φ0 + is a random
bi and B
bα are random vectors.
vector and also the rates of change A
3.3.2
JPDF Transport Equation
Let fL = fL (V, Ψ, x; t|V0 , Ψ0 , x0 ) be the Lagrangian JPDF of the fluid
particle properties conditioned upon the properties V0 , Ψ0 , x0 at an earlier
24
3 JOINT PDF FRAMEWORK
time t0 . This quantity is of importance, since it is the transition density of
turbulent reactive flows, connecting the mass density function F(V, Ψ, x; t)
with the MDF from the initial time F(V0 , Ψ0 , x0 ; t0 ) by the following relation
Z Z Z
F(V, Ψ, x; t) =
fL (V, Ψ, x; t|V0 , Ψ0 , x0 )
· F(V0 , Ψ0 , x0 ; t0 ) dV0 dΨ0 dx0
.
(47)
For the derivation of the corresponding transport equation it is referred to
[72, 19], where the transport equation of the transition density reads
∂fL
∂Vj fL
∂
bj V, Ψ, x, V0 , Ψ0 fL
+
= −
A
∂t
∂xj
∂Vj
∂
b
.
(48)
Bα V, Ψ, x, V0 , Ψ0 fL
−
∂Ψα
The two conditional expectations on the right hand side are conditioned
upon the state {U+ (x0 , t) = V, Φ+ (x0 , t) = Ψ, x+ (x0 , t) = x} and upon
the initial state {U+ (x0 , t0 ) = V0 , Φ+ (x0 , t0 ) = Ψ0 }.
3.3.3
Relation to Eulerian Joint PDF
Differentiating equation (47) and substituting ∂fL /∂t with equation (48) one
gets a transport equation for the Eulerian MDF F
∂F
∂Vj F
∂
b
Aj V, Ψ F
+
= −
∂t
∂xj
∂Vj
∂
bα V, Ψ F
.
(49)
−
B
∂Ψα
This transport equation for the Eulerian MDF F has been derived from
Lagrangian evolution equations for fluid particle properties and is equal to
equation (42). Thus starting from the PDF governing the fluid particle evolution one can derive a MDF transport equation, which is identical to the
Eulerian MDF transport equation. More details on the relation between
Eulerian and Lagrangian PDFs are given in [72].
Furthermore, for the simplified case of constant density flows, a direct
relation between the Lagrangian and Eulerian PDFs is given by
Z
f (V, Ψ; x, t) =
fL (V, Ψ, x; t|x0 )dx0 .
(50)
3.4 Stochastic Systems
3.4
3.4.1
25
Stochastic Systems
Markov Process
A system, which evolves probabilistically in time, is a stochastic system or
a stochastic process. Such a system is completely described by a timedependent random variable X(t) and the corresponding JPDF f . For the
values x1 , . . . , xN of X(t) at times t1 ≤ . . . ≤ tN , the joint probability
density reads f (xN , tN ; . . . ; x1 , t1 ). Moreover, using Bayes Theorem, one can
also define conditional probability densities
f (xN , tN ; . . . ; xM +1 , tM +1 |xM , tM ; . . . ; x1 , t1 )
= f (xN , tN ; . . . ; x1 , t1 )/f (xM , tM ; . . . ; x1 , t1 ) ,
(51)
where 1 < M < N . These conditional probabilities allow to describe the
evolution of the system based on the knowledge of the past. But the concept
of such a general stochastic process is very loose, why Markov processes are
considered next.
A Markov process can be formulated in terms of conditional probability densities by
f (xN , tN ; . . . ; xM +1 , tM +1 |xM , tM ; . . . ; x1 , t1 )
= f (xN , tN ; . . . ; xM +1 , tM +1 |xM , tM ) ,
(52)
where the conditioning is solely upon (xM , tM ), but not upon earlier states.
Therefore, the future of the Markov system is fully determined by the present
state together with the knowledge of the conditional probability density function f . A special formulation of conditional probability functions is the
Chapman-Kolmogorov equation (CKE), which is defined as
Z
f (x3 , t3 |x1 , t1 ) =
f (x3 , t3 |x2 , t2 )f (x2 , t2 |x1 , t1 )dx2 .
(53)
It is a complex nonlinear functional equation and relates all conditional probabilities to each other. It has many solutions, but usually this equation is
employed in its differential form.
A stochastic process can be continuous with continuous sample paths or
discontinuous with discontinuous sample paths. For a continuous process,
the displacement of X tends to zero as the time step tends to zero, whereas
for discontinuous processes, the displacement of X may remain non-zero.
26
3 JOINT PDF FRAMEWORK
More precisely, for a Markov process it was shown (e.g. in [20]), that with
probability 1 the sample paths are continuous functions in time, if
Z
1
∀ε > 0 : lim
f (x, t + ∆t|y, t)dx = 0 .
(54)
∆t→0
∆t |x−y|>ε
The equation above is often called the Lindeberg condition.
3.4.2
Differential Chapman-Kolmogorov Equation
Under appropriate assumptions given next, the Chapman-Kolmogorov equation (eq. (53)) can be reduced to its differential form. For the conditional
PDF f (x, t + ∆t|y, t), three such conditions exist and they read: ∀ε > 0
f (x, t + ∆t|y, t)
= J (x|y, t) for |x − y| ≥ ε ,
(55)
∆t
Z
1
lim
(xi − yi )f (x, t + ∆t|y, t)dx = Ai (y, t) + O(ε) (56)
∆t→0
∆t |x−y|<ε
lim
∆t→0
and
lim
∆t→0
1
∆t
Z
(xi − yi )(xj − yj )f (x, t + ∆t|y, t)dx = Bij (y, t) + O(ε)
|x−y|<ε
(57)
uniformly in x, t and y. The first condition is due to jump processes, where
J (x|z, t) is the jump coefficient. The second and third conditions are due
to drift and diffusion processes, where the corresponding coefficients Ai (x, t)
and Bij (x, t) are called drift and diffusion coefficients, respectively. Note that
all higher order coefficients of the form described in equation (57) have to
vanish. A necessary condition for continuous paths is that the jump coefficient J (x|y, t) vanishes for all x 6= y.
The corresponding evolution equation for the conditional PDF f (x, t|y, t0 )
(with t = t0 +∆t) is the differential Chapman-Kolmogorov equation (DCKE),
which reads
X ∂
∂
f (x, t|y, t0 ) = −
{Ai (x, t)f (x, t|y, t0 )}
∂t
∂x
i
i
X 1 ∂2
+
{Bij (x, t)f (x, t|y, t0 )}
2
∂x
∂x
i
j
i,j
Z
+
{J (x|z, t)f (z, t|y, t0 ) − J (z|x, t)f (x, t|y, t0 )} dz .
(58)
3.4 Stochastic Systems
27
The coefficients in eq.(58), i.e. Ai (x, t), Bij (x, t) and J (x|z, t), are determined through the three conditions (eqs. (55)-(57)). Given these coefficients,
the evolution of the PDF f (x, t|y, t0 ) is fully described by the differential
Chapman-Kolmogorov equation (eq.(58)). Since the DCKE is of fundamental importance for Markov processes, two special forms are considered next.
3.4.3
Jump Process
Let us consider the case, where the random variable X only evolves by jumps.
Thus no diffusion processes are present, i.e. Ai (x, t) = Bij (x, t) = 0, and the
DCKE reduces to the so-called Master equation
Z
∂
0
f (x, t|y, t ) =
{J (x|z, t)f (z, t|y, t0 ) − J (z|x, t)f (x, t|y, t0 )} dz . (59)
∂t
The paths of the random variable X(t) are discontinuous, since they only
contain jumps. For the case where the random variable X(t) can reach only
integer values, the integral in equation (59) can be replaced by a summation.
Thus, the Master equation becomes
X
∂
{J (n|m, t)f (m, t|n0 , t0 ) − J (m|n, t)f (n, t|n0 , t0 )} ,
f (n, t|n0 , t0 ) =
∂t
m
(60)
0
where n, m and n are random integer variables.
The univariate Random Telegraph Process is a special case of the integral
formulation of the Master equation, where the random variable can only
reach two possible integer values. Thus, the Master equation reads for this
particular case
∂
f (0, t|n0 , t0 ) = J (0|1, t)f (1, t|n0 , t0 ) − J (1|0, t)f (0, t|n0 , t0 )
∂t
∂
f (1, t|n0 , t0 ) = J (1|0, t)f (0, t|n0 , t0 ) − J (0|1, t)f (1, t|n0 , t0 )
∂t
(61)
where it holds that f (0, t|n, t0 ) + f (1, t|n, t0 ) = 1. This process has its importance, since the evolution of the progress variable also consists of a jump
process with only two possible values 0 and 1.
3.4.4
Diffusion Process
The differential Chapman-Kolmogorov equation without jump processes (i.e.
J (x|z, t) = 0) is called Fokker-Planck equation. It represents a continuous
28
3 JOINT PDF FRAMEWORK
Markov process and the corresponding PDF evolves according to
X ∂
∂
f (x, t|y, t0 ) = −
{Ai (x, t)f (x, t|y, t0 )}
∂t
∂x
i
i
X 1 ∂2
+
{Bij (x, t)f (x, t|y, t0 )} .
2
∂x
∂x
i
j
i,j
(62)
A particular form of a diffusion process is the Wiener process, where the drift
coefficient Ai is 0 and the diffusion coefficient Bij is 1. Thus, the FokkerPlanck equation modifies to a multivariate Wiener process
∂
1 ∂2
f (x, t|x0 , t0 ) =
f (x, t|x0 , t0 ).
∂t
2 ∂xi ∂xi
(63)
The initial condition of this process is f (x, t|x0 , t0 ) = δ(x − x0 ), and its
solution is a multivariate Gaussian with the two properties
hX(t)i = x0
h(Xi (t) − x0,i )(Xj (t) − x0,j )i = (t − t0 )δij
.
Finally, an analytical expression exists for the PDF, which reads
1 (x − x0 )2
−n/2
f (x, t|x0 , t0 ) = {2π(t − t0 )}
exp −
.
2 (t − t0 )
3.4.5
(64)
(65)
(66)
Stochastic Differential Equations (SDE)
So far, stochastic processes were considered with the help of conditional probabilities, i.e. the evolution of the conditional PDF was described by the differential Chapman-Kolmogorov equation. Here, sample paths described by
stochastic differential equations (SDE) are considered to describe the stochastic process. Thus, single realizations of the stochastic process are considered.
In general, the SDE of a process X(t) can be written as a Langevin
equation
dX
= a(X, t) + b(X, t)ξ(t) ,
(67)
dt
where a(X, t) and b(X, t) are deterministic and known functions. ξ(t) is a
rapidly fluctuating random term, e.g. white noise, which obeys the following
rules
(i)
(ii)
(iii)
t 6= t0 → ξ(t), ξ(t0 ) statistically independent
hξ(t)i = 0
hξ(t)ξ(t0 )i = δ(t − t0 ) .
(68)
3.5 Modeled Joint PDF
29
It has to be mentioned that the random process ξ(t) has infinite variance. On
the other hand, by integrating ξ from 0 to t one obtains the Wiener process
W (t)
Z t
ξ(t0 )dt0 = W (t) ,
(69)
0
which allows to rewrite the Langevin equation in its integral form
Z t
Z t
X(t) − X(0) =
a(X[s], s)ds +
b(X[s], s)dW (s) ,
0
(70)
0
where dW (s) = W (s + ∆s) − W (s) = ξ(s)ds. The first term in equation (70)
can be written as A(X, t) and the second term becomes B(X, t)dW (t). Thus,
the evolution of a random variable can be expressed in terms of a stochastic
differential equation.
Moreover, for a multivariate random vector X(t), such a system of stochastic
differential equations has the form
dX = A(X, t)dt + B(X, t)dW(t) ,
(71)
where A is the drift vector and B is the diffusion matrix. Following the
Kramers-Moyal expansion and the theorem of Pawula, the corresponding
conditional PDF f (x, t|x0 , t0 ) is governed by a Fokker-Planck equation, which
reads
X ∂
∂
f (x, t|x0 , t0 ) = −
(Ai (x, t)f (x, t|x0 , t0 ))
∂t
∂xi
i
1 X ∂2
+
[B(x, t)BT (x, t)]ij f (x, t|x0 , t0 ) .(72)
2 ij ∂xi ∂xj
Hence, for the multivariate random variable X(t), which evolves according
to the system of stochastic differential equations (71), its evolution can also
be expressed in terms of the PDF equation (72).
3.5
Modeled Joint PDF
From the stochastic processes described by the system of stochastic differential equations, a transport equation for the corresponding PDF can be
derived, as described at the end of the previous section. Whereas the system
of SDEs includes randomness (due to the Wiener process vector dW(t)), the
evolution equation of the conditional PDF is deterministic. The solution of
equation (72) can be obtained in two ways, i.e. by a deterministic method
30
3 JOINT PDF FRAMEWORK
or by employing a Monte-Carlo particle method. Employing a Monte-Carlo
particle method has many advantages compared to the deterministic solution
approach. The PDF is represented by an ensemble of stochastic particles
evolving according to stochastic processes. Thus, since governed by a system
of SDEs, the particle evolution description is straightforward and computationally more efficient, which is explained in detail in [72]. Nevertheless, the
representation of the PDF by a finite number of particles induces a statistical
error and a bias (deterministic) error. This topic is further discussed in the
next chapter.
In PDF methods, one is interested in the evolution of the Lagrangian PDF
fL of fluid particles (or equivalently the Eulerian PDF f ), that describes the
underlying physics. The key idea is to devise stochastic processes, such that
the corresponding model PDF fL∗ evolves similar as the real PDF fL . In
addition, also the mass density function F and the modeled MDF F ∗ evolve
in the same way.
Let k be the state (random) vector of a stochastic particle and κ its sample
space variable vector. A stochastic process is defined in terms of a system of
stochastic differential equations
dki = ai (k, t)dt + b(i) (k, t)dWi (t) ,
(73)
where a and b are the drift and diffusion coefficient vectors defined by the
stochastic model process. Thus, the evolution of the corresponding model
PDF fk∗ is governed by
∂
1 ∂2
∂ρfk∗
= −
(ai (κ, t)ρfk∗ ) +
[b(i) (κ, t)]2 ρfk∗
∂t
∂κi
2 ∂κi ∂κi
.
(74)
An example of such a model PDF for the case of a joint velocity-composition
probability density function is presented next.
3.5.1
Example Model PDF
In this subsection, the Eulerian joint probability density function f of velocity and composition is considered, where the corresponding exact, but
unclosed, transport equation is given in section 2.4 by equation (19). As
already explained within this chapter, the solution strategy is to devise a
system of stochastic differential equations, where the corresponding model
PDF f ∗ approximates the PDF f .
3.5 Modeled Joint PDF
31
Let the state vector of a stochastic particle describe location, velocity and
composition, i.e. the random variable vector k becomes k = (X∗ , U∗ , Φ∗ ),
and its sample space variable reads κ = (x, V, Ψ). Note that the asterisk
denotes stochastic particle properties. The single equations within the system of stochastic differential equations (73) are explained next, first for the
particle location, then for the velocity and finally for the composition.
The particle location X∗ is simply influenced by the particle velocity U∗ ,
such that evolution equation (75) for the particle location
dXi∗ = Ui∗ dt
(75)
is deterministic. Note that the randomness enters this equation through the
velocity, which is itself described by a random process.
For the particle velocity, one of the simplest models is used here, i.e. the
simplified Langevin model (SLM), which is presented in equation (76). This
model is a simple form of the generalized Langevin model (GLM) [28]. The
equation for Ui∗ reads
1 ∂ hpi ∂ hτij i
1 3
ε
∗
∗
e
dUi = −
−
dt −
+ C0
Ui (t) − Ui dt
ρ ∂xi
∂xj
2 4
k
+ (C0 ε)1/2 dWi
.
(76)
The first term on the right hand side of equation (76) captures the influence
of the mean pressure and mean viscous stress tensor gradients on the velocity.
The second term describes the decrement of the turbulent motions by a drift
term of the particle velocity towards the mean velocity. The responsible
time scale for this turbulence reduction is proportional to k/ε, where the
turbulent kinetic energy can be extracted from the ensemble statistics, but
for the dissipation rate of k, namely ε, an additional evolution equation has to
be solved. The diffusion process is described by the third term, that contains
a Wiener process. The first part of the second term (with the prefactor
1/2) describes the decay of turbulence, whereas the second part (with the
prefactor 3/4 C0 ) together with the last term represents the influence of the
pressure-rate-of-strain, which is responsible for the redistribution of turbulent
kinetic energy among the several Reynolds stresses. The model constant C0
typically has a value of 2.1.
The evolution of the particle composition is modeled by two processes,
namely molecular mixing and chemical source terms. Whereas the chemical
source term is treated exactly within the PDF framework, the molecular
mixing term has to be modeled for example by the interaction by exchange
with the mean (IEM) model of Villermaux and Devillon [99]. The governing
32
3 JOINT PDF FRAMEWORK
equation for the composition reads
1 ε ∗
fα dt + Sα (Φ)dt ,
dΦ∗α = − C0
φα (t) − φ
2 k
(77)
The second term Sα is the source term of composition α. The IEM model
accounts for molecular mixing, i.e. scalar diffusion, as a drift of the particle
value towards the mean value. The characteristic time is again proportional
to k/ε, where for inert scalars, usually a value between 1.5 and 2.5 is chosen.
Note that the model for the reduction of turbulent kinetic energy and the
mixing model act both in the same way, i.e. by a drift of the particle quantity
towards the mean value with a corresponding time scale proportional to k/ε.
The model system of stochastic differential equations (73) now consists of
the three equations (75), (76) and (77). Therefore, the corresponding model
PDF transport equation can be derived by the procedure described above
and the evolution equation reads
∂ρf ∗
∂ρf ∗
= −Vj
∂t
∂xj
i
∂ hpi ∂f ∗
∂ hτij i ∂f ∗
1 3
ε ∂ h ∗
e
+
+ C0
ρf (Vi − Ui )
−
+
∂xi ∂Vi
∂xj ∂Vi
2 4
k ∂Vi
∂ 2 (ρf ∗ )
1
(C0 ε)
+
2
∂Vi ∂Vi
i
∂
1 ε ∂ h ∗
∗
f
−
ρf (ψα − φα )
.
(78)
Sα ρf
+ C0
∂Ψα
2 k ∂Ψα
This equation (78) is the model PDF transport equation approximating the
exact joint PDF transport equation (19).
3.6
Summary
In this chapter, the mathematical framework and the fundamentals of probability density functions were briefly described. Based on the transport equations of momentum and composition, the exact Eulerian joint PDF transport
equation was presented, which is unclosed. Then, the relation between Eulerian and Lagrangian PDF was described. With the introduction of stochastic
systems a tool (i.e. stochastic Monte-Carlo particle methods) arises making
the solution of transported PDF methods numerically feasible. Therefore,
when employing PDF methods, the modeling challenge is to declare stochastic processes that accurately approximate the desired PDF equation.
4
A Hybrid Solution Strategy for PDF Methods
In the previous chapter it was mentioned that in most cases a PDF transport
equation is solved with a Monte-Carlo particle method. This chapter deals
with the implementation of such stochastic particle methods.
During the past 40 years, various approaches have been proposed. In
1981, a first implementation was proposed by Pope [71], where at the beginning of each time step the particles were located at the grid corners and did
evolve from these fixed locations. With the fundamental work of Pope about
PDF methods for reactive flows in 1985 [72], a strategy was introduced, where
computational particles can freely evolve in the computational domain. With
so called stand alone particle methods [3, 83, 51], it is possible to calculate
the joint velocity-composition PDF for turbulent flames. Later, so-called hybrid methods were proposed, where in addition to the Monte-Carlo particle
method a finite-volume method is used to solve the RANS equations. These
two methods are mathematically fully consistent and the required number of
particles to reduce the bias error could be diminished drastically. A variety of
such hybrid approaches were proposed and successfully applied to turbulent
flame simulations [14, 57, 92, 101, 37, 36, 82]. A review on the progress of
PDF methods for turbulent combustion is given by Haworth [29].
In section 4.1, the implementation of the Monte-Carlo particle method as
a hybrid PDF/RANS method is described, followed by section 4.2, which is
dedicated to the algorithmic topic of the number control of computational
particles. In addition, in chapter 5, a stand alone Monte-Carlo PDF method
is presented for simplified flow configurations, i.e. a quasi one-dimensional
setup.
34
4.1
4 HYBRID SOLUTION STRATEGY
Hybrid Method
In this section, a method is described solving the evolution equation of onepoint one-time mass-weighted joint PDFs f˜ of velocity, turbulence frequency
and composition. As mentioned earlier, stand alone particle methods require a large number of computational particles to keep the bias error low.
Here, a stochastic particle method is used to describe the mass-weighted
joint PDF g̃ of fluctuating velocity, turbulence frequency and composition,
and in addition, a conventional finite-volume solver calculates the mean flow
field by solving the RANS equations. This hybrid algorithm [36, 37, 57, 58]
is internally consistent, i.e. the Reynolds stresses, scalar fluxes and other
moments are extracted from the particle ensemble statistics and no additional turbulence model is required to close the RANS equations. Moreover,
the mean velocities from the RANS solution are fed into the particle method.
In variable density flows, one considers rather the mass density function than
the probability density function, as mentioned in the second and third chapters. Next the mass density function G is introduced and its exact transport
equation is presented. In addition, the set of equations is closed by the RANS
and energy equations solved by the finite-volume method. Then, the coupling of the two methods is illustrated, and finally, the employed stochastic
models for the computational particles are presented.
4.1.1
Set of Equations
The mass density function G considered here is defined as the product of the
mean density hρi and of the mass weighted joint PDF g̃ of fluctuating velocities u00 , turbulence frequency ω and scalars Φ describing the composition,
i.e.
G(v00 , θ, Ψ, x, t) = hρi (x, t) · g̃(v00 , θ, Ψ; x, t) ,
(79)
where θ is the sample space variable of the turbulence frequency. The corresponding transport equation of the MDF G reads
∂G
∂t
00 00
fj + vj00 )
∂G (U
∂ Uei ∂Gvj00
1 ∂hρiug
i uj ∂G
−
+
00
∂xj
∂xj ∂vi
hρi ∂xj ∂vi00
∂
1 ∂p
1 ∂hpi
1 ∂τij
1 ∂hτij i 00
=
G
−
−
+
v , θ, Ψ
∂vi00
ρ ∂xi
hρi ∂xi
ρ ∂xj
hρi ∂xj ∂
Dω 00
∂
Dφβ 00
G
v , θ, Ψ
−
G
v , θ, Ψ
. (80)
−
∂θ
Dt ∂Ψβ
Dt +
The first two lines of equation (80) describe the evolution of the MDF in the
position and velocity space, whereas on the third line the first term describes
4.1 Hybrid Method
35
the change of the MDF within the turbulence frequency space. The last term
in equation (80) denotes the influence of the scalar vector on the evolution of
the MDF, where these scalars describe the composition of the flow. Since this
term is described by the combustion models presented in chapters 8 and 9,
it is unspecified at this point and kept in this general form.
Since the mean velocities cannot be obtained from the considered MDF,
in addition, the Reynolds-averaged Navier-Stokes equations and an energy
equation are required to describe the evolution of the mean flow field. Together with the continuity equation, these equations read
∂ f
∂
(hρi) +
hρi Uj
= 0
(81)
∂t
∂xj
∂ ∂ ∂ 00 00
fj + hpi δij
hρi Uei +
hρi Uei U
=
hρi ug
u
(82)
i j
∂t
∂xj
∂xj
∂ f
∂ f f
ė
hρi Es +
Uj hρi Es + hpi
= hρi Q
∂t
∂xj
∂ 00 00
h
hρi ug
−
j s
∂xj
∂
hρi 00g
00 00
−
u u u
∂xi
2 i j j
∂ f
00 00
Uj hρi ug
u
−
i j , (83)
∂xi
where the mean viscous and diffusion terms are ignored, since they are negligible for high Reynolds number shear flows. The mean source term is denoted
ė and the density weighted mean of the total sensible energy is E
fs . The
by Q
00
e
Favre-fluctuating sensible enthalpy is denoted by hs = hs − hs , where hs is
the sensible enthalpy and hes its corresponding density weighted average. The
fs is defined as
quantity E
1 e e
00 00
g
f
Es = ees +
Ui Ui + ui ui
2
,
(84)
where ees is the density weighted sensible energy. The mean pressure occurring
in equations (82) and (83) is calculated through the equation of state, which
reads
hpi = hρi ees (γ − 1) ,
(85)
and γ is defined as γ = heess . Equations (80) - (85) define a closed system.
Note that this system is redundant, i.e. several quantities are calculated in
f
36
4 HYBRID SOLUTION STRATEGY
both, the Monte-Carlo and the finite-volume method. Therefore, an accuracy criterion is that these twice determined quantities have to be consistent,
i.e. the densities and enthalpies from both methods have to be identical up
to a certain level.
The interplay between the two methods is rather simple. The finite-volume
method solves the RANS and energy equations. The right hand sides of equations (82) and (83) are determined through the provided one point statistics
from the Monte-Carlo particle method and do not require any additional
e and the
modeling. From the RANS equations, the averaged velocities U
mean density hρi are fed into the particle method, which is responsible for
the evolution of the MDF G in the sample space. In place of solving the
MDF transport equation (80) directly, a set of stochastic differential equations is solved. These equations and the underlying models are presented in
section 4.1.2.
4.1.2
Particle Method
The composition including species mass fractions and enthalpy is defined by
a point in the Ψ-space. The position of a particle is defined as
∗
∗
∗
e
dXi = Ui (X (t), t) + ui (t) dt ,
(86)
where the total particle velocity U∗ is the sum of the fluctuating particle
e which is interpolated to the particle
velocity u∗ and the mean velocity U,
location.
The model for the fluctuating velocity used here is the simplified Langevin
model (SLM) [28]. Different than in equation (76), where the SLM was
presented for the full particle velocity, here it is the SLM for the fluctuating
velocity component. The model reads
du∗i =
∂ Uei
1 ∂ hρi ug
i uj
dt − u∗j
dt
hρi ∂xj
∂xj
1 3
−
+ C0 Ωu∗i dt + (C0 Ωk)1/2 dWi
2 4
,
(87)
where the second line is identical to the one in equation (76). The quantity
Ω is the conditional turbulence frequency and is defined by
Ω = CΩ
hρ∗ ω ∗ | ω ∗ ≥ ω
ei
hρi
.
(88)
4.1 Hybrid Method
37
CΩ is a model constant and usually has a value within the range from 0.65
to 0.75. Note that using Ω instead of ω
e allows to account for intermittency
effects. A study on the influence of CΩ is presented in [82] and more details
on the conditional quantity are given in [76].
Instead of solving an ε- or ω
e equation to obtain the turbulence frequency,
a stochastic model for the particle turbulence frequency ω ∗ is used here. A
model for ω ∗ was proposed for homogeneous turbulence [35], whereas here the
extension for inhomogeneous turbulence [85] is employed. From experimental
observations, one knows that the shape of the PDF of ω ∗ is approximately a
Gamma distribution. The model reads
dω ∗ = −C3 (ω ∗ − ω
e )Ωdt − Ωω ∗ Sω dt
+ (2C3 C4 ω
e Ωω ∗ )1/2 dW ,
(89)
where Sω is the source/sink term of turbulence frequency and is defined as
Sω = Cω2 − Cω1
Pk
kΩ
.
(90)
The quantity Pk is the production of turbulent kinetic energy and reads
00 00
Pk = −ug
i uj
∂ Uei
∂xj
.
(91)
The first term on the right hand side of equation (89) describes the reduction
of the variance of the turbulence frequency, whereas the third term describes
together with the first one the redistribution of the turbulence frequency,
such that the approximate Gamma PDF remains conserved. The second
term describes the source term of ω ∗ .
Table 1: Model constants for equations (86), (87) and (89).
C0
CΩ
C3 C4 Cω1 Cω2
2.13 0.6893 1.0 0.25 0.7 0.9
Based on these stochastic models described in equations (86), (87) and (89),
38
4 HYBRID SOLUTION STRATEGY
the modeled MDF transport equation reads
∂G
∂t
00 00
fj + vj00 )
∂G (U
∂ Uei ∂Gvj00
1 ∂hρiug
i uj ∂G
−
+
00
∂xj
∂xj ∂vi
hρi ∂xj ∂vi00
∂ Gvi00
∂ 2G
1 3
1
+ C0 Ω
(C
ε)
=
+
0
2 4
∂vi00
2
∂vi00 ∂vi00
∂GΩC3 (θ − ω
e)
∂GSω Ωθ
∂ 2 Gθ
+
+
+ C3 C4 ω
eΩ
∂θ
∂θ∂θ
∂θ Dφβ 00
∂
G
v , θ, Ψ
.
−
∂Ψβ
Dt +
(92)
The considered system is consistent at the level of governing equations, i.e.
without numerical inaccuracies the quantities calculated in both methods
like mean density or mean sensible energy are identical. The main advantage
of such hybrid algorithms compared to stand alone particle mesh methods
is that much fewer particles are required to achieve the same level of accuracy [37, 57]. Moreover, hybrid methods can be combined with moving time
averaging schemes, which are extremely effective in reducing the statistical
and bias errors [37]. Although such hybrid methods are formally consistent,
various correction schemes are required to achieve internal consistency numerically. In [58] a particle position correction scheme was devised to ensure
that the weighted particle number density (represented by the ensemble of
computational particles) equals the density obtained within the finite-volume
method. Moreover, means of the fluctuating velocities have to be zero, which
is enforced by a particle velocity correction scheme [37].
4.2
Particle Number Control
Probability density function (PDF) methods are particular suited for calculating turbulent reactive flows. They provide the advantage of a closed
turbulent transport and a closed chemical source term. From the modeling
side, they are superior to e.g. two equation or Reynolds stress models. However, efficiently solving the high dimensional PDF transport equation is not
a trivial task. Instead of classical finite-volume or finite-element methods,
Monte-Carlo algorithms are employed using notional particles. With the hybrid method presented in the previous section, far fewer particles are used
for the same level of accuracy than in stand alone particle methods, saving a
lot of computational time. An important ingredient to achieve this accuracy
is a moving time averaging scheme [37]. There, all fields β n extracted from
4.2 Particle Number Control
39
the particle ensemble at time step n are replaced by the smoother field
βen = µmem βen−1 + (1 − µmem ) β n
,
(93)
where µmem is a memory factor between 0 and 1 and βen and βen−1 denote
the smoothed values of β at the new and old time levels, respectively. This
procedure significantly reduces statistical and bias errors in statistically stationary simulations. For the efficiency, another very essential algorithmic tool
is the particle number control mechanism, ensuring a approximately constant
number of particles in each cell.
Here, it is focused on the particle number control scheme. The efficient
treatment of computational particles is essential for a fast code, moreover, a
uniform distribution of the statistical error is preferable. One direct conclusion can be drawn from these requirements: a constant number of particles
per computational grid cell is desired. Larger grid cells should contain the
same number of particles than smaller ones. The standard solution for this
problem are clustering and cloning algorithms producing or destroying particles. It is known that particle clustering introduces a bias error, specially
when a low number of particles per cell is used as e.g. in hybrid codes.
The most common existing schemes for clustering can be divided into
three groups: deterministic, random and statistical particle elimination. An
extension of the statistical clustering approach is proposed, but no improvements could be obtained with this ”quant” elimination scheme.
In the following section, the need for particle number control is pointed
out. Then in section 4.2.2, existing particle number control algorithms together with a new approach are explained. In section 4.2.3, numerical results
of a generic jet flame are presented and, finally, conclusions are given in
section 4.2.4.
4.2.1
Motivation
Representing the MDF by a finite ensemble of computational particles and
employing numerical grids for the discretization, lead to a numerical error.
This error N can be decomposed into three parts [19, 76, 101] and reads
(eq.(6.200) in [19])
N = BN + ΣN + SN .
(94)
The first error BN is a deterministic error due to the bias when using finite
numbers N of particles. The statistical error ΣN represents the statistical
noise and scales with N −1/2 . The third error SN is the deterministic error
due to discretization, since only a finite number of grid points are used. It
40
4 HYBRID SOLUTION STRATEGY
is often called the spatial discretization error and scales with a power of h,
where h is the size of a grid cell. For more details, it is referred to [101].
In PDF methods, particles are realizations. To extract statistical information like mean or higher moments, a ensemble of realizations is needed.
For the efficiency and to achieve a uniformly distributed accuracy of the
extracted statistical moments, a constant number of particles per cell is desirable. Without particle number control, one is left with exorbitantly many
particles in big cells, if non-uniform grids are employed. This is the motivation for particle number control schemes, which proved to reduce the
computational cost of PDF simulations drastically.
The most common attempts to achieve a better distribution is based on
particle cloning and elimination in order to increase or decrease the number
of particles per cell, respectively. Particle clustering is applied in cells with
too many particles and cloning is used to generate more particles in cells
with an insufficient number of particles. While cloning is straightforward
and introduces no error, particle clustering is more problematic and may
lead to significant bias errors for moderate particle numbers, since particles
have to be destroyed without changing the statistics of the ensemble.
4.2.2
Clone/Cluster - A Review
It has to be mentioned, that to the authors best knowledge the literature on
particle cloning/clustering in the field of Monte-Carlo methods for turbulent
reactive flows is rare and numerical studies are missing on that topic. A brief
overview on the achievements in this field is given in [29].
In 1990, Haworth and Tahry [27] proposed a first cluster/clone strategy to
keep the particle number density distribution uniformly and to obtain nearly
constant particle weights in a control volume. The generation of particles is
rather simple; a particle is split into two, where the new particles have half
of the weight as the parent particle and the properties (including physical
location) of the parent particle are copied to the new ones. Particles are
merged in the following fashion: particles with a weight m∗ smaller than
a preferred minimal weight m̄ can be altered by 2 strategies. A particle is
removed with probability 1 − m∗ /m̄ or it is enlarged to a weight of m̄ with
probability m∗ /m̄. This cluster procedure preserves the total mass, but mean
and higher order moments are preserved only in a statistical way. A similar
procedure was described in [90].
In Li and Modest’s work [44], the same cloning procedure is employed for
the heaviest particles, but the clustering strategy is different. Two particles
are clustered, where the new particle gets the mass-weighted location of the
two particles and the properties of particle 1 with probability m∗1 /(m∗1 + m∗2 )
4.2 Particle Number Control
41
and of particle 2 with probability m∗2 /(m∗1 + m∗2 ). This method preserves
mean quantities also in a statistical manner.
In Zhang and Haworth’s paper [103], particle eliminations are performed
by considering three particles and merging them into two new particles. The
total weight of the three particles and the mean particle quantities remain
unchanged, and the reduction of these higher moments (as e.g. the turbulent
kinetic energy) should be minimized. This procedure ensures that mean particle properties are conserved, but the distribution of the particle properties
is changed; this change can be described as ”artificial mixing”.
Rembold and Jenny [80] used the same clone strategy as the authors
above, where always the heaviest particle is cloned. For clustering, the two
lightest particles are considered, where one of them is deleted. The choice
of the particle to delete is given by their weights m∗1 and m∗2 ; particle 1
is removed with probability m∗2 /(m∗1 + m∗2 ) and particle 2 with probability
m∗1 /(m∗1 + m∗2 ). This ensures that the procedure statistically conserves any
statistical moment from the particle properties.
Motivated by the fact that with a removed particle some information is
lost, Naud et al [59] proposed an extension to the existing cluster schemes.
Thus, two particles are only merged together, having a similar history, e.g.
if they have the same location of entrance into the computational domain.
From this review, it can be summarized that increasing the particle number in an underpopulated grid cell is straightforward and can simply be done
by cloning, i.e. the heaviest particle in an ensemble is replaced by two new
ones with the same properties but bisected weight. Note that cloning particles is conservative and no bias error occurs following such cloning strategies.
On the other hand, consistently decreasing the number of particles is not
straightforward and problematic. Next, the three most common approaches
to reduce the particle number in grid cells are explained, followed by a new
particle reduction scheme called ”quant elimination”.
Deterministic Elimination
In this approach, the number of particles in a cell is reduced by merging two
(or three) particles into a single one (or into two, respectively). The new
weight is the sum of the old ones and some other properties like position, velocity and composition are averaged. It is not possible, however, to conserve
all statistical moments, e.g. one cannot honor the previous mean momentum and the turbulent kinetic energy at the same time. Note that with this
strategy, it is not straightforward to treat binary particle quantities.
42
4 HYBRID SOLUTION STRATEGY
Random Elimination
Here, a randomly selected particle is erased and the weights of the other ones
in the same cell are scaled up to preserve the total mass is. It can be shown
that this very simple method is statistically consistent and works well with
large numbers of particles, but it leads to large bias errors using low numbers
of particles per cell.
Statistical Elimination
In this widely used method, the two lightest particles in a cell are selected.
Then particle 1 is eliminated with probability m∗2 /(m∗1 + m∗2 ) and vice versa,
particle 2 with probability m∗1 /(m∗1 + m∗2 ). The weight of the removed particle is added to the weight of the surviving one. A single particle elimination
step is not conservative and all properties of the destroyed particle except
its weight are lost. But in the limit of infinitely many particles this elimination becomes conservative and furthermore, higher order statistics are not
destroyed.
Quant Elimination
The weights of all particles are divided into parts of a certain size, where these
parts are denoted as ”quants” of equal weight mq . Iteratively, one quant
after the other is destroyed and the weight is removed from its corresponding
particle until one particle has weight zero. This particle is then destroyed
and the weights of the remaining particles are scaled, such that the total
weight remains conserved.
4.2.3
Results
For this study, numerical calculations for an axisymmetric piloted jet flame
are performed. The flame consists of three inflow streams: in the center an
unburnt reactive jet encircled by a hot pilot, both surrounded by an inert
air co-flow. The axisymmetric computational domain with length 0.6m and
width 0.1m is resolved with a 50 × 50 grid. Jet and outer pilot diameters are
0.006m and 0.0034m, respectively; the bulk velocity of the jet is 30m/s, the
pilot velocity is 5m/s and the co-flow velocity 3m/s. Fluctuating and mean
velocities of the jet are taken from flame F3 in [16]. Figure 1 shows a sketch
of the computational setup of this test case.
The composition of a computational particle is described by two variables,
i.e. the progress variable c∗ and a flame residence time τ ∗ . c∗ has only two
4.2 Particle Number Control
43
0.6 m
Hot Pilot
U = 5 m/s
0.1 m
Co−flow
U = 3 m/s
Jet
Ubulk = 30 m/s
Figure 1: Sketch of the geometry of the generic piloted jet flame.
possible states, where a value of c∗ = 0 describes a fully unburnt cold particle and a value of c∗ = 1 represents a burning or burnt one. During a small
time step of size ∆t, a particle changes its progress variable with a certain
probability P , whereas this process is irreversible. A simple formulation for
P is employed, i.e. P = 0.4 ω
e hci ∆t. When a progress variable switches from
0 to 1, the flame residence time τ ∗ (which was zero so far) starts to run and
the composition of a particle is retrieved from a precomputed laminar flame
table. Moreover, no molecular mixing model is employed. Note that this is
a simplified formulation of the combustion models presented in chapters 8
and 9; for further explanations of the model it is referred to these chapters.
Note that for these kinds of combustion models (using binary scalars), the
deterministic elimination procedure is not directly applicable, and therefore
it is not considered in this study.
The simulations of the different cluster schemes are compared on the basis of mean downstream velocity, Favre averaged turbulent kinetic energy
and mean temperature measured at four locations; these locations are declared in table 2. One location (P1) is upstream and one is downstream (P4)
of the flame, and the two others (P2 and P3) lie within the turbulent flame
brush.
The hybrid method presented in chapter 4.1 was used for the calculations,
where in addition the moving time averaging scheme [37] was applied with a
memory factor of µmem = 0.9999. Simulations were performed without generation and elimination of particles (denoted as ”reference” simulations). For
the simulations of the three clustering schemes, also cloning was employed;
the heaviest particle was chosen for splitting, if the number of particles per
44
4 HYBRID SOLUTION STRATEGY
Table 2: Locations, at which the quantities are measured for the convergence
study.
Location x [m] y [m]
P1
0.1
0.006
P2
0.1
0.012
P3
0.2
0.012
P4
0.4
0.012
cell NNPC is lower than the desired one (Ninit ). Per time step one particle
per cell is cloned. In the random elimination simulations, if NNPC > Ninit ,
once per time step a randomly chosen particle is destroyed and its weight is
distributed among the remaining particles. The statistical particle reduction
scheme picks the two lightest particles for merging; also one particle per time
step is eliminated. For the quant simulations, at most NNPC − Ninit quants
get iteratively destroyed within a cell, but this iterative process is interupted
when the first particle has been removed. Thus, one obtains an averaged
number of particles per cell NANPC approximately equal for all three clustering schemes. The weight of a quant was chosen as mq = Vol · ρFV /Ninit ,
where Vol is the cell volume and ρFV is the mean density obtained from the
finite-volume solution. Moreover, it is ensured that the total weight in a
cell is at least divided into 10 quants. For these three clustering strategies,
simulations have been performed for Ninit = 10, 20, 40 and 80. In table 3,
the averaged numbers of particles per cell are shown for all performed simulations.
Table 3: Averaged number of particles per cell (NANPC ) for all simulations.
Cluster Type
Reference
Quant
Random
Statistical
Ninit = 10 Ninit = 20 Ninit = 40 Ninit = 80
30
59
119
10
20
41
83
9
20
42
85
10
21
42
84
In figure 2, the mean downstream velocities at the four locations are plotted
against the inverse of NANPC . At locations P1 and P2, all simulations show
approximately the same mean velocity. A different picture is given at location P3; for statistical clustering the best convergence results with respect
to NANPC are obtained, and for quant elimination the slowest convergence
4.2 Particle Number Control
45
P1
P2
40
30
35
25
30
20
25
0
0.05
0.1
15
0
P3
20
20
15
15
10
0
0.05
1/NANPC
0.1
P4
25
10
0.05
0.1
5
0
0.05
1/NANPC
0.1
Figure 2: Bias convergence study: Mean downstream velocity against
1/NANPC at four locations; reference (◦), statistical elimination (∗), random
elimination (+) and quant elimination ().
can be observed. In addition, at location P4, the large bias error for random
clustering for low NANPC can be observed. Favre averaged turbulent kinetic
energies are presented in figure 3, where for locations P1 and P2 approximately the same values are obtained for all simulations. For locations P3 and
P4, the best overall agreement is obtained for statistical clustering, whereas
for random and quant clustering larger values of NANPC are required for the
same level of accuracy obtained by statistical elimination. Mean temperatures are presented in figure 4. The dependence of the mean temperature on
NANPC is negligible for location P1, but increases slightly for P2 except for
the statistical approach. For P3 the random strategy is comparable with the
statistical elimination only for NANPC = 80, whereas for low NANPC random
46
4 HYBRID SOLUTION STRATEGY
P1
P2
10
16
8
14
6
12
4
10
2
8
0
0
0.05
0.1
0
P3
0.05
0.1
P4
25
20
18
16
20
14
12
15
0
0.05
1/NANPC
0.1
10
0
0.05
1/NANPC
0.1
Figure 3: Bias convergence study: Favre averaged turbulent kinetic energy
against 1/NANPC at four locations; reference (◦), statistical elimination (∗),
random elimination (+) and quant elimination ().
elimination leads to a large bias. On the other hand, all elimination schemes
underpredict for all chosen NANPC the mean temperature at P3. At location
P4, statistical elimination delivers the best results and the strong bias error
of the random approach for low NANPC can be observed.
4.2.4
Summary
To efficiently treat Monte-Carlo particle methods and to obtain a uniformly
distributed bias error, particle number control algorithms are required. While
particle generation is straightforward and does not lead to conservation violation of any statistical moments, particle number reduction is more crucial.
4.2 Particle Number Control
47
P1
P2
2000
2500
1800
1600
2000
1400
1200
1000
0
0.05
0.1
1500
0
P3
1600
2000
1400
1800
1200
1600
1000
1400
800
0
0.05
1/NANPC
0.1
P4
2200
1200
0.05
0.1
600
0
0.05
1/NANPC
0.1
Figure 4: Bias convergence study: Mean temperature against 1/NANPC at
four locations; reference (◦), statistical elimination (∗), random elimination
(+) and quant elimination ().
Three major strategies for particle number reduction within a cell are proposed in the literature: deterministic, random and statistical elimination
approaches. In addition, a new clustering scheme based on so called quants
is presented here. Overall, the statistical approach delivers the best results
with respect to the averaged number of particles per cell; the quant cluster
scheme is not able to improve these results.
5
One-Dimensional Setup for Simplified Flow
Simulations
For premixed laminar flames one very important characteristic is the laminar flame speed sL . On the other hand, for a premixed turbulent flame,
the turbulent flame speed sT is the corresponding velocity. While the laminar flame speed can easily be computed in a one-dimensional setup, turbulent flames are subject to three-dimensional turbulent structures. Therefore,
various experiments were designed to measure sT , e.g. weak swirl burners
[15, 5, 68, 17, 40, 84]. There the mean temperature field is approximately
planar close to the nozzle at the centerline.
The goal of this chapter is to derive a quasi one-dimensional flow configuration, which is applicable for Monte-Carlo particle simulations of turbulent
premixed planar flames. In addition, the setup allows for simplified calculations of weak swirl burners. It is postulated that this setup (a) is numerically
efficient and (b) shows less uncertainties regarding the hydrodynamic solution, such that one can focus on the investigation of the effect of turbulent
fluctuations on the premixed flame.
Assumptions and simplifications for this setup are presented next. Then,
the governing equations are presented and finally, the functionality and convergence are shown in the result section. Note that this chapter is focused
on the numerical functionality of the flow setup, wherefore a simplified combustion mechanism is used.
5.1
Idea
This one-dimensional configuration has been devised in analogy to the structure along the symmetry axis of a low swirl burner. It allows for computationally very efficient simulations with little uncertainty regarding the hydrodynamic solution. The basic idea of the setup is to assume homogeneous
50
5 1D SETUP
f1 in downturbulence and to employ the decaying momentum profile hρi U
stream direction as flame stabilizer, where the governing equations have no
direct dependence on the momentum profile. This situation is illustrated in
figure 5, where the shaded area represents the flame. It is a one-dimensional
< ρ U1>
< ρ>
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
T
U1
urms
x1
Figure 5: Schematic illustration of the simplified quasi one-dimensional simulation setup.
setup for the mean velocities, i.e. the mean radial and tangential velocities
f1 decays in downstream direction (with increasing
are set to zero. While hρi U
x1 ), the turbulence is approximately homogeneous, but possibly anisotropic.
As the flame propagates into the fresh gas and competes with the mean downstream velocity, it stabilizes at the location where the turbulent flame speed
f1 . Here, the turbulent flame
is matched by the mean downstream velocity U
speed is defined as the mean velocity at this specific location. Note that the
mean density and the mean temperature are obtained from the PDF method.
This setup is employed to perform simulations of experimentally measured
weak swirl burners. To be more precise, the region close to the centerline is
simulated, where the mean flow field is assumed to be one-dimensional, i.e.
the downstream momentum profile is chosen according to the experiment and
is fixed for the entire simulation. The measured turbulent fluctuations of the
experiment are approximated by homogeneous fluctuations. Here, even the
5.1 Idea
51
hydrodynamic flow field is prescribed, i.e. the momentum profile is fixed,
whereas the mean velocity evolves through the flame with respect to this
profile. The considered domain within the weak swirl burner is sketched in
figure 6. The computational domain is represented by the shaded area, where
the upper and lower boundaries are mean flow stream lines, i.e. the mean
mass fluxes across these boundaries are zero.
quasi
planar
flame
vortex
breakdown
region
11111111
00000000
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
Figure 6: Schematic illustration of the simplified quasi one-dimensional simulation of a weak swirl burner. The shaded area is the computational domain,
the dashed lines are iso-temperature levels and the arrows represent mean
flow field stream lines.
An alternative application of this configuration is illustrated in figure 7,
where the turbulent flame speed is obtained in a pseudo pipe flow configuration, i.e. a planar premixed flame is simulated. The momentum profile
is as flat as possible in a certain range (”flame layer”), where the flame is
expected. The ”buffer layers” are used for flame stabilization due to the
statistical behavior of the flame, and therefore the momentum profile gets
steeper towards the in- and outflow boundaries. Within the centered domain,
the flame structure becomes similar to the one of planar premixed flames,
and therefore the turbulent flame speed can be determined.
52
5 1D SETUP
buffer
layer
flame
layer
buffer
layer
T
< ρU1>
x1
Figure 7: Schematic illustration of the simplified configuration to simulate
planar turbulent flames, where the turbulent flame speed is computed.
5.2
Governing Equations
The presented method is a stand alone particle method describing the joint
PDF of fluctuating velocity, turbulence frequency and scalars. The mean
downstream velocity is determined as the ratio of the predefined momentum
f1 and of the mean density from the simulation hρi
profile hρi U
calculation
f1 =
U
f1
hρi U
.
f ixed
hρicalculation
.
(95)
The quantity hρicalculation is calculated as
hρiJcalculation
J ∗ ∗
P
∀particles ĝ (x )m
≈ P
J
∗
∗
∗
∀particles {ĝ (x )m /ρ }
(96)
at fix grid points xJ and from there interpolated to the particle positions. The
particle weight is denoted as m∗ and ĝ J (x) is the basis function associated
with grid node J (used for both extraction and interpolation). The particle
property ρ∗ is obtained from its composition and enthalpy, i.e.
ρ∗ =
pref γ ∗
h∗ γ ∗ − 1
,
(97)
5.2 Governing Equations
53
which is a valid approximation for low Mach numbers, where the pressure
variations are much smaller than the reference pressure pref . Note that equations (95) - (97) define the mean downstream velocity. In addition, the two
f2 = U
f3 = 0) . Thus, no
other mean velocity components are set to zero (U
RANS solver is required for this setup.
Turbulence is postulated to be homogeneous within the whole computational domain. The turbulent fluctuations as well as the turbulence frequency
have to be spatially independent. Therefore the corresponding stochastic
models for the fluctuating velocity (equation (87)) and for the turbulence
frequency (equation (89)) have to be modified. The simplified Langevin
model (SLM) modifies to
q
3
C0 k(i) ΩdWi ,
(98)
du∗i = − C0 Ωu∗i dt +
4
where the index in brackets indicates that the Einstein summation convention
is circumvented. C0 is a model constant (standard value of 2.1) and Wi
represents a Wiener process. Equation (98) contains the two terms of the
original SLM, that are responsible for the redistribution of turbulent kinetic
energy. These redistribution terms ensure the homogeneity of the turbulent
fluctuations. Since for the three fluctuating velocity components different
values for k(i) are adaptable, it is possible to describe anisotropic turbulence.
Without loss of generality, it is assumed that the two averaged turbulent
fluctuations in the non-downstream directions are identical. Therefore, the
anisotropy coefficient aaniso is introduced, defining the ratio of the Reynolds
stresses in downstream and cross stream directions
aaniso =
00 00
00 00
ug
u
ug
2 u2
= 3 3
00 00
00 00
ug
ug
1 u1
1 u1
.
(99)
Thus, the Reynolds stresses can be expressed as a function of the turbulent
kinetic energy k and the anisotropy coefficient aaniso . They become
2k
1 + 2aaniso
2kaaniso
=
1 + 2aaniso
2kaaniso
=
1 + 2aaniso
= 0
00 00
ug
1 u1 =
00 00
ug
2 u2
00 00
ug
3 u3
00 00
ug
1 u2
g
u001 u003 = 0
00 00
ug
2 u3 = 0 .
(100)
54
5 1D SETUP
Moreover, the modified turbulent kinetic energies k(i) of equation (98) are
defined as
2k
1 + 2aaniso
2kaaniso
00 00
=
1.5
u
= 1.5 ug
2 2
1 + 2aaniso
2kaaniso
00 00
=
1.5
u
= 1.5 ug
3 3
1 + 2aaniso
00 00
k(1) = 1.5 ug
1 u1 = 1.5
k(2)
k(3)
.
(101)
The stochastic model for the turbulence frequency is modified in a similar
way, i.e. it only contains the redistribution terms of the turbulence frequency.
Thus equation (89) reduces to
p
e )Ωdt +
2C3 C4 ω
e Ωω ∗ dW ,
(102)
dω ∗ = − C3 (ω ∗ − ω
where C3 and C4 are model constants with standard values of 1.0 and 0.25,
respectively.
For the extraction of particle properties, a moving time averaging scheme
was applied [37]. The extraction of a particle property κ∗ is done in the
following way. From the instantaneous set of particles at a fixed grid point
xJ , an instantaneous massweighted value is extracted by
J ∗ ∗ ∗
P
ĝ (x )m κ
∀particles
.
(103)
κJinst = P
J
∗
∗
∀particles {ĝ (x )m }
Finally, the extracted value κ
e is obtained from a weighted composition of its
old
previous value κ
e and the instantaneous value κJinst
κ
e = µmem κ
eold + (1 − µmem ) κJinst
.
(104)
The memory factor µmem has usually a value close to (but smaller than) 1.
For further explanation of this scheme, it is referred to [37].
Together with equation (86) for the particle location, equations (98) and (102)
describe the evolution of the particle in a homogeneous (but anisotropic) turbulent flow field.
5.3
Convergence Study
In the previous section, the simplified simulation setup has been explained,
whereas in this section, a convergence study is performed to show the correct
functionality of the setup. To do so, a very simple combustion model is used,
5.3 Convergence Study
55
where the motivation and further explanations about the model are given in
the two chapters about modeling of turbulent premixed flames.
The composition of a computational particle is described by a single variable, i.e. the progress variable c∗ . It has only two possible states, where a
value of c∗ = 0 describes a fully unburnt cold particle with enthalpy h∗u and a
value of c∗ = 1 stands for a fully burnt hot particle with enthalpy h∗b . Thus,
the flame is assumed to be infinitely thin, i.e. the fast chemistry assumption
is employed. During a small time interval of length ∆t, a particle changes
its state from unburnt to burnt with a certain probability P , whereas this
process is irreversible. The ignition probability P is modeled as
P = 1 − exp {200.0 hci ∆t}
.
(105)
For the convergence study, the turbulent flame speed sT is identified as the
mean downstream velocity at the location of an averaged progress variable
value of 0.05, and the corresponding physical location in the computational
domain is denoted as xT . The turbulence frequency was set to ω
e = 100.0 for
the following simulations.
4.5
4
sT [m/s]
3.5
3
2.5
2
1.5
0
0.5
1
NTime Steps
1.5
2
6
x 10
Figure 8: Evolution of the turbulent flame speed sT for different turbulent
intensities over a period of 2 million time steps: urms = 0.2m/s (solid line),
urms = 0.8m/s (dashed line), urms = 1.4m/s (dotted line) and urms = 2.0m/s
(dashed-dotted line).
In figure 8, the evolution of the turbulent flame speed for several turbulence levels is compared. The root-mean-square (rms) of the fluctuation
56
5 1D SETUP
14
12
10
8
6
4
2
0
0
0.5
1
1.5
x1
Figure 9: Simulation result for an rms velocity fluctuation of 2.0m/s: fixed
f1 (solid line), mean downstream velocdownstream momentum profile hρi U
f1 (dashed line), normalized density 10.0 hρi (dotted line) and
ity profile U
00 00
Reynolds stresses ug
1 u1 (dashed-dotted line).
linearly increases from 0.2m/s to 2.0m/s, and as expected the turbulent
flame speed increases too. This result reveals that the one-dimensional setup
works and that the simplified combustion models show the correct tendencies for the turbulent flame speed. Moreover, from the simulation with a rms
of the fluctuating velocity of 2.0m/s, the following quantities are depicted
f1 (solid line), the
in figure 9: the fixed downstream momentum profile hρi U
f1 (dashed line), the normalized density
mean downstream velocity profile U
00 00
10.0 hρi (dotted line) and the Reynolds stresses ug
1 u1 (dashed-dotted line).
This demonstrates the correct functionality of the stabilization mechanism
of this configuration.
In the figures 10 - 14, the numerical stability of this configuration is
investigated. Starting with the investigation how the initial flame position
influences the final result. In figure 10 the flame location xT is depicted over
a period of 2 million time steps. The initialization of the flame at different
locations is observable on the left side of figure 10. Ultimately, the flame
stabilizes for all four simulations at the same location. The same result is
observable in figure 11, where the turbulent flame speed sT is shown for the
same set of simulations.
5.3 Convergence Study
57
1.2
xT [m]
1
0.8
0.6
0.4
0.2
0
0.5
1
NTime Steps
1.5
2
6
x 10
Figure 10: Evolution of the location of the flame front xT over a period of 2
million time steps: xT,init = 1.2m (solid line), xT,init = 0.9m (dashed line),
xT,init = 0.6m (dotted line) and xT,init = 0.3m (dashed-dotted line).
8
sT [m/s]
7
6
5
4
3
0
0.5
1
NTime Steps
1.5
2
6
x 10
Figure 11: Evolution of the turbulent flame speed sT over a period of 2 million
time steps: xT,init = 1.2 (solid line), xT,init = 0.9 (dashed line), xT,init = 0.6
(dotted line) and xT,init = 0.3 (dashed-dotted line).
58
5 1D SETUP
9
8
7
sT [m/s]
6
5
4
3
2
1
0
0
2
4
6
NTime Steps
8
10
5
x 10
Figure 12: Evolution of the turbulent flame speed sT over a period of 1
million time steps: Nx = 100 (solid line with circles), Nx = 200 (dashed
line), Nx = 400 (dotted line), Nx = 800 (dashed-dotted line) and Nx = 1600
(solid line).
The grid convergence is analized in figure 12. The number of grid cells has
been increased from a value of Nx = 100 towards a value of Nx = 1600 by doubling Nx . Obviously a good spatial resolution is required for accurate simulations. For Nx = 100 a turbulent flame speed of approximately sT = 7.5m/s
is obtained, whereas the simulations with Nx = 800 and Nx = 1600 predict
an approximate value of sT = 1.7m/s. The reason for the observed discrepancies lies in the formulation of the combustion process (equation (105)). P
depends on the averaged progress variable, and therefore in a coarse grid,
the profile of hci is underresolved, which leads to a wrong increase of sT .
In the last two figures, the influence of the particle number and of the
memory factor is analyzed. In figure 13, the averaged number of particles
is kept constant at 10, where the memory factor µmem increases from 0.99
to 0.99999. For a low value of µmem , the resulting profile of sT is very
noisy, whereas for the highest value of µmem , the evolution of the turbulent
flame speed is very flat and the bias error is reduced. This influence is
further investigated in figure 14. An averaged number of particles NN P C
of 10 together with a memory factor of µmem = 0.999 is compared with a
configuration with NN P C = 100 and µmem = 0.99. Both simulations lead to
the same turbulent flame speed, but the solution of the second one is more
5.3 Convergence Study
59
3.2
sT [m/s]
3
2.8
2.6
2.4
2.2
1.5
1.6
1.7
1.8
NTime Steps
1.9
2
6
x 10
Figure 13: Evolution of the turbulent flame speed sT over a period of 0.5
million time steps: µmem = 0.99 (solid line), µmem = 0.999 (dashed line),
µmem = 0.9999 (dotted line) and µmem = 0.9999 (dashed-dotted line).
3.2
3.1
sT [m/s]
3
2.9
2.8
2.7
2.6
1.5
1.6
1.7
1.8
NTime Steps
1.9
2
6
x 10
Figure 14: Evolution of the turbulent flame speed sT over a period of 0.5
million time steps: µmem = 0.999 and NN P C = 10 (solid line) and µmem =
0.99 and NN P C = 100 (dashed line).
60
5 1D SETUP
noisy. Whether one increases NN P C or 1/(1 − µmem ) by a factor of 10 (to
increase the accuracy of the simulation) does not matter, but increasing the
memory factor smoothes out the profiles.
5.4
Summary
In this chapter, a quasi one-dimensional flow configuration has been presented, where the joint PDF of fluctuating velocity, turbulence frequency and
scalar vector is solved with a stand alone Monte-Carlo particle method. The
aim of the configuration is to perform numerically efficient simulations of low
swirl burners with little uncertainties regarding the hydrodynamic solution.
This setup has been devised in analogy to the structure along the symmetry
axis of a low swirl burner. In a homogeneous, possibly anisotropic turbulence
field, the decaying but prescribed momentum profile in downstream direction
acts as flame stabilizer. The governing stochastic differential equations of the
computational particles have been presented and the configuration has been
examined for its computational use.
Part II
Modeling of Turbulent
Premixed Combustion
6
Turbulent Premixed Combustion
Most combustion devices operate under turbulent flow conditions. Examples
are gas turbines, internal combustion engines and spark-ignition engines.
Thus, reaction kinetics known from laminar flames are modified by the turbulence and, vice versa, the released heat modifies the flow characteristics.
In particular, the interaction of reaction kinetics and turbulent fluid motions
plays an important role in the understanding of the underlying physical and
chemical processes. These turbulence-chemistry-interactions are rather complex and far from completely understood, even the two single fields are still
subject of intense research. Hence, no general satisfying approach to treat
turbulent reactive flows exists so far.
As already mentioned above, turbulence itself is still a huge research field.
In the past century, progress has been made in the general understanding of
turbulent flows. It is well established to consider a turbulent flow field as
an ensemble of superimposed eddies of different sizes. The largest energycontaining eddies decay into smaller ones, these decay into even smaller ones
and so on, where the smallest eddies dissipate due to viscous forces into
heat. This process is called cascade process of turbulent flows. Therefore, the
range of length, velocity and time scales in a turbulent flow may span several
orders of magnitude, where the ratio of smallest-to-largest length scales can
be expressed as a function of the Reynolds number Re and scales with Re−3/4 .
The largest length scales lI are usually of the size of the geometrical dimension
of the flow, whereas the smallest ones are of the size of the Kolmogorov length
scale lη , which is calculated as a function of the dissipation rate of turbulent
kinetic energy ε and the viscosity ν based on a dimensional analysis.
The first modeling approaches for the treatment of turbulent flows are
mainly based on mean field quantities. Evolution equations are solved for
mean velocities (RANS equations), where the unknown Reynolds-stresses are
estimated under the assumption of the Boussinesq eddy-viscosity-hypothesis.
There, the six Reynolds-stresses are related to one unknown, the so-called
eddy-viscosity νT . Algebraic models for νT have been derived, e.g. the
Prandtl’s mixing length model, but also closures based on a certain number
64
6 TURBULENT PREMIXED COMBUSTION
of additional equation have been proposed. Among these approaches, the
k − ε model is the most common technique for industrial applications, also
since they are quite robust. An alternative to the Boussinesq eddy-viscosityhypothesis are the Reynolds-stress models, where transport equations for
the Reynolds-stresses are derived, modeled and solved. But this method is
numerically less stable. While the computational resources have been amplified, more sophisticated approaches could be considered for flow simulations.
Large eddy simulations (LES) employ a low-pass filter to the exact governing
equations. The filtered equations resolve the large scale turbulent motions
within the flow field and only the effect of the small (subgrid scale (SGS))
structures has to be modeled, in contrast to RANS, where the full range
of turbulent fluctuations has to be captured. The price for the higher level
of closure is the increased numerical cost limiting such kinds of simulations
to smaller and geometrically simpler configurations than feasible in RANS
simulations. With the appearance of probability density function (PDF)
methods, a more rigorous statistical inspection of turbulent flows showed up.
Detailed statistical information is obtained by solving a transport equation
for the one-point one-time PDF of certain flow properties. Initially, this
evolution equation was solved by stand alone Monte-Carlo particle methods,
but high numbers of particles were required to keep the statistical and bias
errors at a low level. With the introduction of so-called hybrid methods and
further algorithmic improvements, e.g. moving time averaging of extracted
quantities or local particle time stepping, PDF simulations are considered
more and more for the study of turbulent reactive flows. A more elaborate
description of the different approaches to model turbulent flows may be found
in [43, 100, 76, 19, 31], but is not a topic of this work.
During the last 40 years, the description of gaseous reactions has been
improved and many reduced mechanisms have been proposed, that are able
to accurately describe laminar flames. Such reduced mechanisms usually
consist of a set of chemical species and a set of elementary (forward and
backward) reactions. Note that these sets of species and elementary reactions are often called elementary reaction mechanisms. A wide range of
elementary reactions can be found in the literature. One detailed mechanism
is the GRI-2.11 mechanism [18], it comprises 277 reactions and 49 chemical
species. On the other hand, also one-step mechanisms are employed in combustion simulations. In particular, in direct numerical simulations (DNS) of
the flow field, chemistry is often treated by just a few reactions. Using a large
elementary reaction mechanism, the governing system of ordinary differential
equations is highly non-linear and stiff. Hence, the wide range of chemical
time scales prohibits a direct numerical solution of the system. This rises the
demand of simplified kinetic mechanisms. One possible simplification is to
6.1 Laminar Premixed Flames
65
assume a quasi steady state for the fast species or reactions, and therefore
the stiffness of the system can be reduced. Nevertheless, the treatment of
reduced elementary chemical mechanisms is not subject of this work, and for
details on that field it is referred to [42].
First, in section 6.1 reaction kinetics are explained on the basis of laminar
one-dimensional premixed flames. Different specific length and time scales of
laminar premixed flames are presented. Next, in section 6.2 turbulent flows
interacting with combustion kinetics are discussed and the different regimes
of premixed turbulent combustion are explained. These regimes depend on
certain dimensionless variables like Karlovitz or Damköhler number Ka and
Da, respectively, that are based on a time and length scale comparison for
turbulence and chemistry. Moreover, a brief overview of the existing modeling approaches is given in chapter 7.
6.1
Laminar Premixed Flames
Laminar premixed flames are an important topic in combustion research.
From one-dimensional flow configurations, e.g. pipe flows, some fundamental
understanding can be gained. Furthermore, for such flows some analytical
or semi-analytical solutions exist, making them very attractive to investigate
the accuracy of reduced and simplified mechanisms by means of laminar
flames. The premixed flamelet approach is based on such laminar flame
profiles (flamelets). In general, as already mentioned, the propagation is
mainly driven by thermal diffusion into the fresh gas. The corresponding
heat release and the propagation speed of the flame depend on the underlying
reaction kinetics, on the thermodynamics and on the surrounding flow field.
The structure of a one-dimensional laminar premixed methane-air flame
is sketched in Figure 15. The profile is usually subdivided into three layers:
the preheat zone (I), the inner or reaction layer (II) and the oxidation layer
(III).
In the preheat zone, no major reactions take place and the temperature
remains relatively low. Nevertheless, in this zone the radical pool of unstable
free or active molecules starts to grow by chain branching reactions, initiated by the free radicals from the reaction zone. In addition, due to the
diffusion of the high temperature in the reaction zone, the temperature is
increased, which further enhances the production of free radicals; free radicals are important to initialize the main chemical reactions. Therefore, this
layer is also called convective-diffusive layer, since the dominating processes
are convection and diffusion.
Layer II can be denoted as motor of the reaction, since it is responsible to
66
6 TURBULENT PREMIXED COMBUSTION
I
111
000
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
II
O2
CH4
lF
III
T
U
CO 2
NO
H2
lδ
Figure 15: Sketch of a steady laminar premixed flame containing the preheat
zone (I), the inner or reaction zone (II) and the oxidation layer (III).
keep the reactions alive. There, the fuel is consumed by the flame and the free
radicals are depleted. The chain branching reactions compete with the chain
breaking ones, whereas both are very sensitive to temperature variations and
radical concentrations. Therefore, the inner layer is very sensitive to flow
perturbations. Heat release due to combustion occurs mainly within this
layer and this heat is transported towards the two neighbouring ones by
diffusion. Thus, it also called the reactive-diffusive layer, which is dominated
by the fast chemical reactions.
In the oxidation layer, the chain breaking process of the flame determines
the reactions. Slow reactions, like e.g. the recombination of free radicals,
consume the remaining active radicals without generating new ones, until the
reaction chain is terminated. In addition, slow processes like the formation
of N O can still be active.
The laminar flame thickness lF is regarded as the thickness of the preheat
zone plus the inner reaction zone. But various kinds of definitions [73, 65, 26]
exist for lF : the thermal thickness lF,T H , the diffusive or Zeldovich thickness
lF,D and the total thickness lF,T . In [70], the definitions are presented in
6.2 Regimes of Turbulent Premixed Combustion
67
section 2.5 and read
(Tb − Tu )
max (∂T /∂x)
D
=
sL T − Tu
T − Tu
= 0.95 − x
= 0.05 = x
Tb − Tu
Tb − Tu
lF,T H =
lF,D
lF,T
,
(106)
where D is the thermal diffusion coefficient and Tb and Tu are the burnt and
unburnt temperatures. The total thickness is much larger than the thermal
thickness, since it also captures the possibly long tail of the laminar flame.
The most physical definition of the laminar flame thickness is the thermal
one, but lF,T H can only be determined a posterior. The diffusive thickness can
be determined from the thermal diffusion coefficient and the laminar flame
speed, but it has less physical meaning. Nevertheless, for the theoretical
reflection of the different combustion regimes for premixed turbulent flames,
the diffusive thickness lF,D is employed due to simplicity, without loss of
generality.
The thickness of the inner or reaction layer is denoted as lδ and it is related
to the laminar flame thickness by lδ = δ lF , where it is usually assumed that
δ = 0.1. Another important quantity is the laminar flame speed sL . It is
defined as the velocity, with which the flame propagates, i.e. in figure 15 the
gas velocity left of the preheat zone.
6.2
Regimes of Turbulent Premixed Combustion
The interaction of reaction kinetics and turbulent fluctuations is complex and
far from being completely understood. Let us consider small eddies influencing the flame. The competition between chain breaking and chain branching
reactions is modified by these processes, where this competition is sensitive
with respect to temperature changes. Thus the reaction kinetics may be enhanced or extinction may occur. In addition, the heat release can lead to flow
instabilities that further enhance the turbulent fluctuations. On the other
hand, the density decreases such that the viscosity gets increased, resulting
in a reduction of the Reynolds number. This effect is called laminarization
by the flame. Note that these processes occur on a wide range of scales, i.e.
from the integral down to the Kolmogorov and the chemical length scales.
Therefore, a more detailed look at these effects is required, where the different responsible length and time scales are considered for these interactions.
68
6 TURBULENT PREMIXED COMBUSTION
The classification of turbulence-chemistry interaction is based on two assumptions [65], that simplify the derivation and illustration of the different
regimes without loss of generality. First, a Lewis number Le of unity is assumed. It is defined as the ratio Dth /Dα , where Di is the diffusion coefficient
of species α and Dth is the thermal diffusion coefficient. Note that in the
following, D is used for the diffusion coefficient, since D = Dα = Dth . In
the second assumption, a Schmidt number of unity is presumed, where the
Schmidt number Sc = ν/D is defined as the ratio of the kinematic viscosity
ν and the diffusivity coefficient D. Therefore, the diffusive laminar flame
thickness, considered here as the characteristic chemical length scale of the
flame, is determined by
ν
,
(107)
lF =
sL
which follows from equation (106) and unity Schmidt number. Supplementary, the corresponding chemical time scale is defined as
ν
lF
.
(108)
=
sL
sL 2
The range of the turbulence scales is bounded by two characteristic scales,
i.e. the integral scales (denoted by the subscript I) describing the large
scale turbulent motions and the Kolmogorov scales (denoted by the subscript
η) describing the size of the smallest eddies. Based on Kolmogorovs first
similarity hypothesis, the Kolmogorov length, time and velocity scales can
be determined as a function of the kinematic viscosity ν and the dissipation
rate of turbulent kinetic energy ; they read
3 1/4
ν
lη =
ν 1/2
tη =
0
uη = (ν)1/4 .
(109)
tF =
The integral length scale lI is of the order of the system dimension, the corresponding velocity scale uI is in the order of the root mean square of the
fluctuating velocity and the integral time scale tI is given by the ratio of lI
and uI .
A first dimensionless variable is the turbulent Damköhler number Da, comparing the integral time scale tI with the chemical time tF . Under the assumptions of unity Lewis and Schmidt numbers, one can write Da as
Da =
tI
lI sL
=
tF
lF uI
.
(110)
6.2 Regimes of Turbulent Premixed Combustion
69
If Da 1, the turbulent integral times are shorter than the chemical times.
The educts and products are mixed in a much shorter time than required
for the chemical reactions to take place. Thus, if Da 1, one talks of
”perfectly stirred reactor” conditions, where the mean reaction rate can be
approximated by the reaction rate of mean quantities. On the other hand, for
Da 1 the turbulence time scales are much larger than these of the chemistry and thus the turbulence is too slow to perturb the inner flame structure.
Another important dimensionless quantity is the Karlovitz number Ka, describing the ratio of the laminar flame time tF and the time tη of the smallest
eddies. It reads
2
lF
tF
=
.
(111)
Ka =
tη
lη
In addition, a second Karlovitz number Kaδ exists, that is defined with
respect to the inner reaction zone of the flame instead of the full flame. It is
related to the first Karlovitz number and it reads
2
2
δ lF
lδ
=
= δ 2 Ka .
(112)
Kaδ =
lη
lη
The turbulent Reynolds number Re can be rewritten by replacing the kinematic viscosity by equation (107). It becomes
Re =
lI uI
lF sL
,
(113)
and it can be expressed as a function of the Damköhler and Karlovitz number
Re = Da2 Ka2 , where the relation ε ∝ u3I /lI is used.
Based on these dimensionless quantities, many different types of combustion
diagrams have been proposed, e.g. by [63, 2, 69]. Here the one presented in
[64] is considered. In Figure 16, this diagram for turbulent premixed combustion is illustrated. It contains four regimes, the laminar flame-, the corrugated
flamelet-, the thin reaction zone- and the broken reaction zone regimes. The
first regime considering laminar flames is characterized by Re < 1 and was
already described in the previous section. The three other regimes are briefly
explained next, based on the detailed description given in [65].
The corrugated flamelet regime is characterized by a Karlovitz number
smaller than 1, such that the smallest eddies of Kolmogorov size are still
larger than the laminar flame thickness. The flame is embedded within eddies
of size lη , where the flow remains quasi laminar. Therefore, the reactivediffusive flame structure is not perturbed by the turbulence and remains
70
6 TURBULENT PREMIXED COMBUSTION
u’rms
sL
103
Kaδ = 1
Broken Reaction Zone
10
2
Thin Reaction Zone
Ka = 1
10
Re = 1
Corrugated Flamelets
1.0
Laminar
Flames
0.1
0.1
1.0
10
102
103
lI
lF
Figure 16: Regimes of premixed turbulent combustion by [64].
quasi steady. Kinematic interaction occurs between turbulent eddies and the
laminar flame. But not all eddies are able to interact, only those with a
turnover velocity close to the laminar flame speed sL . By that reason, the
Gibson scale lG was proposed [63] to be the size of eddies with a turnover
velocity equal to the laminar flame speed and it is determined by
lG =
sL 3
.
(114)
Only eddies close to the size lG can directly interact with the flame front. Eddies, that are smaller than lG have a corresponding turnover velocity smaller
than sL and accordingly are not able to wrinkle the flame front. The laminar
flame propagation dominates over the wrinkling of the flame front by the
turbulence. On the other hand, if an eddy is larger than lG , it has a turnover
velocity larger than sL . Hence, such an eddy can compete with the flame
propagation and is able to push the flame front around, i.e. the flame front
gets corrugated and wrinkled.
The thin reaction zone regime is bounded by Ka = 1 and Kaδ = 1, where
6.2 Regimes of Turbulent Premixed Combustion
71
the corresponding eddies are able to interact with the propagating flame. A
consequence of Karlovitz numbers larger than one is that the smallest eddies
are smaller than the laminar flame thickness. Note that if Kaδ < 1, these
eddies are still larger than the inner reaction zone. Therefore, the smallest
eddies are able to penetrate into the reactive diffusive flame structure, but not
into the inner layer. Eddies, entering the preheat zone, transport preheated
fluid ahead of the reaction zone and vice versa reactive fluid into the preheat
zone. By that reason, scalar mixing gets enhanced. An important time scale
associated with the thin reaction zone regime is the quenching time tq [64],
defined as the inverse of the strain rate required to extinguish a flame. It is
of the same order than the laminar flame time tF . With this time
p scale, one
can introduce a diffusion thickness lD , which is defined as lD = D tq . The
corresponding length scale of an eddy with turnover time tq is defined as
q
lq =
tq 3 ,
(115)
which is also called mixing length scale [64]. Eddies of this size interact
with the flame front and are able to transport preheated fluid structures
of thickness lD away from the reaction zone towards the fresh gas during
the time tq . On the other hand, eddies larger than lq also have a bigger
turnover time. Thus they are only able to transport structures that are
thicker than lD , such that they corrugate the flame, but they do not perturb
its entire structure. But the maximum distance by which fluid structures are
transported away from the reaction zone is lq , which is larger than lD . Thus
one talks of so called ”pockets” of the flame that occur due to the above
described phenomena. These ”pockets” are of size close to lD , that is in the
same order of magnitude as the laminar flame thickness lF . Therefore, in the
thin reaction zone regime, variations in the thickness of the flame structure
occur that are of the same size as the thickness itself.
The last regime is called broken reaction zone regime and occurs for
Kaδ > 1. Thus there exist eddies that are smaller than the reaction zone.
These eddies enter the inner layer and perturb its reactive diffusive structure. The strong turbulent diffusion, induced by eddies smaller than lδ , is
responsible for the strong heat loss within the reaction layer and also for the
loss of radicals that are both transported towards the preheat zone. The
reduced temperature within the ”motor” of the flame and the absence of a
sufficiently large radical pool reduce the reaction rates further and further,
which then finally leads to local extinction of the flame. The reaction zone
can then be regarded as a thickened layer containing holes where the flame
is locally extinct. In the extreme case this leads to global extinction of the
flame.
72
6 TURBULENT PREMIXED COMBUSTION
With the classification of these regimes, the main effects of the turbulent
eddies on the premixed flame are considered. But how does the flame influence the turbulent flow field? Several phenomena are known, which are
responsible for the effects of the flame on the turbulence and the interaction of
eddies with the propagating flame. In the next paragraph, three well known
phenomena are explained, i.e. laminarization by the flame, enhancement of
turbulence by the flame and counter-gradient diffusion.
When crossing a turbulent premixed flame from the cold to the hot side,
the turbulence intensity decreases. This phenomenon is called relaminarization by the flame and its origin lies in the temperature or density dependence
of the viscosity. The temperature increase induces a growth of the viscosity,
resulting in a lower Reynolds number. On the other hand, density variations
induced by the heat release may lead to flow instabilities, which enhance the
turbulence. Thus these two effects, i.e. relaminarization and flame induced
turbulence compete with each other.
The third effect explained here is the so called counter-gradient diffusion
due to gas expansion. Since density variations from cold to hot gases may become large, i.e. ratios larger than 5 are common, the corresponding pressure
gradients act different for hot and cold gas. This can lead to turbulent scalar
fluxes with a direction opposed to the one prescribed by the corresponding
gradients. Therefore, this phenomenon is called counter-gradient diffusion
and it mainly occurs within ducted flows. Most modeling approaches for
turbulent scalar fluxes are based on a gradient-diffusion ansatz, which does
not work here. Thus the modeling of counter-gradient diffusion is still a
challenging task. A more detailed discussion on counter-gradient diffusion is
given in [45, 97, 4, 104, 102].
7
Existing Modeling Approaches for Turbulent Premixed Flames
In this chapter, an overview is given on existing modeling approaches for
the prediction of premixed turbulent flames. This overview is far from presenting a complete state-of-the-art review on modeling of premixed turbulent
flames, but it shows the basic ideas of the most common approaches.
For non-premixed combustion, various models exist that are based on
the mixture fraction [62, 30, 6], where the basic idea is the one of laminar
flamelets. There the flame is considered to consist of an ensemble of such
flamelets, while it is assumed that the inner reactive-diffusive flame structure
remains unperturbed by the turbulence. Numerical simulations show the
applicability and the entitlement of such models [41, 101, 66]. On the other
hand, for turbulent premixed combustion no such general approach exists,
which is pointed out in [65, 73].
In the first section, models based on the BML approach are presented,
followed by the G-equation approach in the second section. Note that these
approaches are explained in the RANS context, but are also applicable in
the context of large eddy simulations (LES), where the models only have to
capture the effects of the subgrid scales (by subgrid scale models). Modifications of these models for LES are not discussed in the overview given here.
In the third section, PDF methods are briefly explained and, finally, in the
last section, several reduction schemes are sketched.
7.1
BML Approach
The basic principle of many approaches within an Eulerian description of
turbulent premixed flames is the employment of a progress variable. Usually the progress variable is a normalized temperature or a normalized mass
fraction and it is bounded between 0 and 1. The chemistry is treated by a
single variable one-step irreversible reaction. Within this Eulerian context,
a well known model is the one by Bray, Moss and Libby (BML) [12]. The
74
7 EXISTING MODELING APPROACHES
progress variable describes a normalized temperature and is denoted as c.
The probability density function fc at a certain location x at a certain time
t reads
fc (ĉ) = α δ(ĉ) + β δ(1 − ĉ) + γ h(ĉ) ,
(116)
R
where h(0) = h(1) = 0, the integral of h is unity ( h(ĉ)dĉ = 1) and the sum
of the three probabilities is one, i.e. α + β + γ = 1. With the assumptions
of large Reynolds and Damköhler numbers (Re 1 and Da 1), the
probability γ of the intermediate state tends towards zero, since it scales
with Da−1 . Therefore, the intermediate states can be neglected under these
assumption, which was also done in the original version of the BML model,
where the gaseous fluid is either fully burnt
R or unburnt. Thus the PDF
fc attains a bimodal shape. Since hci = ĉfc (ĉ)dĉ = β, α and β can be
expressed as functions of the averaged progress variable. Therefore the PDF
fc becomes
fc (ĉ) = (1 − hci)δ(ĉ) + hci δ(1 − ĉ) ,
(117)
where this (bimodal) PDF is fully determined by the Reynolds-averaged
quantity hci.
A transport equation for the Favre-averaged progress variable c̃ can be
derived and reads
00 00
∂ hρi Ueie
c
∂ hρi ug
∂ hρi e
c
ic
+
= −
+ hρSc i
∂t
∂xi
∂xi
,
(118)
where Uei and u00i are the Favre-averaged and the fluctuating velocities. The
mean density is denoted as hρi, the fluctuating progress variable as c00 and the
mean source term of the progress variable as hρSc i. Note that the averaged
thermal diffusion term is neglected in this equation. The right hand side
terms in equation (118) describe turbulent convection and mean progress;
both require modeling.
The turbulent convection term was originally modeled by a gradientdiffusion assumption, i.e. as
∂e
c
00 00
hρi ug
= − hρi Dm
ic
∂xi
,
(119)
where Dm is the model diffusion constant. For counter-gradient diffusion,
however (see e.g. [45, 97, 104, 4, 102]), this simple approach fails. Progress
to capture counter-gradient diffusion effects has been made by taking conditional velocity statistics into account.
Research related to the closure of the mean reaction source term (the
second term on the rhs of equation (118)) started approximately 40 years ago,
7.1 BML Approach
75
and since then various models have been proposed [8, 11, 45, 70]. Two models
for mean source term are presented in the following subsections, first the
flame crossing frequency approach and then a flame surface density closure.
7.1.1
Flame Crossing Frequencies
Based on the assumption that the mean reaction rate at a certain location
depends more on the passage frequency of instantaneous flame fronts than
on the local mean temperature, a model was proposed by Bray, Libby and
Moss [10, 9]. The mean reaction rate hρSc i is modeled with a separation
approach, where hρSc i is written as the product of flame crossing frequency
qc and the reaction rate per flame crossing ωc
hρSc i = qc ωc
.
(120)
Turbulence controls the mean reaction rate (governed by qc ), whereas the
chemistry enters through the reaction rate per flame crossing ωc . The flame
crossing frequency is estimated as
qc = 2
hci (1 − hci)
tI
.
(121)
The intergal time scale tI is usually determined as k/, where these two
quantities (the turbulent kinetic energy k and the dissipation rate of k) are
known from e.g. a k − model. The model for the source term per flame
crossing reads
tt
,
(122)
ωc = ρu sL
lF
where tt is the transit time to cross one flame front. Finally, the mean
reaction source term (equation 5.64 in [70]) becomes
hρSc i = 2 ρu sL
7.1.2
tt hci (1 − hci) .
lF k
(123)
Flame Surface Density Approach
This approach is valid under the flamelet assumption and expresses the mean
source term as a function of the flame surface density Σ, which is the flame
surface area per unit volume [74, 93, 8, 52]. Similar as in the previous approach, complex chemistry and turbulence-chemistry interactions are treated
in a distinct fashion. Thus, the mean source term hρSc i is modeled as
hρSc i = Σ ρu hsL,s i
,
(124)
76
7 EXISTING MODELING APPROACHES
where ρu is again the density of the unburnt fresh gas and hsL,s i is the consumption rate of a flame surface element. The calculation of hsL,s i includes
complex chemistry mechanisms as well as the influence of flame stretching,
and it is stored in look-up tables. On the other hand, the model for the
flame surface density has to capture the effects of turbulence-flame interactions. There are basically two ways to determine Σ: algebraic expressions
and evolution equations. Various modeled transport equations have been
derived [93, 94, 96, 74], but this topic is beyond the scope of this work.
An algebraic expression for the flame surface density was proposed by [8],
where Σ is modeled as
Σ =
g
hci (1 − hci) .
σ y Ly
(125)
The flamelet orientation σy describes the angle between instantaneous and
averaged flame front, g is a model constant and Ly is the wrinkling length
scale, which has to be modeled. Note that this model (eq. (125)) shows
similarities to the model (eq. (123)) presented earlier and the eddy break-up
model [87].
7.2
G-Equation Model
Another approach are flamelet models [63] based on the level-set formulation
[38, 64]. In general, the level-set formulation is used to track interfaces. An
isosurface (G(x, t) = G0 ) of a non-reacting scalar G describes the position of
the flame front, for which a transport equation, the so called G-equation, has
to be solved. At a certain location, a G-value smaller than G0 defines cold
unburnt gas, whereas a value larger than G0 represents hot burnt gas. But
beside the value G0 , the G-field has no meaning. The transport equation for
G reads
∂G
∂G
+ ρUi
= ρu sd |∇G| ,
(126)
ρ
∂t
∂xi
where sd is the displacement speed. Note that the displacement speed depends on the laminar flame speed sL , strain rate and the curvature of the isosurface. Since G is a non-reactive scalar, different than the progress variable,
this approach has the advantage that no source term has to be modeled. In
addition, complications due to counter-gradient diffusion are avoided, which
makes it very attractive to study instantaneous flame dynamics.
But equation (126) can only be solved directly, if the wrinkling of the
isosurface is captured in full detail, which is done in a DNS. In general,
however, the averaged G equation (in the RANS context, whereas filtered in
7.3 PDF Approach
77
the LES context) is considered, which reads
hρi
00 00
e
e
∂G
∂G
∂ hρi ug
iG
+ hρi Uei
= ρu hsd |∇G|i −
∂t
∂xi
∂xi
.
(127)
In spite of the mentioned advantages above, the closure of equation (127)
is not straightforeward. Overviews on turbulent flame speed models for the
different combustion regimes and corresponding discussions are given in [23,
48].
7.3
PDF Approach
Two kinds of probability density functions are covered here: first the presumed PDF approach for a single variable reaction and second a joint composition PDF method.
In the progress variable consideration, for example in the BML context,
the chemistry is represented by one single variable reaction, and the mean
reaction rate is unclosed and has to be modeled. Since the mean reaction rate
hρSc i usually is a highly non-linear function of c, the assumption Sec = Sc (e
c)
is poor. Knowing the probability density function fc of the progress variable
c, the mean source term can be determined exactly as
Z
hρSc i =
1
ρ(ĉ)Sc (ĉ) fc (ĉ) dĉ ,
(128)
0
where Sc (ĉ) is the reaction rate for the specific value ĉ. Such a PDF contains
all statistical one-point one-time information about the progress variable.
Under the assumption that the shape of the PDF is known as a function of
a finite number of parameters, one can solve transport equations for those
parameters and obtains the full (presumed) PDF [50, 7]. The simplest example of a presumed PDF is the bimodal PDF used in the BML model;
the only parameter is the mean progress variable. In more sophisticated
approaches, transport equations for mean and variance, or for even higher
moments, are solved. The main advantage of this method is its numerical
efficiency compared to transported PDF methods.
The other type of PDF methods are transported PDF methods, which
have already been presented in chapters 2 and 3. They provide a very general statistical description of turbulent reactive flows, considering one-point
one-time joint statistics of velocity and composition. A transport equation
describes the evolution of the PDF, where turbulent convection appears
in closed form and the mean source term can be treated exactly. On the
78
7 EXISTING MODELING APPROACHES
other hand, the closure of molecular mixing remains a challenging task. Often transported PDF methods are employed in conjunction with detailed
chemistry together with an efficient implementation like in-situ-adaptivetabulation (ISAT) [75] or with a reduced reaction mechanism. Such chemical
reduction mechanisms are presented next.
7.4
Reduced Chemical Mechanisms
Intrinsic Low-Dimensional Manifolds (ILDM)
Intrinsic low-dimensional manifolds (ILDM) [49] is a general procedure to
simplify chemical kinetics. Based on an eigenvalue analysis of the local Jacobian of the chemical source terms, the processes associated with large eigenvalues are assumed to be in steady state, whereas the slow processes span a
manifold in composition space, which can be stored as a look-up table. This
reduction strategy separates fast from slow processes; the only required input
is the desired order of reduction, i.e. the number of control variables. For homogeneous cases, this method works well with a small number of parameters
(1 or 2), but for low temperature regions (as the preheat zone of premixed
flames), a large number of parameters is needed, since the fast processes are
assumed to be in steady state. For steady flames, this is not crucial, but for
transitional phenomena it may become crucial.
Flame Prolongation of Intrinsic Low-Dimensional Manifolds (FPI)
The flame prolongation of ILDM (FPI) [21] approach has been developed
to overcome the drawback of ILDM. This approach is physically motivated
and is based on a less mathematical background as ILDM. The basic idea
is to employ one-dimensional laminar premixed flames for the look-up table. For these calculations complex chemistry can be considered. Then, the
source terms and species concentrations are stored as a function of a certain number of control variables, e.g. mixture fraction and progress variable.
This methodology recovers the results of ILDM for high temperatures and
improves the predictions in the preheat zone.
Flamelet-Generated Manifolds (FGM)
The flamelet-generated manifold (FGM) approach [95] is similar as FPI and
has been derived independently. The manifold is considered as an ensemble of laminar premixed flamelets, calculating and tabulating for each gas
composition one laminar premixed one-dimensional flame as a function of
one progress variable and the enthalpy (to account for heat losses). Thus,
7.4 Reduced Chemical Mechanisms
79
transport equations for the controlling variables have to be solved. This
model overcomes the lack of accuracy in the prediction of colder (preheat)
zones, if local chemical equilibrium is assumed (e.g. in ILDM), such that less
controlling variables are needed, for which a transport equation has to be
solved.
Multidimensional Flamelet-Generated Manifold (MFM)
The multidimensional flamelet-generated manifold (MFM) method [60] is an
extension of FGM. There the manifold is not considered as a sum of single
independent flamelets, but the fluxes through isomixture fraction surfaces
are taken into account. The chemical state can be tabulated as a function of
controlling variables (usually mixture fraction and a reactive progress variable) and of scalar dissipation rates of the control variables, for which an
accurate formulation has to be specified.
Reactive-Diffusive Manifolds (REDIM)
To overcome the disadvantage of the intrinsic low-dimensional manifolds approach, the ILDM methodology was extended such that not only the chemical source terms are considered (as in ILDM), but also convective diffusive
processes. This results in reactive-diffusive manifolds (REDIM) [13]. In
equation (129), the evolution of the composition vector Φ (including species
mass fractions and enthalpy) is described in terms of reaction and diffusion
processes:
1 ∂
∂φi
∂φi
∂φi
= Si (Φ) − Uj
−
ρDjl
.
(129)
∂t
∂xj
ρ ∂xj
∂xl
The chemical source term of composition i is denoted as Si (Φ) and D is the
transport coefficient tensor. In the ILDM approach, equation (129) contains
only the time derivative and chemical source term Si (Φ), whereas REDIM
also includes diffusion and convection. This approach overcomes the most
drawbacks of ILDM in a more mathematical manner as for example MFM,
which relies on physical consideration. Nevertheless, the manifolds obtained
from REDIM and MFM are very similar.
8
Novel Joint Probability Density Function
Model for Turbulent Premixed Combustion
In this chapter, a novel model for turbulent premixed combustion in the
corrugated flamelet regime is presented, which is based on a transported
joint probability density function (PDF) of velocity, turbulence frequency
and scalars. Due to the high dimensionality of the corresponding sample
space, the PDF equation is solved with a Monte-Carlo method, where individual fluid elements are represented by computational particles. Unlike in
most other PDF methods, the source term not only describes reaction rates,
but accounts for ”ignition” of reactive unburnt fluid elements due to propagating embedded quasi laminar flames within a turbulent flame brush. By
that reason, the following properties are introduced: a flag indicating whether
a particle represents the unburnt mixture (not yet reached by the flame) and
a flame residence time that allows to resolve the embedded quasi laminar
flame structure together with precomputed one-dimensional laminar flame
tables. For the probability of an individual particle to ”ignite” during a time
step, an empirical model is proposed. The corresponding model constants
are obtained from one-dimensional simulations of a weak swirl burner. This
model predicts the dependency of turbulent flame speed on the turbulent intensity very accurately. In addition to the turbulent flame brush, molecular
mixing of the products with a co-flow has to be modeled. Therefore a modified interaction by exchange with the mean (IEM) mixing model is employed,
but it has to be emphasized that this is not critical for the propagation of the
turbulent premixed flame. For example, no mixing model is required, if the
co-flow consists of the hot products. To validate the proposed PDF model,
simulation results of a piloted methane-air Bunsen flame are compared with
experimental data and very good agreement is observed.
The motivation for this approach is given in section 8.1 and a general outline of the joint PDF method of velocity, turbulence frequency and scalars
is briefly presented in section 8.2. Section 8.3 deals with the combustion
82
8 NOVEL JPDF COMBUSTION MODEL
model, followed by a closure for the ignition probability in section 8.4. The
construction of the manifold is presented in section 8.5 and the modification
of the IEM mixing model in section 8.6. Numerical simulations of a piloted
Bunsen flame as well as one-dimensional simulations of a weak swirl burner
are presented in section 8.7, and finally conclusions are given in section 8.8
8.1
Motivation
As explained in the chapter about turbulent premixed flames, the accurate
description of turbulence, chemistry and their interaction is essential for reliable predictions. For non-premixed combustion, various models exist that
are based on the mixture fraction [62, 30, 6]. There it is assumed that the
inner reactive-diffusive flame structure is not disrupted by the turbulence.
Numerical simulations show the applicability and the entitlement of such
models [51, 41, 101, 66]. For premixed turbulent combustion, no generally
valid approach exists [73, 65].
One existing approach is the model by Bray, Moss and Libby [12], more
details are given in chapter 7.1. In the transport equation (118) for the
averaged progress variable, turbulent convection and mean source term are
unclosed. Although progress has been made in modeling the mean source
term [10, 9, 8, 70], it still is an issue in that context, as well as a general
closure for turbulent convection capable to properly account for countergradient diffusion [104]. In [77], the transport equation of the joint PDF of
velocity and a progress variable was solved by a Monte-Carlo method for
flamelet and distributed combustion. With the help of idealized premixed
turbulent flame simulations, they have compared a standard PDF closure
with combined reaction-diffusion formulation. Their results confirm the bimodal distribution of the progress variable for fast reactions made in the
original BML approach.
Another approach are flamelet models [63, 64] based on the level-set formulation [38], which are presented in chapter 7.2. They allow to study instantaneous flame dynamics, but it is not straightforward to achieve closure
in the RANS context.
A third approach is to employ probability density function (PDF) methods [72], which allow a very general statistical description of turbulent reactive flows. Whereas closure of molecular mixing remains a challenging task,
turbulent transport appears in closed form and the reaction source term can
be averaged without additional assumptions. PDF methods have been widely
used for diffusion flames [54, 101, 41], but simulations of premixed flames are
rare [3, 88, 46].
8.2 JPDF Method
83
Here, a novel model for turbulent premixed combustion [79, 24, 25] is presented for the corrugated flamelet regime. It is based on a joint velocityturbulence frequency-scalar PDF method described in chapter 4, and it combines both the ideas of the BML progress variable and the flamelet approach
within a scale separation consideration. The progress variable in this Lagrangian joint PDF framework becomes a computational particle property
c∗ , indicating the arrival of the embedded quasi laminar flame front at the
particle’s location. It can only attain two states, i.e. 0 for not yet arrived
and 1 for already passed by the flame. In addition, the flame residence time
τ ∗ is introduced, which allows to resolve the embedded flame structure in
the turbulent flame brush. It represents the time which elapsed since the
corresponding particle has been reached by the embedded flame sheet, i.e.
the time since the particle has been ”ignited”. Assuming a composition
evolving similarly as the one in a laminar flame, which is valid in the corrugated flamelet regime, the residence time allows to retrieve the composition
of a particle from precomputed one dimensional laminar premixed flame tables. The remaining challenge is to accurately describe when the flame front
reaches the particle, which is modelled with an ignition probability P , i.e.
the probability that a particle ”ignites” during a time step of size ∆t. Here,
a simple empirical closure for P is presented to show the applicability of
the underlying modeling concept; a more general description is the topic of
future work.
The transport equation for the Favre-averaged progress variable c̃, derived from the PDF equation, is consistent with the one employed by the
BML model. The advantage of using a joint PDF method, however, is that
turbulent convection appears in closed form. As a consequence, countergradient diffusion effects are automatically recovered. Moreover, as already
described, the Lagrangian modelling framework allows to resolve the embedded flame structure. The new framework allows for a natural description
and modeling of turbulent premixed combustion. Numerical simulations of
a turbulent premixed piloted Bunsen flame (the Aachen flame F3 [16]) show
good agreement with experimental data.
8.2
JPDF Method
In this section, a brief outline of the PDF modeling framework used here is
presented; more details on PDF methods for turbulent reactive flows are given
e.g. in [72] or in chapter 4. The one-point one-time Eulerian mass-weighted
joint PDF ge of (Favre) fluctuating velocity u00 = (u001 , u002 , u003 )T , turbulence
frequency ω and the scalar vector Φ = (Φ1 , . . . , ΦNs ) (Ns is the number of
scalars) is considered. The corresponding sample space variables are v00 =
84
8 NOVEL JPDF COMBUSTION MODEL
(v100 , v200 , v300 )T for the fluctuating velocities, θ for the turbulence frequency and
Ψ = (Ψ1 , . . . , ΨNs ) for the scalars. Note that the composition including
species mass fractions, temperature and enthalpy are described as a function
of the scalar vector Φ. The mass density function (MDF) G is defined as
G(v00 , θ, Ψ, x, t) = hρi (x, t) g̃(v00 , θ, Ψ; x, t),
(130)
where the mean density hρi at location x and time t is only a function of
the scalar PDF. Here, the first scalar represents the (inert) mixture fraction
Z, i.e. Φ1 = Z, the second scalar the progress variable c and the third one
the flame residence time τ . From the Navier-Stokes and scalar conservation
equations, the transport equation for G can be derived exactly; details are
found in [72]. The transport equation for the MDF reads
00 00
fj + v 00 )
∂G (U
∂ Uei ∂Gvj00
1 ∂ hρi ug
∂G
j
i uj ∂G
+
−
+
∂t
∂xj
∂xj ∂vi00
hρi
∂xj
∂v 00
i
∂
1 ∂p
1 ∂ hpi
1 ∂τij
1 ∂ hτij i 00
v , θ, Ψ
=
G
−
−
+
∂vi
ρ ∂xi
hρi ∂xi
ρ ∂xj
hρi ∂xj *
+!
∂
Dω 00
∂
1 ∂Jiβ 00
−
v , θ, Ψ
+
G
v , θ, Ψ
G
∂θ
Dt ∂Ψβ
ρ ∂xi −
∂GSβ
+ S c (G) .
∂Ψβ
(131)
Favre-averaged quantities are denoted as e·, Reynolds-averaged quantities as
h·i and volume weighted conditional expectations as h·|·i. The variable p
means pressure, ρ density, τij is the viscous stress tensor, Jβ is the molecular
diffusion flux of scalar β and Sβ is the source term of scalar β. Moreover, the
source S c describes discontinuous evolutions; here in particular transitions
(jumps) of Φ2 = c from zero to one. In the case of a continuously evolving
progress variable, the source S c would vanish. Note that the left-hand side of
e is provided); the conditional
eq. (131) is closed (below it is explained how U
expectations on the right-hand side (rhs) on the other hand require modeling.
Here, the simplified Langevin model (SLM) [28] is used to close the first rhsterm and another stochastic model is employed for the turbulence frequency
[85] in the second rhs-term. It will become clear later that the molecular
diffusion fluxes in the third rhs-term are non-zero only for the scalars Φ1 and
Φ3 , for which a modified IEM mixing model [99] is devised; the corresponding
model is presented in section 8.6. The fourth rhs-term is non-zero only for
scalar Φ3 = τ , i.e. Sβ = δ3β Ψ2 , where δαβ is the Kronecker delta. Therefore τ
(which by definition is zero for c = 0) represents the time, which elapsed since
8.3 Combustion Model
85
c switched from zero to one. Note that equation (131) is not complete, since
e cannot be extracted from the MDF G. Therefore, simultaneously with the
U
modeled version of eq. (131), the Reynolds averaged Navier-Stokes (RANS)
equations are solved to provide the mean velocity. Vice versa, the unclosed
terms in the RANS equations are obtained from G. For the RANS equations,
a finite-volume solver is employed and due to the high dimensionality of the
sample space, a Monte-Carlo method is used to solve eq. (131). In the MonteCarlo method, Lagrangian particles consistently evolve in the v00 -θ-Ψ-space
according to stochastic differential equations (SDE), such that the MDF
is represented by the particle ensemble density. Such internally consistent
hybrid particle/finite-volume PDF solution algorithms proved to be much
more efficient than stand alone particle methods; more details are provided
in [37, 57] or in chapter 4.
8.3
Combustion Model
The novel model for turbulent premixed combustion is based on a scale separation approach [79, 24, 25]. The idea is based on the model by Bray,
Moss and Libby, which is translated into the Lagrangian JPDF context and
combined with the flamelet approach. The progress variable c∗ indicating
whether the flame front has already reached the particle or not, is a computational particle property like e.g. the particle position X∗ , . It has two
possible states, i.e. 0 for an unburnt particle and 1 for a burnt or burning
one. The mass-weighted PDF h̃ of the progress variable at a given location
in space and time can be written as
h̃(ĉ) = (1 − c̃) δ(ĉ) + c̃ δ(1 − ĉ) .
(132)
Note that the bimodal PDF h̃ is fully determined knowing the Favre-averaged
progress variable. An example of such a bimodal PDF is sketched in Figure
17. Within our scale separation approach, the progress variable c∗ is responsible to determine the location of the turbulent flame front. Different than
in the BML model, however, here a flame residence time τ ∗ is introduced
for each particle, which allows to resolve the embedded quasi 1D flame profile. It represents the elapsed time since a computational particle has been
”ignited” (i.e. the time since it has been reached by the embedded flame
when c∗ switched from 0 to 1). The assumption, that the embedded flame
structure can be approximated by a quasi 1D laminar flame profile, allows
to determine the particle’s composition and temperature based on the flame
residence time τ ∗ and corresponding look-ups from precomputed steady 1D
laminar flames. These laminar 1D flame profiles depend on the equivalence
86
8 NOVEL JPDF COMBUSTION MODEL
h(c)
0
c
1
Figure 17: Sketch of a possible bimodal mass weighted PDF h̃ of the progress
variable c.
ratio of the gas mixture and for their computation, complex reaction mechanisms can be considered. Note that this idea is similar to the one used in
flamelet modeling of premixed combustion. Figure 18 shows a sketch of the
T
c
Tb
1
Tu
0
x
0
τ
Figure 18: Sketch of the laminar 1D flame profile showing T ∗ , c∗ and τ ∗ .
temperature profile in a laminar premixed 1D flame. The dashed line indicates the value of c∗ , triggering the beginning of the residence time τ ∗ and
anchoring the 1D flame. Since a unique transformation from the τ ∗ - to the
8.4 Ignition Probability
87
x-space exists, h∗ and other scalars can be retrieved.
When, at the beginning of the time step, a particle with c∗ = 0 ignites
(with the probability P ), its c∗ -value switches to 1. From then on, the residence time τ ∗ , which was zero so far, gets incremented as the simulation time
proceeds. Based on the approximation that the fluid particle evolves along
a quasi 1D flame profile, its residence time allows to determine the particle’s
relative location and thus (from precomputed 1D flames) its composition.
The decision, whether an unignited particle ignites during a time step or
not, is modeled by an ignition probability P described next.
8.4
Ignition Probability
The decision of an unburnt particle (c∗ = 0) to ignite is captured by a model
for P , which is discussed in this section for the corrugated flamelet regime.
In general, the ignition probability is defined as
P = 1 − exp (−F ∆t)
,
(133)
where F is the ignition probability per unit time and ∆t the size of the time
step. This formulation ensures that P fulfills the conditions for a probability
(P ∈ [0, 1]), even for large time steps. Note that F is determined as
P
∆t→0 ∆t
F = lim
.
(134)
From the MDF equation one can derive a transport equation for the Favreaveraged progress variable, which reads
hρi
00 00
∂c̃
∂hρiug
∂c̃
ic
+ hρiŨi
= −
+ hρi (1 − c̃)F
∂t
∂xi
∂xi
.
(135)
Besides the last term on the right hand side of equation (135), this equation
(135) is consistent with the transport equation for e
c of the model by Bray,
Moss and Libby (BML), presented in equation (118) in chapter 7.1. However,
different than in the classical BML model, in the PDF context c is not interpreted as a normalized temperature (see explanations above), but as trigger
variable for the arrival of the flame at the particle location. In addition,
the turbulent convection term (first right-hand side term) appears in closed
form, such that issues due to counter-gradient diffusion are circumvented.
Moreover, if one sets F equal to
F =
Sec
(1 − c̃)
,
(136)
88
8 NOVEL JPDF COMBUSTION MODEL
one obtains a formulation for the transport equation (135) that is equal to
the transport equation in the BML context (equation (118)).
For the closure of the ignition probability, F has to be modeled. There
are several ways to close F . A first one is suggested by equation (136), where
a direct relation between mean source term and ignition density is given.
With the help of equation (136) it is possible to link the ignition probability
to existing combustion models within the BML context, which are presented
in chapter 7 or in [8, 11, 45, 70], e.g. the flame crossing frequency model.
This leads to an ansatz for F which reads
F = α hci
,
(137)
where α is a variable depending on various parameters. It is consistent with
e . In [24, 25], models
the flame crossing frequency model, if α = 2ρu sL ltFt ω
for α were proposed that depend on the mean turbulence frequency ω
e . One
simple formulation reads
(
2 )
ω
e − ωmin
α = H(e
ω − ωmin ) αmax min 1 ,
,
(138)
ωc − ωmin
where H(·) is the Heavyside function, ωmin was set to 200s−1 , ωc to 860s−1
and αmax to 1000s−1 . This dependency of α(ω), which is illustrated in figure 19, represents the enhancing influence of the turbulence on the turbulent
flame propagation in the corrugated flamelet regime, where α is limited by
αmax . This closure already showed very promising results.
Alternatively, here a more general approach for F is proposed. The ansatz
reads
F = CF ω
e m hcin ,
(139)
whereas CF , m and n are model constants to be determined. These parameters are determined from simplified quasi one-dimensional simulations of a
weak swirl burner [68] and are then employed for the simulation of a piloted
Bunsen flame [16]. Both simulation results are presented in the result section. It has to be mentioned that the ignition probability model may be
further optimized and refined to cover a wider range of regimes. Here, the
prime goal is to demonstrate the potential of the described Lagrangian joint
PDF framework combined with a reasonable ignition probability model for
the corrugated flamelet regime.
8.5
Tabulation
To generalize the combustion model for scenarios of varying mixture fractions of the unburnt reactive mixture, multiple laminar 1D flames have to be
8.5 Tabulation
89
α(ω)
αmax
0
0
ωmin
ωc
ω
Figure 19: Sketch of α(ω) used for the simulations in [25].
precomputed and tabulated; i.e. for an adequate number of mixture fraction
values in the flammable range. Between these selected Z values linear interpolation is applied. Outside the flammable range, i.e. where the mixture
fraction is smaller than an appropriately specified Zf , diffusion dominates
and therefore the species mass fractions and enthalpy are linearly interpolated between the values corresponding to Zf and the ones in the co-flow
stream where Z = 0. Other extrapolation schemes, mainly for rich conditions, are discussed in [39], but for lean conditions the extrapolation is not
crucial. A sketch of the resulting temperature manifold, i.e. of Tm (Z, τ ), is
depicted in figure 20. Temperature, and similarly also species mass fractions
and enthalpy, can be tabulated as functions of mixture fraction and flame
residence time. Thus they can be retrieved during PDF simulations by simple and cheap look-up operations. For each pair (Z ∗ ,τ ∗ ) the composition ϕ∗
is uniquely defined by
ϕ∗ = ϕm (Z ∗ , τ ∗ ) ,
(140)
where ϕm represents the manifold for the composition ϕ∗ as a function of Z ∗
and τ ∗ . Here, the laminar one-dimensional flames were computed using the
Cantera package [22].
These precomputed tables have some similarities with the ones used in the
flamelet generated manifolds (FGM) method [95] or in the flame prolongation
of the ILDM method (FPI) [21]. The manifolds are also constructed as
ensembles of laminar one-dimensional flames; different ”control variables”
are employed, however. As an example, in FGM transport equations for
the reactive control variables have to be solved, whereas here the ignition
90
8 NOVEL JPDF COMBUSTION MODEL
T
Z
1.0
1.0
Zf
0.0
τ max
τ
Figure 20: Sketch of the normalized temperature T̂ as a function of the
mixture fraction Z and of the flame residence time τ .
probability has to be determined.
In the extension of FGM, i.e. in MFM [60], the calculation of the manifold
contains the fluxes through iso-mixture fraction surfaces. These fluxes are
missing in the presented model, but this limitation seems not to be crucial for
the calculated flames, since the mixture is close to stoichiometric conditions.
In addition, it was shown by [61] that laminar triple flames and premixed
laminar flames behave similarly in the vicinity of stoichiometric conditions.
8.6
Mixing Model
To close the third rhs-term in equation (131), affecting the Z ∗ and τ ∗ values
of the computational particles, a variety of micro-mixing models have been
proposed in the past [99, 33, 91, 55]. In the context of the proposed PDF
method, the interaction by exchange with the mean (IEM) model is employed
to account for molecular mixing of mixture fraction and enthalpy
dφ∗β
1 ∂Jiβ
1
∗
˜
=
≈ − Cφ Ω φβ − φβ
dt
ρ ∂xi
2
.
(141)
The standard value of 2.0 is used for Cφ and Ω is the conditional turbulence frequency [37]. Equation (141) is applied for the particle temperature
and mixture fraction. Then, under the assumption that the temperature
must lie on the manifold Tm (Z, τ ) (equation (140) with ϕm = Tm ), and since
8.7 Results
91
it is a monotonous function of both Z and τ , the new pair (Z ∗ , τ ∗ ) is determined from the new values of Z ∗ and T ∗ applying the unique mapping
(Z ∗ , T ∗ ) → (Z ∗ , τ ∗ ). While it is clear that the ”effective” flame residence time
of a particle changes due to molecular mixing, it is questionable whether a
mixing model designed for inert scalars like mixture fractions can directly be
employed to describe the effect on τ ∗ .
During each time step the position X∗ of a computational particle evolves
e ∗ ) + u∗ , whereas the fluctuating veaccording to its individual velocity U(X
locity u∗ is updated according to the simplified Langevin model (SLM) [28]
and the turbulence frequency ω ∗ by solving another stochastic model equation [85]. Then, if the value of c∗ is zero, it is set to one with probability P
and if c∗ = 1, the flame residence time is incremented by the time step size.
Now, mass fractions and temperature are retrieved from the precomputed
tables and a modified IEM model is employed to account for micro-mixing
of the products with a potential co-flow stream.
8.7
Results
To validate the presented model for turbulent premixed combustion, a low
swirl burner [68] and a piloted Bunsen flame [16] are considered. The low
swirl burner simulations are used to obtain the model constants in the ansatz
for F (equation (139)), whereas these model constants are used for the simulation of the piloted Bunsen flame.
8.7.1
Quasi 1D Swirl Burner Simulation
For the swirl burner simulations, the simplified one-dimensional configuration
of the standalone PDF method was used, presented in detail in chapter 5,
was used. This simplified setup (along the symmetry axis of the experiment)
allows for computationally very efficient simulations with little uncertainty
regarding the hydrodynamic solution.
The low swirl burner was experimentally investigated and is well documented [68]. For the same burner configuration, measurements with six
different turbulence levels and flow rates (indicated with ’FR’) corresponding to flames 1 to 6 in [68] are provided. Flame 1 corresponds to the lowest
and flame 6 to the highest turbulence intensity. The flames range from the
corrugated flamelet to the thin reaction zone regimes, i.e. the Karlovitz
number Ka ranges from 0.96 (flame 1) to 12.33 (flame 6), and all have an
equivalence ratio φ of 0.7. From the mean experimental temperature and
velocity profiles, the required mean momentum profiles are obtained. The
92
8 NOVEL JPDF COMBUSTION MODEL
measured root-mean-square velocity components ui,rms have been averaged
over the flame region and are set constant over the whole computational
domain (confirmed by the measurements). Note that the anisotropy of the
turbulence is captured by the quasi one-dimensional
setup. At the inflow,
√
the turbulence frequency is determined as ω = 2 uI /lI , where lI = 0.015
is the measured integral length [5] and uI = (2k/3)0.5 the integral velocity
scale.
e decays in downstream direction (x-direction), the turbulence
While hρiU
is approximately homogeneous. As the flame propagates into the fresh gas, at
some point, where the turbulent flame speed is matched by the mean velocity
e , it stabilizes. Here, the turbulent flame speed is defined as the mean
U
e at the position, where the mean temperature equals the average
velocity U
of fully burnt and unburnt temperatures. All particles have an equivalence
ratio of 0.7 and the surrounding air is neglected in these simulations, i.e. no
mixing model is employed for these one-dimensional simulations.
The computational domain has a length of 0.08m and the grid contains
200 equidistant cells; the average particle number per cell is 20. Simulations
with a higher grid resolution (400 cells) and more particles per cell (40) yield
the same results.
Numerical simulations have been performed for all six flames with different
values for the model constants CF , m and n (ignition probability; equation
(139)). The best results were obtained for
CF = 1.0 , m = 1.0 , n = 1.5 .
(142)
In Figure 21, comparisons of the normalized mean temperature profiles are
depicted for all six flow rates. The solid lines show the numerical simulation
results and the dashed lines represent the experimental measurements. For all
six flames, very good agreement for the mean temperature could be achieved,
except for a flow rate of 5m/s, where the temperature is underpredicted.
Figure 22 shows the turbulent flame speed sT as a function of the turbulent
intensity. The solid line represents the theoretical predictions of [64], the
dashed line with the crosses the experimental measurements [68] and the
dashed-dotted line with the circles the PDF results. The numerical results
are in good agreement with the experiments and the theoretical model. It has
to be mentioned once more that these model constants were chosen to obtain
the result presented in the figures 21 and 22. The goal of these simplified onedimensional simulations was to find the set of model constants that deliver
the best match with the experimental measurements. Next, this set of model
constants is employed for simulations of a piloted jet flame.
8.7 Results
1.0
93
FR=5m/s
1.0
0.5
0.5
0.0
0.00
1.0
0.02
0.04
0.06
FR=15m/s
0.0
0.00
1.0
0.5
0.02
0.04
0.06
0.02
0.04
0.06
0.04
0.06
FR=20m/s
0.5
0.0
0.00
1.0
FR=10m/s
0.02
0.04
0.06
FR=25m/s
1.0
0.5
0.0
0.00
0.0
0.00
FR=30m/s
0.5
0.02
0.04
0.06
x
0.0
0.00
0.02
x
Figure 21: Axial profiles of the normalized mean temperature for all six flow
rates FR: experiments (dashed line) and simulations (solid line).
8.7.2
Piloted Bunsen Burner
The well documented Aachen flame [16] is a piloted Bunsen flame with three
inflow streams: in the center an unburnt premixed gas jet, around it an
already burnt premixed gas pilot and outside the surrounding air co-flow.
A schematic illustration of the axisymmetric flame is shown in Figure 23.
The jet bulk velocity U0 is 30m/s, the reference turbulent kinetic energy
is k0 = 3.82 m2 /s2 and the temperatures of burnt and unburnt gas are
Tb = 2248 K and Tu = 298 K, respectively. The mean and root mean square
e1 and urms are adopted from the experimental values from
(rms) velocities U
1
[16]. The laminar pilot stream has a uniform velocity of 1.3m/s and the
surrounding co-flow is assumed to be laminar too with a uniform velocity of
0.25 m/s. For the simulation, a computational domain of 0.6 m length and
0.1 m width was considered to represent the axisymmetric configuration, and
a 50 × 50 grid with 20 particles per cell (in average) was used. At the inflow
94
8 NOVEL JPDF COMBUSTION MODEL
20
18
16
14
sT/sL
12
10
8
6
4
2
0
0
2
4
6
8
10
ν‘/sL
12
14
16
18
20
Figure 22: Validation of the turbulent flame speed sT as a function of the
turbulence intensity ν 0 = (2k/3)0.5 , both scaled with the laminar flame speed
sL = 0.2: theoretical model [64] with lt /lF = 43 (solid line), experiments
(dashed line with crosses) and numerical simulations (dashed-dotted line with
circles).
boundaries, mean and rms velocities are taken from the measured profiles,
rms rms
00 00
ug
and ω is determined under the assumption
1 u2 is set equal to 0.4u1 u2
of turbulence in equilibrium resulting in
f
f
g
rms ∂ U1
00 00 ∂ U1 .
(143)
ω k = −u1 u2
≈ −0.4urms
1 u2
∂x2 ∂x2
In the jet, the normalized temperature is set to T̂ = 0 and the mixture
fraction to Z = 1, in the co-flow T̂ = 0 and Z = 0, and in the pilot T̂ = 0.8
and Z = 0.8.
The combustion model employed for the low swirl burner has not been
developed for high Karlovitz-numbers (e.g. the broken reaction zone regime),
8.7 Results
95
J
C
P
J
12 mm
P
C
68 mm
Figure 23: Sketch of the Aachen flame with the unignited jet (J), the hot
pilot (P ) and the co-flow (C).
but in the Aachen flame configuration, close to the nozzle at the interface
between jet and pilot, high flame stretch rates occur. Therefore, to perform
simulations of this flame, the model for F has been modified to
F = CF hci1.5 min ωmax , ω
e 1{Z ∗ >Zf lamable } ,
(144)
|{z}
=1.0
where ωmax is 900s−1 and Zf lamable is the border of the flamable mixture
fraction range. The last extension is necessary to account for the different
gas mixtures, i.e. jet, pilot and co-flow. The one-dimensional flame profile
is anchored at a normalized temperature of 0.05, and for molecular mixing
with the co-flow, the standard value of 2.0 was used for Cφ .
Radial profiles of the normalized mean axial velocity Û = Ũ /U0 , the
normalized turbulent kinetic energy k̂ = e
k/k0 , the normalized mean temperature T̂ = (T̃ − Tu )/(Tb − Tu ) and the normalized root-mean-square tem-
96
8 NOVEL JPDF COMBUSTION MODEL
perature T̂ rms = (T rms )/(Tb − Tu ) are presented at four axial locations, i.e.
at x/D = 2.5, 4.5, 6.5 and 8.5, where D = 12mm is the jet diameter. In
general, good agreement with the experiment was achieved, except for the
fluctuating temperature.
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
Figure 24: Radial profiles of the normalized mean downstream velocity Û
at several downstream locations in the piloted Bunsen flame: experiments
(circles) and numerical simulations (solid lines).
15.0
15.0
x/D = 2.5
15.0
x/D = 4.5
15.0
x/D = 6.5
x/D = 8.5
10.0
10.0
10.0
10.0
5.0
5.0
5.0
5.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
Figure 25: Radial profiles of the normalized turbulent kinetic energy k̂
at several downstream locations in the piloted Bunsen flame: experiments
(circles) and numerical simulations (solid lines).
In figures 24 and 25, the mean downstream velocity and the turbulent
kinetic energy are presented. The shear layer between jet and pilot is shifted
outwards due to gas expansion resulting from chemical reactions; this can
be observed in the mean velocity and turbulent kinetic energy profiles. The
velocity Û is in very good agreement with the experiment at all four locations.
An overprediction of k̂ can be observed at x/D = 4.5, whereas the three other
predictions are very accurate. The mean temperature is shown in figure 26,
where the prediction of T̂ is quite good except close to the inflow, where it
8.7 Results
97
x/D = 2.5
x/D = 4.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
Figure 26: Radial profiles of the normalized mean temperature at several
downstream locations of the piloted Bunsen flame: experiments (circles) and
numerical simulations (solid lines).
0.6
0.6
x/D = 2.5
0.6
x/D = 4.5
0.6
x/D = 6.5
x/D = 8.5
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
Figure 27: Radial profiles of the normalized rms temperature at several
downstream locations of the piloted Bunsen flame: experiments (circles) and
numerical simulations (solid lines).
is overpredicted. At all four locations the mean temperature gradients are
in good agreement with the experiment. The rms temperature is presented
in figure 27. The location of the peaks are predicted well, whereas the peak
values are overpredicted by approximately 50%.
In the previous figures, the applicability of the presented modeling approach for turbulent premixed flames in the corrugated flamelet regime is
demonstrated. Next, numerical convergence and model parameter sensitivity analysis are presented to demonstrate robustness of algorithm and model.
8.7.3
Sensitivity and Convergence Study
In this section, convergence of the PDF solution algorithm and sensitivity
regarding model parameters are investigated. Results of this study are presented in the figures 28 - 32. The first row of each figure shows the mean
98
8 NOVEL JPDF COMBUSTION MODEL
normalized downstream velocity Û , the second row the normalized mean turbulent kinetic energy k̂, the third row the normalized mean temperature T̂
and the fourth row the normalized rms temperature T̂ rms . The experimental measurements (circles) and the results presented in the previous section
(”reference solution”, solid lines) are compared with modified simulations,
where one numerical or model parameter is modified.
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
Figure 28: Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the piloted Bunsen
flame: experiments (circles), reference simulations (solid lines) and 80 × 80
grid with 20 particles per cell (dashed lines).
8.7 Results
99
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
Figure 29: Radial profiles of Û (first row), k̂ (second row), T̂ (third row) and
T̂ rms (last row) at several downstream locations of the piloted Bunsen flame:
experiments (circles), reference simulations (solid lines), CF = 0.8 (dashed
lines) and ωmax = 700.0s−1 (dotted lines).
First, in figure 28 grid convergence of the approach is demonstrated. The
reference solution (simulation with 50 × 50 grid) is compared with the results
from a simulation with a 80×80 grid; the same average number of particle per
cell (20) was employed. The mean velocities are in good agreement, whereas
the larger grid delivers a slightly higher peak value of k̂ than the reference
case. Besides a small difference close to the inflow near the interface between
100
8 NOVEL JPDF COMBUSTION MODEL
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
Figure 30: Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the piloted Bunsen
flame: experiments (circles), reference simulations (solid lines), zf lamable =
0.5 (dashed lines) and zf lamable = 0.9 (dotted lines).
pilot and co-flow, also the mean temperature compares well. Equally good
agreement can be observed for the rms temperature. In general, differences
between the results obtained with the 50 × 50 and 80 × 80 grids are very
small.
Next, the sensitivity of the ignition probability P on the model constants
CF and ωmax is summarized in figure 29. The prefactor CF is reduced by
8.7 Results
101
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
Figure 31: Radial profiles of Û (first row), k̂ (second row), T̂ (third row) and
T̂ rms (last row) at several downstream locations of the piloted Bunsen flame:
experiments (circles), reference simulations (solid lines) and the manifold
constructed out of 7 one-dimensional laminar profiles between a mixture
fraction of 0.7 and 1.0 (dashed lines).
20% in one simulation (dashed lines) and in another simulation, the maximum ω
e -value was reduced from ωmax = 900 s−1 to ωmax = 700 s−1 (dotted
lines). Both parameters are reduced by approximatly 20%, but the effect
on the simulation results is much smaller, i.e. the values of Û , T̂ and T̂ rms
are slightly lower, whereas k̂ is slightly increased in average. These studies
102
8 NOVEL JPDF COMBUSTION MODEL
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
Figure 32: Radial profiles of Û (first row), k̂ (second row), T̂ (third row)
and T̂ rms (last row) at several downstream locations of the piloted Bunsen
flame: experiments (circles), reference simulations (solid lines) and Cφ = 4.0
(dashed lines).
demonstrate the robustness of the proposed modeling approach with respect
to the main combustion model constants.
The robustness of the combustion model is further investigated by varying
the model parameter defining the flamable range, namely Zf lamable . In the
reference solution, a value of 0.7 was used; in figure 30 additional results with
Zf lamable = 0.5 (dashed lines) and Zf lamable = 0.9 (dotted lines) are presented.
8.7 Results
103
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
Figure 33: Radial profiles of Û (first row), k̂ (second row), T̂ (third
row) and T̂ rms (last row) at several downstream locations of the piloted
Bunsen flame: experiments (circles), reference simulations (solid lines) and
e1,co−f low = 1.0m/s (dashed lines).
U
In the jet and central pilot regions, hardly any influence can be observed, and
only small differences can be detected towards the co-flow region (outer pilot
region). Hence, it can be concluded that Zf lamable is not a sensible parameter
for the prediction of this type of flames.
In the reference solution, the manifold was constructed (motivated by the
Burke-Schumann solution) based on a single one-dimensional laminar flame
104
8 NOVEL JPDF COMBUSTION MODEL
together by linear interpolation between this profile and the composition of
the surrounding air. In figure 31, the reference solution is compared with
a result obtained from a simulation, where a manifold consisting of 7 onedimensional laminar flames was employed. The flame profiles were computed
for mixture fractions between Z = 0.7 and Z = 1.0 (the constant Z-interval
between two flames is ∆Z = 0.05). Between these flames, linear interpolation was employed to obtain the manifold. Mean velocity and turbulent
kinetic energy profiles are in almost perfect agreement, whereas the mean
temperature is by trend slightly higher in the modified case.
As already mentioned, the reference simulation overpredicts the rms temperature by up to approximately 50%. The reason therefore may be due
to the IEM mixing model with a mechanical-to-scalar-time scale ratio of
Cφ = 2.0, which is a standard value for inert scalars. To investigate this, a
simulation with a higher value of Cφ = 4.0 was performed. The results of
these simulations are compared in figure 32. As expected, a higher Cφ value
leads to a reduction of the rms temperature. But in general, the agreement
with the experimental data does not improve significantly.
Finally, the influence of the mean co-flow velocity is summarized in figure 33. The reason for this investigation is the uncertain specification of
this velocity. In the reference simulation, a value of 0.25 m/s was employed;
here, the result based on a simulation with an increased value of 1.0 m/s is
presented too. The influence of the co-flow velocity can only be observed in
the mean and rms temperature profiles; mean downstream velocity and the
turbulent kinetic energy profiles are in almost perfect agreement. The main
differences can be detected towards the co-flow, i.e. beyond the temperature
peaks.
8.8
Summary
A new model for turbulent premixed combustion in the PDF modeling context is proposed. The approach is based on a joint PDF method and employs a Lagrangian interpretation of the BML model. Compared to the BML
model it has the advantage of closed turbulent convection. Furthermore, the
Lagrangian framework allows to introduce a flame residence time resolving
the quasi laminar flame structure. An approach to model the probability
P of a computational particle to ”ignite” during a time step is presented.
Based on simplified simulations of a low swirl burner, the model constants
have been determined to accurately predict the dependency of the turbulent
flame speed on the turbulence intensity compared with measured data and
a theoretical model. This set of model constants was used to perform simulations of a piloted Bunsen burner. Good agreement between PDF results
8.8 Summary
105
and corresponding experimental data could be achieved, demonstrating the
applicability of the presented approach for premixed turbulent flame predictions.
Future work will mainly deal with a generalization of the presented model
for the ignition probability, such that the more challenging flames F2 and F1
(same flame configuration) as well as other hydrodynamically more complex
flames can be predicted. The model for P leads to good results for one particular flame in the corrugated flamelet regime. Since this empirical ansatz
was formulated and tuned for this regime, it is not straightforward to extend
this formulation to the thin reaction zone regime. In the next chapter, a
closure for P is provided based on a statistical fine scale picture of turbulent
reactive flows. There, P is calculated based on an estimate of the mean flame
surface density.
9
Extension of the Novel JPDF Combustion
Model
In this chapter, the model for turbulent premixed combustion presented in
the previous chapter is extended. The empirical closure for the ignition probability (eq.(139)) leads to good results for the considered piloted jet flame,
but the formulation for P has to be generalized. The ignition probability
describes the rate at which unburnt particles get ”ignited” by the embedded propagating quasi laminar flame, i.e. it reflects the coupled fine-scale
convection-diffusion-reaction dynamics in the flame. Let us consider again
the corrugated flamelet regime, where two assumptions are made: on one
hand the laminar flame speed remains constant and on the other hand the
embedded flame structures remain unperturbed by the turbulence. Then,
the probability for an individual particle to ”ignite” during a time step is
calculated based on an estimate of the mean flame surface density (FSD).
Latter gets transported by the PDF method, whereas a model has to be
employed to account for local production and dissipation of the FSD. The
following particle properties are introduced: a flag indicating whether a particle represents the unburnt mixture; a flame residence time which allows to
resolve the embedded quasi laminar flame structure; and a flag indicating
whether the flame residence time lies within a specified range. Latter is used
to transport the FSD, but to account for flame stretching, curvature effects,
collapse and cusp formation, a mixing model for the flame residence time
is employed. The same mixing model also accounts for molecular mixing of
the products with a co-flow. Numerical results of three piloted premixed jet
flames and comparisons with corresponding experimental data demonstrate
the applicability of this new approach.
In section 9.1, a brief outline of the combustion model is presented. Then the
closure for the ignition probability based on the mean flame surface density
is explained in sections 9.2 and 9.3, and the modified (IEM) mixing model,
capturing also production and dissipation of the flame surface, is presented
108
9 COMBUSTION MODEL EXTENSION
in section 9.4. Numerical validation studies are shown in section 9.5 and finally, conclusions are given in section 9.6. For the underlying PDF transport
equation it is referred to section 8.2.
9.1
Combustion Modeling Review
In this section, the general framework of the combustion model for turbulent
premixed combustion in the corrugated flamelet regime is presented. To
simplify the explanations, one considers computational particles in the PDF
solution algorithm, which can also be viewed as representative fluid elements.
Essential for the proposed modeling approach are the individual particle
properties Z ∗ , c∗ ∈ {0, 1} and τ ∗ ≥ 0 representing the mixture fraction,
a binary progress variable and a flame residence time. Moreover, in the
extended version of the combustion model, we introduce the function d(τ ∗ )
(from now on denoted as d∗ ), which is one for 0 ≤ τa < τ ∗ ≤ τa + τd and
c∗ = 1 and zero otherwise; τd is a specified small time constant and τa defines
the flame surface. The scalars c∗ and τ ∗ are crucial to model the turbulent
flame brush; the mixture fraction on the other hand quantifies the level of
molecular mixing. Next, the roles of these particle properties are further
detailed, where for the mixing and tabulation procedure it is referred to the
previous chapter.
The scalar c∗ is a flag indicating whether a particle represents the unburnt
reactive mixture; in that case c∗ = 0, else c∗ = 1. In the case of infinitely thin
embedded flames, c∗ can be interpreted as a normalized temperature, similar
as in the BML model [12]. Here, however, the embedded flame structure is
not infinitely thin and to account for that, the flame residence time τ ∗ is
useful. As already mentioned, it is non-zero only if c∗ = 1 and reflects the
time which elapsed since c∗ switched from zero to one, i.e. since the particle
was ”reached” by the embedded flame surface (marking the very front of the
embedded flame). Since one considers the corrugated flamelet regime, the
embedded flame structure and the laminar flame speed sL are assumed to
remain unaffected by the turbulent eddies and can be obtained from precomputed steady laminar 1D flames. Note that for these calculations complex
mechanisms can be considered. Now it is straightforward to consistently map
τ ∗ onto the spatial coordinate of that 1D flame and to retrieve species mass
fractions, enthalpy and temperature via cheap table look-up.
Hence, a time step of a computational particle is briefly summarized: during
each time step the position X∗ of a computational particle evolves according
e ∗ ) + u∗ , whereas the fluctuating velocity u∗ is
to its individual velocity U(X
updated according to the simplified Langevin model (SLM) [28] and the tur-
9.2 Ignition Probability
109
bulent frequency ω ∗ by solving another stochastic model equation [85]. Then,
if the value of c∗ is zero, it is set to one with probability P and if c∗ = 1,
the flame residence time is incremented by the time step size. Now species
mass fractions, enthalpy and temperature are retrieved from precomputed
tables (see section 8.5). A modified IEM model is employed to account for
micro-mixing of the composition as well as for the production and dissipation
of the flame surface; details are provided in section 9.4.
The crucial remaining question is: when does the c∗ -value of a particle
switch from zero to one? During a given time step, this occurs with the
probability P , for which the new formulation is explained in the next section.
9.2
Ignition Probability
In the corrugated flamelet regime, the ignition probability P is a function of
the mean flame surface density hΣi and the laminar flame speed sL (which
is assumed to be constant here). For infinitesimal small time steps one can
write P = F ∆t, but to always ensure that P ∈ [0, 1] the formulation
P = 1 − e−F ∆t
(145)
is employed. Note that F is the ignition probability density. To derive
Ωu
fresh gas
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Ωd
burnt gas
Figure 34: Sketch of an instantaneous flame surface with the volumes Ωu
(left) and Ωd (shaded).
an expression for P , an ergodic statistical fine scale picture of turbulent
110
9 COMBUSTION MODEL EXTENSION
premixed flames is considered. A sketch of an instantaneous snapshot is
depicted in figure 34, where the solid line represents the embedded flame
front at time t within a control volume Ω. The shaded area Ωd represents
the fluid volume ”consumed” by this flame front since the time t − τd . Note
that Ωd is approximately equal to AF ld , where AF is the flame surface area
and ld the average separation distance between the flame front and a fluid
element, which was located on the flame front at time t − τd . Taking the
ensemble average of many such realizations leads to
hΩd i ≈ hAF ld i = hAF ild1D + hAF δld i ,
(146)
whereas the decomposition ld = ld1D + δld is employed with the corresponding
separation distance ld1D from a laminar 1D flame calculation and the perturbation δld . Note that the last term in equation (146) requires modeling; e.g.
flame stretching results in negative values. For fix τd one now can write
hΩd i − hAF δld i
τd →0
ld1D
hAF i = lim
.
(147)
With these definitions, the mean flame surface density can be expressed as
hΣi =
hΩd i − hAF δld i
hAF i
= lim
τd →0
Ω
ld1D Ω
,
(148)
which requires modeling of hAF δld i; note that hΩd i = hdiΩ can be calculated
in this PDF modeling framework. Thus, from eq. (148) one obtains the
expression
hΣi ≈
hdi
hAF δld i
−
1D
ld
ld1D Ω
.
(149)
Neglecting parts of the flame surface propagating into itself, the probability
for an unburnt fluid element to be ”reached” by the propagating embedded
flame sheet during the next infinitesimal time interval is
hΣisL Ω
1
hAF δld i sL
F dt =
dt ≈
hdi −
dt ,
(150)
Ωu
1 − hci
Ω
ld1D
where Ωu = (1 − hci)Ω is the volume of unburnt gas. Note that the laminar
flame speed sL is assumed to be a function of the unburnt gas composition
and temperature only and that expression (150) is only correct, if all particles
with c∗ = 0 have the same density ρ = ρu , i.e. if the flame sheet marks the
very front of the embedded flame where T ≈ Tu . With this result based on
9.3 Analogy to the G-Equation Approach
111
the above assumptions, for small time steps of length ∆t, one obtains the
expression
P ≈ 1 − e−F ∆t
(151)
for the ignition probability. Here the effect of hAF δld i and that the flame
can run into itself is approximately captured by a mixing model for τ ; more
details are provided in section 9.4.
9.3
Analogy to the G-Equation Approach
The evolution of the flame residence time can be compared with the formulation for the G-equation. For that reason, the mixture fraction is assumed
to be constant and molecular mixing of τ is neglected. Then, integrating
equation (131) over the whole sample space except for Ψ3 = τ̂ , leads to the
transport equation for the MDF of the flame residence time, i.e.
∂ Ui τ̂ , x, t F(τ̂ , x, t)
∂cF(τ̂ , x, t)
∂F(τ̂ , x, t)
+
=−
. (152)
∂t
∂xi
∂ τ̂
On the other hand, the G-equation (126) can be written as
∂G
∂G
DG
ρu
=
+ Ui
sd |∇G|
=
Dt
∂t
∂xi
ρ
,
(153)
where a very small level-set Gls is chosen, i.e. 0 < Gls 1. The corresponding transport equation for the MDF F(Ĝ, x, t) (Ĝ is the sample space
variable of G) reads
∂ Ui Ĝ, x, t F(Ĝ, x, t)
∂F(Ĝ, x, t)
+
∂t
∂x
i
−1 ∂ |∇G| ρ Ĝ, x, t F(Ĝ, x, t)
= − ρu sd
. (154)
∂ Ĝ
Now, the gradient of of the level-set can be approximated under the assumption of constant sd and ρu , i.e. |∇G| ≈ ρuρsd c. Thus, both MDF transport
equations (154) and (152) become identical. However, equation (152) cannot predict the rate at which fluid with τ = 0 gets ignited and therefore an
ignition probability has to be specified.
112
9.4
9 COMBUSTION MODEL EXTENSION
Molecular Mixing
For the closure of the third rhs-term in equation (130), which affects the
Z ∗ and τ ∗ values of the computational particles, a variety of micro-mixing
models have been devised in the past [99, 33, 91, 55]. In the context of the
proposed PDF method, the interaction by exchange with the mean (IEM)
model is employed to account for molecular mixing of mixture fraction and
temperature. Then, under the assumption that the temperature must lie
on the manifold Tm (Z, τ ) and since it is a monotonous function of both Z
and τ , it is straightforward to determine the new pair (Z ∗ , τ ∗ ) employing
the mapping (Z ∗ , T ∗ ) → (Z ∗ , τ ∗ ). Note that changing the statistics of τ ∗
influences the local values of hdi and thus the ignition probability density
F . It is questionable, however, whether a mixing model designed for inert
scalars like mixture fractions can directly be employed to describe the effect
on τ ∗ . Therefore, it is not surprising that the mechanical-to-scalar time scale
ratio CΦ had to be increased from 2 (standard value for inert scalars) to
8. Here, mixing of τ is motivated primarily to account for flame stretching,
curvature effects, collapse and cusp formation, and secondarily to account
for mixing between products and co-flow. This is a different motivation
than in other combustion modeling approaches, where the mechanical-toscalar time scale ratio had to be adjusted for reactive flows, since some scalar
gradients are affected by chemistry. Lindstedt and Vaos [47] investigated
the influence of varying CΦ in the range [2.0, 8.0] for a turbulent premixed
combustion model based on a transported joint composition PDF method
with a reduced chemical reaction scheme. Stöllinger and Heinz [89] showed
good agreement of a piloted premixed burner with a value of Cφ = 12.0;
also within a joint composition PDF framework combined with a skeletal
mechanism. More recently, Rowinski and Pope [82] presented a detailed
study of Cφ and concluded that increasing Cφ improves the prediction of the
flame temperature, whereas a value of approximately 2.0 is most appropriate
for inert scalars like the mixture fraction.
9.5
Results
For validation, numerical calculations for three axisymmetric premixed piloted Bunsen flames [16] were performed. Each of these flames has three
inflow streams, i.e. an unburnt reactive jet encircled by a hot pilot, both surrounded by a slow ambient air co-flow. The jet bulk velocities are U0 = 30m/s
(flame F3), U0 = 50m/s (flame F2) and U0 = 65m/s (flame F1), and the
reference turbulent kinetic energies are k0 = 3.82m2 /s2 , k0 = 10.8m2 /s2 and
k0 = 12.7m2 /s2 , respectively. The adiabatic temperature of the fully burnt
9.5 Results
113
mixture is Tb = 2248K and that of the unburnt jet stream and the co-flow is
Tu = 298K. Profiles at the jet inflow of mean and root mean square (rms)
e1 and urms
velocities U
1 , respectively, are directly adopted from [16] and the
rms
00 00
g
estimation u1 u2 ≈ 0.5urms
is used for the velocity covariance (subscripts
1 u2
1 and 2 indicate axial and radial components, respectively). The turbulence
0.5
00 00
frequency ω
e at the jet inflow is set proportional to (ug
i ui /3) /D, where
D = 0.012m is the jet diameter. Pilot and co-flow have uniform mean velocities, i.e. 1.3m/s in the hot pilot, 1.0m/s in the cold pilot and 0.5m/s
= 0.1m/s for the pilot
= urms
in the co-flow. The rms-velocities are urms
2
1
rms
rms
and u1 = u2 = 0.05m/s for the co-flow. The turbulence frequency for
the pilot is 103 s−1 and 102 s−1 for the co-flow. From now on it is convenient
to consider the following normalized quantities: the normalized mean axial
e 0 , the normalized turbulent kinetic energy k̂ = e
velocity Û = U/U
k/k0 and
the normalized temperature T̂ = (T − Tu )/(Tb − Tu ). Normalized temperature and mixture fraction are T̂ = 0 and Z = 1 in the jet, T̂ = 0 and
Z = 0 in the co-flow, and T̂ = 0.8 and Z = Zp in the pilot, such that
T̂m (Zp , τ → ∞) = 0.8.
For the simulations presented here, the simplified manifold T̂m (Z, τ ) =
T̂st (τ )Z was employed, whereas T̂st (τ ) is the normalized temperature along
the profile of a premixed 1D flame with stoichiometric mixture (Z = 1). In
the manifold, by construction c is one where 0.05 ≤ T̂ and zero otherwise
and the function d(τ ) is chosen such that it is one for 0.2 ≤ T̂ ≤ 0.8 and zero
otherwise; note that this defines τd = 0.227 10−3 s and ld1D = 0.386 10−3 m. For
molecular mixing, Cφ values in the range between 2 and 10 were considered.
The best agreement for flame F3 was obtained for Cφ = 8; the same value
was then also employed for the simulations of flames F2 and F1.
For the computations, a rectangular domain of 0.6m in axial direction
(starting at the nozzle exit) and 0.1m in radial direction (starting at the
symmetry-axis) was considered and a 50 × 50 non-equidistant grid with an
average of 20 computational particles per cell was used. To investigate the
numerical convergence with respect to grid refinement and particle number,
an additional simulation of flame F3 on a 80 × 80 grid and 40 particles per
cell in average was performed; comparison with the result obtained with the
50 × 50 grid and 20 particles per cell shows very little difference.
These piloted Bunsen flames (Aachen flames [16]) have already been simulated by different groups. In 1999, the first simulations of the considered
flames have been presented in [78], where four different flame surface density
models were employed. In 2002, large-eddy simulations (LES) have been
performed solving a level-set formulation for the G-equation [67]. In [32] the
114
9 COMBUSTION MODEL EXTENSION
level-set formulation has been extended to capture effects of ambient cold
air entrainment. In [56], the first transported PDF method was applied to
this flame. The drawback of employing standard mixing models was circumvented by using a splitting procedure. The flame brush is dictated by the
flamelet library and for the outer part a standard mixing model was applied.
The joint composition PDF has been used together with the modified Curl
model [33] and a reduced chemical mechanism to investigate the influence of
the mechanical-to-scalar time ratio [47]. More recently, the flame has been
simulated using a transported joint composition PDF method [88]. A generalization of the scalar mixing frequency model was developed resulting in
the generalized correlation (GC) model.
9.5.1
Non-Reactive Results
In figures 35 and 36, the normalized mean downstream velocity Û1 and the
normalized turbulent kinetic energy k̂ of the non-reactive flow cases are presented for all three flames. Note that for flame F1 no experimental data are
available for the non-reactive case. The solid lines represent numerical results
and the circles experimental data. While for flame F3 the normalized mean
downstream velocity is well predicted, Û1 is slightly underpredicted for flame
F2 close to the centerline. The normalized turbulent kinetic energy is well
predicted for flames F2 and F3, except downstream of x/D = 6.5, where k̂
is underpredicted. But in general, good agreement between experiments and
numerical simulations can be observed.
9.5.2
Reactive Results
For the reactive flow simulations, the same boundary conditions are applied.
In general excellent agreement is obtained between numerical simulation and
experiments [16] for the mean flow quantities. The normalized mean downstream velocity Û (figure 37) is predicted very accurately for all three flames.
The expansion of the fluid due to the chemical reactions is perceivable in the
acceleration of Û compared with the velocities in the non-reacting flows. In
figure 38, the normalized turbulent kinetic energy k̂ is presented, which is in
very good agreement with the experimental measurements. Moreover, the
gas expansion induced by the flame shifts the shear layer outwards resulting
in a shift of the turbulent kinetic energy peaks. This effect is captured very
well by the presented simulations. In figure 39, the normalized mean temperature is presented, which is in very good agreement with the experiments.
The gradients of T̂ from the simulations agree with the measured gradients,
except at X/D = 2.5, where these gradients are overpredicted. In addition,
9.5 Results
115
F3
1.5
F2
1.5
x/D = 2.5
F1
1.5
x/D = 2.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
0.0
0.0
1.5
x/D = 4.5
1.0
2.0
0.0
0.0
1.5
x/D = 4.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
0.0
0.0
1.5
x/D = 6.5
1.0
2.0
0.0
0.0
1.5
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
0.0
0.0
1.5
x/D = 8.5
1.0
2.0
0.0
0.0
1.5
x/D = 8.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
x/D = 10.5
0.0
0.0
1.5
1.0
2.0
x/D = 10.5
0.0
0.0
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
x/D = 12.5
0.0
0.0
1.5
1.0
2.0
x/D = 12.5
0.0
0.0
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
r/D
2.0
x/D = 4.5
1.0
r/D
2.0
x/D = 6.5
1.0
r/D
2.0
x/D = 8.5
1.0
r/D
2.0
x/D = 10.5
1.0
r/D
2.0
x/D = 12.5
1.0
r/D
2.0
Figure 35: Radial profiles of the normalized mean downstream velocity Û1
for the three cold cases at several downstream locations: experiments (circles)
and numerical simulation (solid lines).
at X/D = 2.5 the peak temperature is overpredicted for all three flames.
Similar overpredictions were also reported in [47, 88, 32], where in [88] it is
supposed that the complex interactions of the cold, highly turbulent jet with
116
9 COMBUSTION MODEL EXTENSION
F3
15.0
F2
15.0
x/D = 2.5
F1
15.0
x/D = 2.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
0.0
0.0
15.0
x/D = 4.5
1.0
2.0
0.0
0.0
15.0
x/D = 4.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
0.0
0.0
15.0
x/D = 6.5
1.0
2.0
0.0
0.0
15.0
x/D = 6.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
0.0
0.0
15.0
x/D = 8.5
1.0
2.0
0.0
0.0
15.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
x/D = 10.5
0.0
0.0
15.0
1.0
2.0
x/D = 10.5
0.0
0.0
15.0
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
x/D = 12.5
0.0
0.0
15.0
1.0
2.0
x/D = 12.5
0.0
0.0
15.0
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
r/D
2.0
x/D = 4.5
1.0
r/D
2.0
x/D = 6.5
1.0
r/D
2.0
x/D = 8.5
1.0
r/D
2.0
x/D = 10.5
1.0
r/D
2.0
x/D = 12.5
1.0
r/D
2.0
Figure 36: Radial profiles of the normalized turbulent kinetic energy k̂ for
the three cold cases at several downstream locations: experiments (circles)
and numerical simulation (solid lines).
the hot, laminar pilot stream might be the reason. Further downstream,
at X/D = 10.5 , the mean temperature of the flames F2 and F1 is underpredicted, whereas at X/D = 12.5 for flame F1 the temperature forecast is
9.5 Results
117
F3
1.5
F2
1.5
x/D = 2.5
F1
1.5
x/D = 2.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
0.0
0.0
1.5
x/D = 4.5
1.0
2.0
0.0
0.0
1.5
x/D = 4.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
0.0
0.0
1.5
x/D = 6.5
1.0
2.0
0.0
0.0
1.5
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
0.0
0.0
1.5
x/D = 8.5
1.0
2.0
0.0
0.0
1.5
x/D = 8.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
x/D = 10.5
0.0
0.0
1.5
1.0
2.0
x/D = 10.5
0.0
0.0
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.5
1.0
2.0
x/D = 12.5
0.0
0.0
1.5
1.0
2.0
x/D = 12.5
0.0
0.0
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
r/D
2.0
x/D = 4.5
1.0
r/D
2.0
x/D = 6.5
1.0
r/D
2.0
x/D = 8.5
1.0
r/D
2.0
x/D = 10.5
1.0
r/D
2.0
x/D = 12.5
1.0
r/D
2.0
Figure 37: Radial profiles of the normalized mean downstream velocity Û for
all three flames at several downstream locations: experiments (circles) and
numerical simulation (solid lines).
accurate; otherwise good agreement between simulations and experimental
data is observed. Figure 40 shows the normalized rms-temperature T̂ rms .
For flame F3, the predictions are in good agreement with the experiment,
118
9 COMBUSTION MODEL EXTENSION
F3
15.0
F2
15.0
x/D = 2.5
F1
15.0
x/D = 2.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
0.0
0.0
15.0
x/D = 4.5
1.0
2.0
0.0
0.0
15.0
x/D = 4.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
0.0
0.0
15.0
x/D = 6.5
1.0
2.0
0.0
0.0
15.0
x/D = 6.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
0.0
0.0
15.0
x/D = 8.5
1.0
2.0
0.0
0.0
15.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
x/D = 10.5
0.0
0.0
15.0
1.0
2.0
x/D = 10.5
0.0
0.0
15.0
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
15.0
1.0
2.0
x/D = 12.5
0.0
0.0
15.0
1.0
2.0
x/D = 12.5
0.0
0.0
15.0
10.0
10.0
10.0
5.0
5.0
5.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
r/D
2.0
x/D = 4.5
1.0
r/D
2.0
x/D = 6.5
1.0
r/D
2.0
x/D = 8.5
1.0
r/D
2.0
x/D = 10.5
1.0
r/D
2.0
x/D = 12.5
1.0
r/D
2.0
Figure 38: Radial profiles of the normalized turbulent kinetic energy k̂ for
all three flames at several downstream locations: experiments (circles) and
numerical simulation (solid lines).
i.e. the peak values and the peak locations of the fluctuating temperature are
predicted very accurately. For flames F2 and F1, the predictions are good upstream of x1 /D = 6.5; further downstream the model tends to under-predict
9.5 Results
119
F3
F2
x/D = 2.5
F1
x/D = 2.5
x/D = 2.5
1.0
1.0
1.0
0.5
0.5
0.5
0.0
0.0
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
1.0
0.5
0.5
0.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
0.5
0.5
0.5
1.0
2.0
0.0
0.0
x/D = 8.5
1.0
2.0
0.0
0.0
x/D = 8.5
1.0
1.0
0.5
0.5
0.5
1.0
2.0
0.0
0.0
x/D = 10.5
1.0
2.0
0.0
0.0
x/D = 10.5
1.0
1.0
0.5
0.5
0.5
1.0
2.0
0.0
0.0
x/D = 12.5
1.0
2.0
0.0
0.0
x/D = 12.5
1.0
1.0
0.5
0.5
0.5
1.0
2.0
0.0
0.0
1.0
2.0
2.0
1.0
r/D
2.0
1.0
r/D
2.0
x/D = 12.5
1.0
0.0
0.0
1.0
r/D
x/D = 10.5
1.0
0.0
0.0
2.0
x/D = 8.5
1.0
0.0
0.0
1.0
r/D
x/D = 6.5
1.0
0.0
0.0
2.0
x/D = 4.5
1.0
0.0
0.0
1.0
r/D
0.0
0.0
1.0
r/D
2.0
e
Figure 39: Radial profiles of the normalized mean temperature T̂ for all three
flames at several downstream locations: experiments (circles) and numerical
simulation (solid lines).
T̂ rms .
The dependency of the particle temperature on mixture fraction Z ∗ and
flame residence time τ ∗ from flame F3 is presented in figure 41. The location
120
9 COMBUSTION MODEL EXTENSION
F3
0.6
F2
0.6
x/D = 2.5
F1
0.6
x/D = 2.5
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.6
1.0
2.0
0.0
0.0
0.6
x/D = 4.5
1.0
2.0
0.0
0.0
0.6
x/D = 4.5
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.6
1.0
2.0
0.0
0.0
0.6
x/D = 6.5
1.0
2.0
0.0
0.0
0.6
x/D = 6.5
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.6
1.0
2.0
0.0
0.0
0.6
x/D = 8.5
1.0
2.0
0.0
0.0
0.6
x/D = 8.5
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.6
1.0
2.0
x/D = 10.5
0.0
0.0
0.6
1.0
2.0
x/D = 10.5
0.0
0.0
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.6
1.0
2.0
x/D = 12.5
0.0
0.0
0.6
1.0
2.0
x/D = 12.5
0.0
0.0
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
r/D
2.0
x/D = 4.5
1.0
r/D
2.0
x/D = 6.5
1.0
r/D
2.0
x/D = 8.5
1.0
r/D
2.0
x/D = 10.5
1.0
r/D
2.0
x/D = 12.5
1.0
r/D
2.0
Figure 40: Radial profiles of the normalized rms-temperature T̂ rms for all
three flames at several downstream locations: experiments (circles) and numerical simulation (solid lines).
where Z ∗ ≈ 1.0 and τ ∗ ≈ 0.0 describes particles from the unignited jet.
First, these particles ignite and evolve according to the one-dimensional flame
profile, i.e. the range of the particle distribution is very sharp. When these
9.5 Results
121
T
τ
Z
Figure 41: The manifold T ∗ − Z ∗ − τ ∗ represented by the particle ensemble
in the simulation of flame F3.
particles have passed the reaction zone, i.e. at high temperatures, their
evolution is dominated by molecular diffusion.
Before this result section is concluded, it has to be emphasized that only
flame F3 is operated in the corrugated flamelet regime, for which the modeling assumptions have been proposed. Flames F2 and F1 are subject to the
thin reaction zone regime and to a small extent even to the broken flamelet
regime, which is most likely the reason for most of the observed discrepancies
with measurements, since the assumption of constant laminar flame speeds
and unperturbed embedded flame structures do not hold in these regimes.
9.5.3
Mechanical-to-Scalar-Time Scale Ratio
In figure 42, simulation results for flame F3 are presented, where the influence
of the mechanical-to-scalar time scale ratio is investigated, as already mentioned earlier. Simulations with Cφ = 2.0, Cφ = 4.0, Cφ = 6.0 and Cφ = 8.0
were performed. For Cφ = 2.0, the mean temperature at x1 /D = 8.5 is
under-predicted. On the other hand, the rms-temperature is over-predicted
for positions upstream of x1 /D = 8.5. The best general agreement was found
for Cφ = 8.
It is not surprising that the mechanical-to-scalar time scale ratio CΦ had
to be increased from the standard value for inert scalars of 2 up to 8, since
122
9 COMBUSTION MODEL EXTENSION
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
e
Figure 42: Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream locations: experiments (circles), numerical simulations: Cφ = 2.0 (solid lines), Cφ = 4.0
(dashed lines), Cφ = 6.0 (dashed-dotted lines) and Cφ = 8.0 (dotted lines).
mixing of the flame residence time is primarily motivated to account for flame
stretching, curvature effects, collapse and cusp formation, and secondarily to
account for mixing between hot products and co-flow. Note that the situation
here is different than in other combustion modeling approaches, where the
mechanical-to-scalar time scale ratio had to be adjusted for reactive flows,
since some scalar gradients are affected by chemistry.
9.5 Results
9.5.4
123
Numerical Analysis
In this subsection, the numerical robustness of the proposed PDF combustion
model is investigated. Simulations of the identical combustion model have
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
e
Figure 43: Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream locations: experiments (circles), numerical simulations: Nx = 50, Ny = 50, Np = 20 (solid
lines) and Nx = 80, Ny = 80, Np = 40 (dashed lines).
been performed with varying number of particles per cell (Np ) and with
varying number of grid cells in x1 (Nx ) and x2 -direction (Ny ). In figure 43,
the simulation of flame F3 presented above is compared with a better resolved
simulation, i.e. with a 80 × 80 grid with 40 particles per cell in average.
The mean downstream velocity is almost identical in both cases, but the
124
9 COMBUSTION MODEL EXTENSION
0.1
0.1
0.1
x/D = 4.5
x/D = 4.5
x/D = 4.5
0.05
0.05
0.05
0.0
0.0
0.0
−0.05
−0.05
−0.05
−0.1
0.0
1.0
2.0
0.1
−0.1
0.0
1.0
2.0
0.1
−0.1
0.0
x/D = 8.5
x/D = 8.5
0.05
0.05
0.05
0.0
0.0
0.0
−0.05
−0.05
−0.05
1.0
2.0
−0.1
0.0
2.0
0.1
x/D = 8.5
−0.1
0.0
1.0
r/D
1.0
2.0
−0.1
0.0
1.0
r/D
2.0
MC
Figure 44: Radial profiles of ρF Vρ−ρ
(left), ue001 (center) and ue002 (right) at
FV
two downstream locations in the reactive piloted Bunsen flame.
turbulent kinetic energy gets insignificantly increased with the 80 × 80 grid.
The mean temperature slightly differs at the interface of pilot and co-flow,
but in the shear layer between jet and pilot, both simulations show similar
results. Anyhow, the differences are insignificant and the general agreement
of the simulations with the two different grid resolutions is excellent. This
shows that convergence is obtained with respect to grid resolution and with
respect to the number of particles.
In the hybrid finite-volume/particle method presented in chapter 4, the
mean density is calculated twice, i.e. in each of the two methods separately.
One important criterion to achieve convergence with this hybrid method is
consistency between the two densities. The profiles at the left of figure 44
MC
, where ρF V represents the mean
show the relative density mismatch ρF Vρ−ρ
FV
density obtained from the RANS equations and ρM C is the mean particle
number density, i.e. the mean density represented by the ensemble of computational particles. At both axial positions the discrepancies remain below
a level of 3 %. The radial profiles in the center and at the right of figure 44
are further important consistency criteria. By definition, the mass average
of the fluctuating velocity components should be zero. But due to numerical
errors, this is not automatically guaranteed by the solution algorithm and,
therefore, a correction scheme is applied. Figure 44 shows ue001 and ue002 and it
can be seen that the deviation from 0 m/s remains below an absolut value
9.5 Results
125
111111
000000
000000
111111
000000
0000111111
1111
000000
111111
0000
1111
000000
0000111111
1111
000000
111111
0000
1111
000000
0000111111
1111
0000
1111
0000
1111
11111111
00000000
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
11111111
00000000
00000000
11111111
00000000
11111111
00000000
11111111
00000000
11111111
Figure 45: Sketch of particle trajectories (solid lines) within the numerical
grid (dashed lines). The bullets show the location of the flame surface on the
trajectories and the hatched area indicates the locations where d∗ equals 1
for a particle.
of 0.02 m/s, which is more than one order of magnitude smaller than the
fluctuations itself.
As described earlier, the mean flame surface density hΣi is estimated
as hdi /ld1D plus a modeled correction, whereas this estimation requires a
deeper inspection. In figure 45, four particle trajectories (solid lines) are
shown. The shaded areas represent the locations, where the particle’s dvalue equal 1, and the dashed lines represent the numerical grid. The bullets
on the particle trajectories show where the particle states equal the one on
the flame surface. Note that the model becomes independent of ld1D for
ld1D → 0, but for smaller ld1D , more particles are required to obtain a good
estimate of hdi. For that reason, additional simulations have been performed
to investigate the accuracy of this approximation. In figure 46, three different
approximations of hdi /ld1D are compared: the flame surface trigger d∗ of a
particle is 1, if its normalized temperature lies within [0.2, 0.8] (dashed line),
[0.3, 0.7] (solid line) and [0.4, 0.6] (dotted line); the observable discrepancies
are small. Furthermore, figure 47 is presented, where the intervals [0.1, 0.9],
[0.1, 0.7] and [0.1, 0.5] are considered. For these intervals, convergence can
be observed.
126
9 COMBUSTION MODEL EXTENSION
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
e
Figure 46: Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream locations. d∗
is equal 1, if its normalized temperature lies within [0.2, 0.8] (solid lines),
[0.3, 0.7] (dashed lines) and [0.4, 0.6] (dotted lines); experiments (circles).
9.6
Summary
A novel model for turbulent premixed combustion is presented. The modeled
transport equation for the joint PDF of velocity, turbulence frequency, mixture fraction, a binary progress variable and a flame residence time is solved
with a hybrid particle/finite-volume solution algorithm. Besides other advantages, such joint velocity-scalar PDF methods are not subject to counter
gradient diffusion, since turbulent convection appears in closed form.
During a time step, a computational particle representing a reactive un-
9.6 Summary
127
1.5
1.5
x/D = 2.5
1.5
x/D = 4.5
1.5
x/D = 6.5
x/D = 8.5
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.0
0.0
1.0
2.0
15.0
0.0
0.0
1.0
2.0
15.0
x/D = 2.5
0.0
0.0
1.0
2.0
15.0
x/D = 4.5
0.0
0.0
x/D = 8.5
10.0
10.0
10.0
5.0
5.0
5.0
5.0
1.0
2.0
0.0
0.0
x/D = 2.5
1.0
2.0
0.0
0.0
x/D = 4.5
1.0
2.0
0.0
0.0
x/D = 6.5
1.0
1.0
1.0
0.5
0.5
0.5
0.5
1.0
2.0
0.6
0.0
0.0
1.0
2.0
0.6
x/D = 2.5
0.0
0.0
1.0
2.0
0.6
x/D = 4.5
0.0
0.0
0.4
0.4
0.2
0.2
0.2
0.2
2.0
0.0
0.0
1.0
r/D
2.0
0.0
0.0
1.0
2.0
x/D = 8.5
0.4
1.0
r/D
2.0
0.6
x/D = 6.5
0.4
0.0
0.0
1.0
x/D = 8.5
1.0
0.0
0.0
2.0
15.0
x/D = 6.5
10.0
0.0
0.0
1.0
1.0
r/D
2.0
0.0
0.0
1.0
r/D
2.0
e
Figure 47: Radial profiles of the normalized quantities Û , k̂, T̂ and T̂ rms
(from top to bottom) for the flame F3 at several downstream locations. d∗
is equal 1, if its normalized temperature lies within [0.1, 0.9] (solid lines),
[0.1, 0.7] (dashed lines) and [0.1, 0.5] (dotted lines).
burnt fluid volume is ”reached” by the embedded propagating flame surface with the ”ignition” probability P , calculated based on an estimate of
the mean flame surface density. In the proposed joint PDF framework, the
flame surface density is transported, but effects of flame stretching, curvature, collapse and cusp formation have to be modeled via a mixing model for
the flame residence time. Once ”reached” by the embedded flame surface,
species mass fractions, enthalpy and temperature of a particle are governed
by mixture fraction and flame residence time and can be retrieved by look-up
128
9 COMBUSTION MODEL EXTENSION
from precomputed premixed laminar flame tables. To account for molecular
mixing between hot products and the co-flow (and to account for production
and dissipation of the flame front, as mentioned earlier), the IEM mixing
model is employed for mixture fraction and temperature; the flame residence
time is then obtained via unique mapping. The best results were obtained
for a mechanical-to-scalar time scale ratio of Cφ = 8. As explained above,
here it is different than in other combustion modeling approaches, where the
mechanical-to-scalar time scale ratio was adjusted to account for the impact
of the chemistry on scalar gradients. Numerical validation studies of piloted
premixed Bunsen flames reveal that the proposed model not only delivers excellent results for the corrugated flamelet regime, but seems also applicable
to the thin reaction zone regime and, to a small extent, also to the broken
reaction zone regime. However, for the latter two regimes, more research is
required, since the current modeling assumptions are not valid there.
In the proposed modeling approach, the mixing model has to account for
two effects, i.e. primarily the production and dissipation of the flame surface
and secondarily molecular mixing between hot products and the ambient air.
While the mean temperatures are in good agreement, the rms temperatures
are underpredicted. One reason for this underprediction can be found in the
non-localness of the employed mixing model, since localness in scalar space
is an important requirement. To overcome this issue, a procedure can be
proposed accounting for the localness in scalar space, even if the employed
mixing model does not fulfill this requirement. Assuming a closed flame, a
particle on one side of the flame cannot mix with a particle on the other
side, i.e. the flame acts as barrier separating unburnt from burnt fluid. By
that reason, a conditional mixing model with respect to flame surface would
fulfill this requirement. In the Lagrangian particle method, such a procedure
is straightforward to implement, i.e. the ensemble of computational particles
is divided into two sets, one consisting of particles on the unignited side of
the flame and the other consists of burnt or burning particles. Then, each of
the two sets mix independently of the other set, which is valid under the assumption made for the corrugated flamelet regime. To treat partially broken
flame sheets, this distinct treatment has to be generalized, such that at large
Karlovitz numbers (where the flame consists of holes or is broken) mixing
between both sets is possible.
Finally, the last topic is the generalization of the presented model, such
that more challenging flames within thin and broken reaction zone regimes
can be calculated. To do so, the limiting assumptions of unperturbed local flame structures and constant laminar flame speeds have to be nullified.
Thus, flame stretch and curvature effects have to be taken into account to
9.6 Summary
generalize the formulation for the ignition probability.
129
10
Conclusions
To perform simulations of turbulent reactive flows, various numerical approaches exist that may be categorized in terms of computational cost and
level of closure. On one hand, RANS methods are numerically very efficient,
but only mean quantities are resolved, such that the whole range of turbulent motions has to be modeled. Thus the accuracy strongly depends on the
proposed models. On the other hand, a DNS fully resolves all underlying
physical processes at the price of tremendous numerical cost, limiting DNS
to academic problems. In LES the large scales of turbulent motions are captured, whereas the small ones have to be modeled. These unresolved subgrid
scales behave in a more universal manner than the larger ones, such that
modeling becomes less crucial than in the RANS context, but vice versa,
the required computational time is considerably higher. A very general statistical description of reactive flows is given by transported PDF methods,
which are less expensive than LES. Considering the joint PDF of velocity,
turbulence frequency and composition, turbulent convection and the highly
non-linear mean source term appear in closed form. More recently, the same
methodology has been applied to the LES context, i.e. filtered density function (FDF) methods are employed to model the unresolved subgrid scales.
The PDF transport equation can exactly be derived from the underlying
conservation equations, but various terms require modeling. Moreover, it is
numerically not feasible to solve this equation by a conventional finite-volume
or finite-element method. The reason is the high dimensionality of the sample space, in which the PDF evolves. Therefore, the modeled PDF transport
equation is not directly solved, but an equivalent set of stochastic differential
equations is solved with a Monte-Carlo particle method, where the computational cost linearly increases with the dimensionality of the sample space.
Further improvements were achieved by so-called hybrid PDF/RANS methods, where the mean velocities are obtained from the RANS equations, such
that the number of particles to achieve smooth mean velocity profiles can be
reduced. Even though the method is internally consistent at the level of gov-
132
10 CONCLUSIONS
erning equations, it is not straightforward to achieve numerical consistency.
Therefore, various correction schemes were proposed to ensure consistency
between redundant RANS and PDF quantities. Moreover, strategies to reduce the statistical and deterministic errors were proposed in the past. In the
first part of this work, a hybrid solution strategy to solve the MDF transport equation is presented - followed by a stand alone Monte-Carlo PDF
method for simplified flow configurations. In addition, the influence of different particle number control algorithms for statistically stationary problems
was investigated. A new particle elimination scheme performed similarly for
a generic jet flame as existing ones.
The second part of this work focuses on the modeling of premixed turbulent flames. The first two chapters describe the phenomena occurring in
turbulent reactive flows and the most popular existing modeling approaches
are reviewed. In the following two chapters, a novel model approach for turbulent premixed flames in the corrugated flamelet regime is presented, which
is based on a transported joint PDF of velocity, turbulence frequency, mixture fraction, a binary progress variable and a flame residence time. Besides
other advantages, such joint velocity-scalar PDF methods are not subject to
counter-gradient diffusion, since turbulent convection appears in closed form.
Unlike in most other PDF methods, the source term not only describes reaction rates, but accounts for ”ignition” of reactive unburnt fluid elements
due to propagating embedded quasi laminar flames within a turbulent flame
brush. The ignition probability describes the rate at which unburnt particles
get ”ignited” by the embedded laminar flame, i.e. it reflects the coupled finescale convection-diffusion-reaction dynamics within the flame. The ignition
probability is estimated based on the following particle properties: a flag
indicating whether a particle represents the unburnt mixture; a flame residence time that allows to resolve the embedded quasi laminar flame structure; and a flag indicating whether the flame residence time lies within a
specified range. Latter, together with assumed unperturbed embedded flame
structures and constant laminar flame speeds (as expected in the corrugated
flamelet regime), is used to transport the flame surface density. But to account for flame stretching, curvature effects, collapse and cusp formation, a
mixing model for the flame residence time is employed. Note that the same
mixing model also accounts for molecular mixing of the hot products with a
co-flow. In addition, an alternative formulation for the ignition probability is
given by an empirical ansatz, where the corresponding model constants are
determined by simplified one-dimensional simulations of a weak swirl burner.
Once ”reached” by the embedded flame surface, species mass fractions, enthalpy and temperature of a particle are governed by mixture fraction and
133
flame residence time and can be retrieved by look-up from precomputed
premixed laminar flame tables. To account for molecular mixing, the IEM
mixing model is employed for mixture fraction and temperature; the flame
residence time is then obtained via unique mapping. Numerical validation
studies of piloted premixed Bunsen flames reveal that the proposed model
delivers excellent results for the corrugated flamelet, the thin reaction zone
and, to a small extent, also for the broken reaction zone regime. However, for
the latter two regimes more research is required, since the current modeling
assumptions are not valid there.
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Curriculum Vitae
Personal details
First names
Last name
Date of birth
Place of birth
Citizen of
Mathias Leander
Hack
24.08.1978
Winterthur (ZH), Switzerland
Wolfenschiessen (NW), Switzerland
2006 - present
PhD candidate at the Institute of Fluid
Dynamics, ETH Zurich, Switzerland.
Supervisor: Prof. Dr. Patrick Jenny
Diploma study in Computational Science
and Engineering at ETH Zurich, Switzerland. Major in Computational Fluid Dynamics.
Basic study in Mathematics at ETH
Zurich (1. and 2. intermediate examination)
State high school (Gymnasium Typus C),
Kantonsschule Zürcher Unterland, Bülach
(ZH)
Secondary school in Embrach (ZH)
Primary school in Embrach (ZH)
Education
2002 - 2006
1999 - 2002
1994 - 1999
1991 - 1994
1985 - 1991
Employment
2006 - present
Research and teaching assistant at the Institute of Fluid Dynamics, ETH Zurich,
Switzerland
Mathias Leander Hack
Joint Probability Density Function (PDF) Closure of Turbulent Premixed Flames
The accurate and reliable prediction of turbulent premixed flames is a crucial task and
even after 50 years of intense research this problem could not be solved. One reason is
found in the sophisticated turbulence-chemistry-interaction occurring on a wide range of
length and time scales. With the help of direct numerical simulations (DNS) and
experimental investigations of such flames, the general understanding of the underlying
processes within a premixed turbulent flame could be ameliorated tremendously, but
remains far from being complete. Moreover, the computational effort makes it infeasible to
predict turbulent flames without any modeling assumptions. An optimal balance between
computational efficiency and level of closure has to be found. By that reason a variety of
approaches dealing with the description of phenomena within turbulent flames have been
proposed. The Reynolds-averaged Navier-Stokes (RANS) equations can be solved very
efficiently, but the resolution remains at a level of mean flow and thermodynamic
quantities, such that predictions strongly depend on the disposed models. In large eddy
simulations (LES) the large turbulent scales are resolved, such that only the so-called
subgrid scale (SGS) phenomena have to be modeled. This approach allows to study
instantaneous flame dynamics at the price of increased simulation times. A sophisticated
alternative are transported probability density function (PDF) methods. While providing the
full statistical information of flow and thermodynamic quantities at a certain location, the
numerical effort remains between those of RANS and LES simulations. Moreover,
turbulent convection and mean source terms appear in closed form in the PDF transport
equation. Due to the high dimensionality of the corresponding sample space, the PDF
transport equation is solved by a Monte-Carlo particle method, representing the PDF by an
ensemble of computational particles. For these particles, an equipollent set of stochastic
differential equations (SDE) is solved. In addition, a finite-volume method provides the
mean flow velocity by solving the RANS equations, where the RANS equations get closed
by the joint statistics of the PDF method. Even though consistency is ensured at the level
of governing equations, it is not straightforward to achieve convergence and consistency
between the two methods. Therefore, various correction schemes have been proposed to
achieve consistency and convergence. The first part of this work deals with the hybrid
solution strategy. In addition, a study on the topic of particle number control mechanisms is
presented.
In the second part of the thesis, a novel model for turbulent premixed combustion is
presented for the corrugated flamelet regime. It is based on a transported joint PDF of
velocities, turbulence frequency and scalars. A binary progress variable indicates the
arrival of embedded quasi laminar flames within a turbulent flame brush at the particle
location. In addition, a flame residence time is introduced to resolve the embedded quasi
laminar flame structure. Under the assumption of undisrupted embedded flame structures
and of constant laminar flame speed (i.e. unaffected by strain effects), the particle
composition can be retrieved from precomputed one-dimensional laminar flame tables
knowing its flame residence time. The ''ignition'' of reactive unburnt fluid elements by the
propagating embedded flame is described by the ''ignition'' probability P, which describes
the rate at which unburnt particles get consumed by the flame. First, an empirical ansatz
for P is proposed; second with the help of a flag indicating whether the flame residence
time lies within a specified range, the ignition probability is calculated based on an
estimate of the mean flame surface density. Latter gets transported by the PDF method,
but to account for flame stretching, curvature effects, collapse and cusp formation, a
mixing model for the residence time is employed. The same mixing model also accounts
for molecular mixing of the products with a co-flow. Numerical simulations of three piloted
premixed Bunsen flames show excellent agreement with the experimental measurements
and demonstrate the applicability of the proposed PDF model.
Joint Probability Density
Function (PDF) Closure of
Turbulent Premixed Flames
Mathias Leander Hack
Dissertation ETH No. 19927