Large Eddy Simulation of Premixed
Combustion Using Flamelets
Ivan Langella
Department of Engineering
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Robinson College October 2015

Sometimes I still look up in the sky and wonder what is that force that keeps men going
beyond themselves. This thesis is dedicated to all those who still listen to this force and have
never stopped to believe in their dreams. The world needs these people more than ever. I also
dedicate this work to the people who loved and supported me the most since the day I was
born, my mother Franca, my father Pietro, my sisters and my missing grandfather Giovanni.

Declaration
I hereby declare that except where specific reference is made to the work of others, the contents
of this dissertation are original and have not been submitted in whole or in part for consideration
for any other degree or qualification in this, or any other university. This dissertation is my own
work and contains nothing which is the outcome of work done in collaboration with others,
except as specified in the text and Acknowledgements. This dissertation contains approximately
65,000 words including appendices, bibliography, footnotes, tables and equations and has 112
figures.
Ivan Langella
October 2015

Acknowledgements
I would like to acknowledge Prof. N. Swaminathan for being a brilliant supervisor and also a
great mentor. In these PhD years he has been more than supportive and always available for
any kind of problem I may have had. The opportunity he has given to me with this PhD is
priceless and I will be always thankful for this.
I acknowledge Prof. N. Chakraborty, for his help and advices in developing the models pre-
sented in this paper, and Y. Gao for sharing her DNS results, which have been useful for the
understanding of the models presented in this work. I also want to acknowledge Prof. E.
Mastorakos for his priceless help and several advices during the PhD period, and Profs. R. W.
Pitz and F. A. Williams for their helps and suggestions for some of the simulations presented in
this thesis.
I acknowledge Mr. P. Benie, a formidable expert in computing technologies, which helped me
several times on different issues related with Unix systems and programming languages.
I would like to thank my colleagues and friends at the Hopkinson Lab in Cambridge, the
professors I have been in contact with and the technicians, all of which have contributed to my
personal and professional growth. A particular thought goes to the ‘italian team’ which have
passed the lab across these years: A. Maffioli, D. Cavaliere, G. Borghesi and more recently
Simone Lombardi, A. Giusti and A. Cabboi, but also to Z. Nicolau, I. Ahmed, M. Youstos, G.
Ghiasi and T. Zhi Chen. I wish the best for all of them.
Finally, I would like to give a special thanks to my girlfriend Eli, for giving inspiration and
support during the busy months of my PhD.
I acknowledge the funding through the EPSRC and the partnerships of Siemens and Rolls-
Royce in this PhD project and also the conference grants received from the Combustion Institute
viii
and Robinson College. This work made use of the Darwin high performance computing service
at the University of Cambridge, whose access was partly funded by EPSRC and Siemens.
Publications
In chronological order, results from this work have appeared, or are currently prepared for
publication in
Journals:
1. I. Langella, N. Swaminathan, Y. Gao, N. Chakraborty, Combust. Theory Model., vol. 19,
pp. 628-656. Assessment of dynamic closure for premixed combustion LES.
2. I. Langella, N. Swaminathan, Combust. Theory Model., accepted,
DOI 10.1080/13647830.2016.1140230. Unstrained and strained flamelets for LES of
premixed combustion.
3. I. Langella, N. Swaminathan, Y. Gao, N. Chakraborty, Combust. Sci. Technol., submitted
August (2015) - accepted pending revision. LES of premixed combustion using an
algebraic closure involving scalar dissipation rate.
4. I. Langella, N. Swaminathan, F. Williams, J. Furukawa, Combust. Sci. Technol., sub-
mitted October (2015) - accepted pending revision. LES of premixed combustion in
corrugated flamelets regime.
5. I. Langella, N. Swaminathan, R. Pitz. LES of lean premixed combustion behind a
bluff-body, Combust. Flame, submitted January (2016) - accepted pending revision.
Conferences:
1. I. Langella, N. Swaminathan, N. Chakraborty. Large Eddy Simulation of Turbulent
Premixed Combustion, 6th European Combustion Meeting (ECM), Lund-Sweden, July
(2013).
2. I. Langella, N. Swaminathan, N. Chakraborty, Large Eddy Simulation of Premixed
Combustion Using Scalar Dissipation Rate Approach, 8th Mediterranean Combustion
Symposium (MCS), Ces¸me-Turkey (2013).
x3. I. Langella, N. Swaminathan, LES of Premixed Combustion Using Unstrained and
Strained flamelets, Joint combustion meeting of the Nordic-Scandinavian and British
Sections of the Combustion Institute, Cambridge-UK (2014).
4. I. Langella, N. Swaminathan, LES of Lean Premixed Combustion Using a Flamelet
Approach, Conference on Thermal Energy Systems: Production, Storage, Utilization and
the Environment (ASME-ATI-UIT), Napoli-Italy (2015).
5. I. Langella, N. Swaminathan, R. Pitz, LES of Bluff Body turbulent premixed flame using
Scalar Dissipation Rate Based Modelling, 9th Mediterranean Combustion Symposium
(MCS), Rhodes-Greece (2015).
Abstract
Large Eddy Simulation (LES) has potential to address unsteady phenomena in turbulent
premixed flames and to capture turbulence scales and their influence on combustion. Thus,
this approach is gaining interest in industry to analyse turbulent reacting flows. In LES, the
dynamics of large-scale turbulent eddies down to a cut-off scale are solved, with models to
mimic the influences of sub-grid scales. Since the flame front is thinner than the smallest scale
resolved in a typical LES, the premixed combustion is a sub-grid scale (SGS) phenomenon
and involves strong interplay among small-scale turbulence, chemical reactions and molecular
diffusion. Sub-grid scale combustion models must accurately represent these processes.
When the flame front is thinner than the smallest turbulent scale, the flame is corrugated
by the turbulence and can be seen as an ensemble of thin, one-dimensional laminar flames
(flamelets). This allows one to decouple turbulence from chemistry, with a significant reduction
in computational effort. However, potentials and limitations of flamelets are not fully explored
and understood. This work contributes to this understanding. Two models are identified, one
based on an algebraic expression for the reaction rate of a progress variable and the assumption
of fast chemistry, the other based on a database of unstrained flamelets in which reaction rates
are stored and parametrised using a progress variable and its SGS variance, and their potentials
are shown for a wide range of premixed combustion conditions of practical interest. The
sensitivity to a number of model parameters and boundary conditions is explored to assess the
robustness of these models. This work shows that the SGS variance of progress variable plays a
crucial role in the SGS reaction rate modelling and cannot be obtained using a simple algebraic
closure like that commonly used for a passive scalar. The use of strained flamelets to include
the flame stretching effects is not required when the variance is obtained from its transport
equation and the resolved turbulence contains predominant part of the turbulent kinetic energy.
Thus, it seems that SGS closure using unstrained flamelets model is robust and adequate for
wide range of turbulent premixed combustion conditions.

Table of contents
List of figures xvii
List of tables xxix
Nomenclature xxxi
1 Introduction 1
1.1 The role of turbulent combustion . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modelling of turbulent premixed combustion . . . . . . . . . . . . . . . . . 3
1.3 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Background on premixed combustion 7
2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Essential parameters in premixed flames . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Progress variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Flame thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Flame speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Turbulent premixed flames . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Turbulence-combustion interaction . . . . . . . . . . . . . . . . . . . 11
2.3.2 Combustion regimes . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Modelling for turbulent premixed combustion . . . . . . . . . . . . . . . . . 13
2.4.1 LES filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Favre filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.3 Governing equations for LES . . . . . . . . . . . . . . . . . . . . . 16
2.4.4 Chemistry modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.5 Modelling for turbulent-flame interaction . . . . . . . . . . . . . . . 18
2.4.6 Laminar flamelets approach: range of validity and limitations . . . . 24
2.5 Scalar dissipation rate modelling in turbulent premixed combustion . . . . . . 26
xiv Table of contents
2.5.1 SDR model for LES . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Numerical modelling 31
3.1 Residual stresses and common practices in LES . . . . . . . . . . . . . . . . 31
3.1.1 Eddy diffusivity models . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 One equation model for LES of premixed combustion . . . . . . . . 34
3.1.3 Relation between LES filter, grid spacing and numerical scheme . . . 34
3.1.4 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . 36
3.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Numerical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Filtered reaction rate modelling . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 Presumed PDF approaches . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.3 Estimation of sub-grid scale velocity . . . . . . . . . . . . . . . . . . 50
3.3.4 Dynamic formulations . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.5 Laminar flame computation . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Mathematical model and equations summary . . . . . . . . . . . . . . . . . 59
4 Validation cases 63
4.1 Propane-air mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Computational model . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Piloted jet burner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 Computational model . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Cold flow results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Bluff body burner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Computational model . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Cold flow results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Non piloted jet burner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Piloted flames - Algebraic closure 79
5.1 Experimental Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Specific numerical detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Results for static βc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.1 General flame features . . . . . . . . . . . . . . . . . . . . . . . . . 83
Table of contents xv
5.3.2 Sensitivities to SGS velocity modelling . . . . . . . . . . . . . . . . 85
5.3.3 Analysis of flame brush structure . . . . . . . . . . . . . . . . . . . 90
5.3.4 Analysis of SGS kinetic energy equation . . . . . . . . . . . . . . . 101
5.4 Assessment of dynamic procedure for βc . . . . . . . . . . . . . . . . . . . . 104
5.4.1 Flame structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.2 The PDF of βc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6 Piloted flames - Presumed PDF closures 121
6.1 Specific numerical detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Filtered reaction rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 Grid sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Unstrained vs Strained Flamelets . . . . . . . . . . . . . . . . . . . . . . . . 125
6.4.1 Major species comparison . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.2 Minor species comparison . . . . . . . . . . . . . . . . . . . . . . . 134
6.5 Analysis of Da∆ and Ka∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6 Comparison of static and dynamic βc . . . . . . . . . . . . . . . . . . . . . . 140
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Bluff body stabilised flame 143
7.1 Experimental case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 Flame and recirculation region . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.1 2% approach turbulence . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.2 24% approach turbulence . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3 Post-flame region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.4 Implication of local Damköhler and Karlovitz numbers . . . . . . . . . . . . 157
7.5 Species comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.5.1 Sensitivity to choice of progress variable . . . . . . . . . . . . . . . 170
7.6 Analysis of βc PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8 Non piloted jet flame 177
8.1 Experimental case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.2 Numerical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.3.1 Role of SGS velocity modelling . . . . . . . . . . . . . . . . . . . . 181
8.3.2 Comparison of statistics . . . . . . . . . . . . . . . . . . . . . . . . 184
xvi Table of contents
8.3.3 Bimodal of radial velocity . . . . . . . . . . . . . . . . . . . . . . . 187
8.3.4 Influence of numerical grid . . . . . . . . . . . . . . . . . . . . . . . 191
8.3.5 Influence of bimodal PDF on statistics . . . . . . . . . . . . . . . . . 193
9 Concluding remarks 199
9.1 Summary and final considerations . . . . . . . . . . . . . . . . . . . . . . . 199
9.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
References 205
Appendix A Database 221
List of figures
2.1 Typical behaviour of progress variable, θ , in freely propagating unstrained
laminar premixed flame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Regimes diagram for premixed turbulent combustion [180]. The subscript
δ indicates that Karlovitz and Damköhler numbers are computed using the
Zel’dovich thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Comparison of different terms of Eq. (3.30) for increasing values of u′+∆ =
u′∆/sL, and for (a) ∆= 4δth and (b) ∆= 8δth. . . . . . . . . . . . . . . . . . . 41
3.2 β -PDF of c, obtained using Eq. (3.34) for c˜= 0.01 and σ2c,sgs = 0.0089. Arrows
indicate presence of very large values. The approximated PDF, obtained using
Eq. (3.38), is shown with dashed line in the inset. . . . . . . . . . . . . . . . 44
3.3 Isolines of c˜, obtained using Eq. (3.33) (with c integrated instead of W˙ ) and
(3.38) (continuous lines), are shown on the plane c˜-g˜, where g˜ = σ2c,sgs/[c˜(1−
c˜)]. Isolines obtained using Eq. (3.39) are also shown (dashed lines). . . . . . 44
3.4 Scatter plot of computed σ2c,sgs and its modelled value using A ∆2 |∇c˜|2 with
A = 1 in the top row. The results for the revised model, A1
(
c˜Da∆+∆2|∇c˜|2
)
with A1 = 0.15, are shown in the bottom row. The first and second columns
are for two different grid sizes, of 1.5M and 4.2M cells, for F1 flame of [46]
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Scatter plot of computed σ2c,sgs and its modelled value using A ∆2 |∇c˜|2 with
A = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Comparisons of functions fI and fII in Eqs (3.64) and (3.65) respectively. The
dashed line represents fIFb, where Fb is the bridging function discussed
in [61, 77, 79]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 Strained flamelet configuration. . . . . . . . . . . . . . . . . . . . . . . . . . 56
xviii List of figures
3.8 Variation of consumption speed, sc (normalised with the unstrained flamelet
consumption speed, s0c), with characteristic strain rate for stoichiometric strained
flamelets. The strain rate is estimated as a≈ (|Ur|+ |Up|)/L. Three particular
values of strain rate are marked with circles. . . . . . . . . . . . . . . . . . . 57
3.9 Profiles of (a) axial velocity, U , and (b) progress variable, c, along the centreline
for the three values of strain rate marked in Fig. 3.8. x0 is the location at which
c = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.10 Profiles of (a) reaction rate, ω˙c, and (b) scalar dissipation rate, Nc, in c space
for the three strain rates marked in Fig. 3.8. . . . . . . . . . . . . . . . . . . 58
3.11 Profiles of (a) major and (b) minor species mass fractions in c space for the
three strain rates marked in Fig. 3.8. . . . . . . . . . . . . . . . . . . . . . . 58
3.12 Mass flux profiles in c space for the three strain rates marked in Fig. 3.8. The
arrows at c = 0 and c = 1, and the corresponding labels, indicate the mass
fluxes at the boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Sketch of the numerical domain used for the Sandia propane-air non-reacting
case in [214]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Mean axial velocity centreline profiles from LES, obtained using cases A ( ),
B ( ), C ( ) and D (•••) of Table 4.2, are compared with experimental
results [214] (symbols). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Radial profiles of mean (a) axial and (b) radial velocity from LES are compared
with experimental results. Legend is as for Fig. 4.2. . . . . . . . . . . . . . . 66
4.4 Radial profiles of (a) axial and (b) radial velocity variance from LES are
compared with experimental results (symbols). Legend is as for Fig. 4.2. . . . 66
4.5 Mixture-fraction half radius predictions from LES of case B ( ) and C ( )
of table 4.2 are compared with experimental data [214] (symbols). . . . . . . 67
4.6 A schematic of (a) the experimental [46] and (b) computational setup of piloted
stoichiometric methane-air Bunsen flames. . . . . . . . . . . . . . . . . . . . 68
4.7 Numerical grid used for piloted flame case [46] for 1.5M case. An axial slice
through the mid-plane (z = 0) is shown in (a) for a region of 0≤ x≤ 15D and
0≤ y≤ 5D. The grid distribution in a radial slice of size y× z = 5D×5D is
shown for axial locations of (b) x = 0, (c) 4.5D and (d) 10.5D. . . . . . . . . 70
4.8 Centerline profiles of (a) mean normalised axial velocity, ⟨U⟩/Ub, and (b)
turbulent kinetic energy, ⟨k⟩/U2b , computed using LES on 4.2M grid for cases
B ( ), C ( ), SM1 (+), SM2 (×) and SC ( ) of Table 4.2 are compared
with experimental results (circles) for the F2 cold flow configuration [46]. . . 71
List of figures xix
4.9 Comparison of measured [46] (symbols) and computed (lines) normalised
mean axial velocity and turbulent kinetic energy in cold flow of flame F2. The
radial variations are shown for 5 axial locations and two numerical grids, 1.5M
(solid line) and 4.2M (dotted line). . . . . . . . . . . . . . . . . . . . . . . . 72
4.10 A sketch of (a) the experimental burner configuration and (b) its numerical model. 73
4.11 Computational mesh in the mid-plane (z = 0) around the bluff body for (a)
1.8M and (b) 2.2M grids. An enlargement of the 2.2M grid is also shown in (c). 74
4.12 The variation of computed (lines) and measured [176] (symbols) ⟨U⟩/Uref
along the centreline for cold flow with (a) 2% and (b) 24% TI. The results
are shown for both 1.8M ( ) and 2.2M ( ) grid. The sensitivity to wall
functions is shown for 1.8M grid ( ). . . . . . . . . . . . . . . . . . . . . . 76
5.1 Combustion regime diagram showing conditions of flames F1, F2 and F3. . . 80
5.2 The spatial variation of normalised filtered temperature, T˜+, field in z = 0
plane for the three flames, obtained using algebraic model for ω˙ with βc = 7.5.
The result from 4.2M grid is shown only for F2 to depict the grid influence. . 83
5.3 The spatial variation of averaged and normalised temperature, ⟨T˜+⟩, in three
flames, obtained using algebraic model for ω˙ with βc = 7.5. . . . . . . . . . 84
5.4 Histogram of normalised filter width, ∆+, for (a) 1.5M and (b) 4.2M numerical
grids. Different domain sizes have been used: whole computational domain
( ), 0≤ x≤ 20D, 0≤ r ≤ 10D ( ), and 0≤ x≤ 11D, 0≤ r ≤ 2D ( ). . 85
5.5 Colour map of normalised filtered reaction rate, ω˙+, for flame F2, computed
for various u′∆ models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 Colour map of normalised mean reaction rate, ⟨ω˙+⟩, for flame F2, computed
for various u′∆ models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.7 Influence of SGS velocity modelling on averaged streamwise velocity, ⟨U⟩,
and turbulent kinetic energy, ⟨k⟩, obtained using algebraic model for ω˙ with
βc = 7.5. Symbols (◦) are experimental results [46], lines are from simulations
using Eq. 3.50 ( ), Eq. 3.51 ( ), Eq. 3.52 (•••) and Eq. 3.55 ( ). . . . . 88
5.8 Influence of SGS velocity modelling on averaged fuel mass fraction and nor-
malised temperature, obtained using algebraic model for ω˙ with βc = 7.5. The
legend is as in Fig. 5.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.9 Comparison of radial variation of normalised axial velocity, ⟨U⟩/Ub, in exper-
iments [46] (symbols) and simulations ( ) using Eqs. (2.19), (2.22) (with
βc = 7.5) and (3.51) on 1.5M grid. Results obtained using 4.2M grid (•••) are
shown only for F2. Previous RANS [122] ( ) and LES [240] ( ) results are
also shown for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xx List of figures
5.10 Comparison of measured [46] (◦ ◦ ◦) and computed ( ) normalised mean
TKE, ⟨k⟩/k0, variation with r/D, obtained using algebraic model for ω˙ with
βc = 7.5. The computational results are for the 1.5M grid. Results of the 4.2M
grid (•••) are shown only for F2. Previous results [53] (△), [122] ( ) and
[240] ( ) are shown for comparison. . . . . . . . . . . . . . . . . . . . . . 92
5.11 The mean fuel mass fraction computed on the 1.5M grid ( ), obtained using
algebraic model for ω˙ with βc = 7.5, is compared to the measurements (◦ ◦ ◦).
Previous results [53] (△), [56] (□), [122] ( ) and [240] ( ) are shown for
comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.12 The mean normalised temperature computed on the 1.5M grid ( ), ob-
tained using algebraic model for ω˙ with βc = 7.5, is compared to the measure-
ments (◦ ◦ ◦). Previous results [53] (△), [56] (□), [122] ( ) and [240] ( )
are shown for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.13 Comparison of mean mass fraction of O2: experimental data (◦ ◦ ◦), results
obtained using algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□),
and [122] ( ). The dash-dotted line ( ) for F1 at x/D = 2.5, 8.5 and 10.5
shows the value obtained without using Eq. (3.70). . . . . . . . . . . . . . . 96
5.14 Comparison of mean mass fraction of H2O: experimental data (◦ ◦ ◦), results
obtained using algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□),
[122] ( ) and [240] ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.15 Comparison of mean mass fraction of CO2: experimental data (◦ ◦ ◦), results
obtained using algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□),
and [122] ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.16 Comparison of mean mass fraction of OH: experimental data (◦ ◦ ◦), results ob-
tained using algebraic model for ω˙ with βc = 7.5 ( ), [56] (□) and [122] ( ). 99
5.17 Comparison of mean mass fraction of H2: experimental data (◦ ◦ ◦), results
obtained using algebraic model for ω˙ with βc = 7.5 ( ), and [122] ( ). . . 100
5.18 Comparison of mean mass fraction of CO2: experimental data (◦ ◦ ◦), results
obtained using algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□),
and [122] ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
List of figures xxi
5.19 Flame brush thicknesses, δt (mm), computed here using the radial variation
of averaged temperature (⋄) and fuel mass fraction (□) reported in [46] are
compared to δt values reported (◦) in [46] (top row). The values of δt obtained
from LES using algebraic model for ω˙ with βc = 7.5 and SGS velocity model
in Eq. (3.51) (+) and ksgs transport equation (×) on 1.5M grid are compared
with experimental data [46] (⋄) and past RANS [122] (•) and LES [240] (△)
studies in the bottom row. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.20 Radial profiles of mean normalised velocity, ⟨U⟩/Ub, normalised TKE, ⟨k⟩/k0,
methane mass fraction, ⟨CH4⟩ and normalised temperature, ⟨T+⟩, obtained
using algebraic model for ω˙ with βc = 7.5 and models K1 ( ), K2 ( ) and
K3 (•••) of table 5.1, are compared with LES results obtained using Eqs. (2.22)
and (3.51) ( ) and with experimental data [46] (symbols) for flame F2 and
1.5M grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.21 The spatial variation of mean progress variable field for flame F1 obtained
using algebraic model for ω˙ with static and dynamic approaches for βc. This
comparison is shown for both 1.5M and 4.2M grids. . . . . . . . . . . . . . . 105
5.22 Comparison of measured [46] (symbols) and computed (lines) radial variation
of normalised axial velocity, ⟨U⟩/Ub, obtained using algebraic model for ω˙
with dynamic ( , •••) and static ( , •••) βc models for flames F1 and F3.
The results are shown for 1.5M (solid lines) and 4.2M (dotted) grids. . . . . . 106
5.23 Comparison of measured [46] (symbols) and computed (lines) radial variation
of normalised mean turbulent kinetic energy, ⟨k⟩/k0. The legend is as in Fig. 5.22.108
5.24 Comparison of measured [46] (symbols) and computed (lines) radial variation
of mean fuel mass fraction. The legend is as in Fig. 5.22. The dash-dotted line
is for the power-law based model in Eqs. (3.60) and (3.61) for 4.2M grid. . . 109
5.25 Measured [46] (symbols) and computed (lines) radial variations of normalised
mean temperature are compared above. The legend is as in Fig. 5.22. . . . . . 110
5.26 Comparison of measured [46] (symbols) and computed (lines) mean O2 mass
fractions. The legend is as in Fig. 5.22. . . . . . . . . . . . . . . . . . . . . . 112
5.27 Comparison of measured [46] (symbols) and computed (lines) mean H2O mass
fractions. The legend is as in Fig. 5.22. . . . . . . . . . . . . . . . . . . . . . 113
5.28 Comparison of measured [46] (symbols) and computed (lines) mean CO2 mass
fractions. The legend is as in Fig. 5.22. . . . . . . . . . . . . . . . . . . . . . 114
5.29 Comparison of measured [46] (symbols) and computed (lines) mean OH mass
fractions. The legend is as in Fig. 5.22. . . . . . . . . . . . . . . . . . . . . . 115
xxii List of figures
5.30 Comparison of measured [46] (symbols) and computed (lines) mean H2 mass
fractions. The legend is as in Fig. 5.22. . . . . . . . . . . . . . . . . . . . . . 116
5.31 Comparison of measured [46] (symbols) and computed (lines) mean CO mass
fractions. The legend is as in Fig. 5.22. . . . . . . . . . . . . . . . . . . . . . 117
5.32 The PDF of βc obtained from LES of flames (a) F1 and (b) F3 using algebraic
model for ω˙ with dynamic computation of βc with 1.5M grid. The average
value and the lower limit imposed by Eq. (3.59) are also shown. . . . . . . . 118
5.33 The recomputed (⋄) and LES values of flame brush thickness, δt , obtained
using algebraic model for ω˙ with dynamic computation of βc, are compared
for various cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.1 Typical variation of normalised filtered reaction rate in unstrained and strained
flamelets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2 Comparison of measured [46] (symbols) and computed (lines) radial variation
of ⟨U⟩/Ub for F1 and F3 flames. Unstrained flamelet result is shown for
1.5M ( ) and 4.2M (•••) grids, and strained flamelets result is shown for
1.5M ( ) and 4.2M ( ) grids. The results of 4.2M grid is shown only for
the F1 flame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3 Comparison of measured [46] (symbols) and computed (lines) radial variation
of ⟨k⟩/k0 for F1 and F3 flames. The legend is as in Fig. 6.2. . . . . . . . . . . 125
6.4 Comparison of measured [46] (◦ ◦ ◦) and computed mean fuel mass fraction.
The computational results are shown for the 1.5M grid using unstrained ( )
and strained ( ) flamelets. Previous results [53] (△), [56] (□), [122] ( )
and [240] (•••) are shown for comparison. . . . . . . . . . . . . . . . . . . . 126
6.5 The normalised mean temperature computed on the 1.5M grid using un-
strained ( ) and strained ( ) flamelets is compared to the measurements [46]
(◦ ◦ ◦). Previous results [53] (△), [56] (□), [122] ( ) and [240] (•••) are shown
for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.6 Spatial variation of log(100ω˙+) in flame F3 obtained at an arbitrarily chosen
time, t1, using (a) unstrained and (b) strained flamelets with 1.5M grid. These
results at 20 ms later are shown in (c) and (d). The contours are shown for
ω˙+ > 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 Spatial variation of log(100ω˙+) in flame F1 obtained at an arbitrarily chosen
time, t1, using (a) unstrained and (b) strained flamelets with 1.5M grid. These
results at 20 ms later are shown in (c) and (d). The contours are shown for
ω˙+ > 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
List of figures xxiii
6.8 Comparison of measured [46] and computed H2O mass fraction. The legend is
as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.9 Comparison of measured [46] and computed O2 mass fraction. The legend is
as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.10 Comparison of measured [46] and computed CO2 mass fraction. The legend is
as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.11 Comparison of measured [46] and computed OH mass fraction. The legend is
as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.12 Comparison of measured [46] and computed H2 mass fraction. The legend is
as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.13 Comparison of measured [46] and computed H2 mass fraction. The legend is
as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.14 PDFs of Da∆ and Ka∆ from flames F1 (top row) and F3 (bottom row). The
results are shown for unstrained ( ) and strained ( ) flamelets. . . . . . . 138
6.15 The PDF of βc from flame (a) F1 and (b) F3 obtained using unstrained ( )
and strained ( ) flamelets with 1.5M grid. . . . . . . . . . . . . . . . . . . 139
6.16 Comparison of measured [46] (◦ ◦ ◦) and computed mean fuel mass fraction.
The computational results are shown for the 1.5M grid using unstrained flamelet
model with dynamic ( ) and static ( ) formulation for βc. . . . . . . . . 141
7.1 Histograms of ∆+ = ∆/δth for (a) 1.8M and (b) 2.2M grids are shown for the
combustor region 0≤ x/D≤ 6 and a fraction of it for r/D< 0.5. . . . . . . 144
7.2 Stream lines of (a) filtered and (b) averaged velocity fields along with contours
of (filtered and averaged) log10(1000 ω˙
+
), where ω˙+ = ω˙δth/(ρusL). The
results are on the mid-plane for unstrained flamelets model and 2% TI. The
contour of log10(1000Da∆) is shown in (c). The three regime regions, R1, R2
and R3, are separated in the figures using dashed lines. . . . . . . . . . . . . 146
7.3 Centreline variation of (a) ⟨U⟩/Uref, (b) ⟨T+⟩, and (c)
√
σ2T for 2% TI. The
results with algebraic model on 1.8M ( ) and 2.2M ( ) grids and with
unstrained flamelets model on 1.8M ( ) and 2.2M ( ) grids are compared
with experimental data [176, 178] (symbols). . . . . . . . . . . . . . . . . . 147
7.4 Radial variation of ⟨T+⟩ and
√
σ2T for 2% TI case with algebraic model using
1.8M ( ) and 2.2M ( ) grids and with unstrained flamelets model using
1.8M ( ) and 2.2M ( ) grids are compared with experimental data [176,
178] (symbols). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
xxiv List of figures
7.5 Radial profiles of (a) rms and (b) mean velocities, obtained using the algebraic
model on 1.8M grid ( ) and unstrained flamelet model on 2.2M grid ( ),
are compared with experimental data (symbols) for 24% TI at x/D = 0. . . . 150
7.6 Contours of filtered normalised reaction rate ω˙+ = ω˙δth/(ρusL) at an arbitrary
chosen time. The results are on the mid-plane for unstrained flamelet and
algebraic models with 24% TI. Isolines having ω˙+ = 0.1 are also shown. . . 152
7.7 Centreline variation of (a) ⟨U⟩/Uref and (b) ⟨T+⟩ obtained with algebraic model
using 1.8M ( ) grid, and with unstrained flamelet model using 2.2M ( )
grid are compared to experimental data [176, 178] (symbols) for 24% TI case.
The influence of ⟨σ2T,SGS⟩ on ⟨T+⟩ is shown for the unstrained flamelet ( )
and algebraic ( ) models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.8 Scatter plots of ⟨c˜⟩ and ⟨c⟩ for unstrained flamelet model with 24% TI. The
Reynolds average is obtained using Eq. (3.18) (a) without and (b) with SGS
variance, ⟨σ2c,sgs⟩. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.9 Scatter plot of mean progress variable and total variance, ⟨σ2c ⟩ (sum of resolved
and SGS parts), for unstrained flamelet model with 24% TI. Colours represent
the axial locations, x/D, for the data. The maximum possible variance, ⟨c˜⟩(1−
⟨c˜⟩) is also shown ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.10 Radial variation of ⟨U⟩/Uref and ⟨Vr⟩/Uref, obtained with algebraic model
using 1.8M ( ) grid and with unstrained flamelets model using 2.2M ( )
grid are compared with measurements [176, 178] (symbols) for 24% TI within
the recirculation region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.11 Radial variation of normalised averaged temperature and its rms, obtained
using the algebraic model with 1.8M ( ) grid and unstrained flamelet model
with 2.2M ( ) grid are compared with measurements [168, 169, 176, 178]
(symbols) for 24% TI case within the recirculation region. The influence of
⟨σ2T,sgs⟩ on ⟨T+⟩ is shown for the unstrained flamelet ( ) and algebraic ( )
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.12 Radial variation of ⟨U⟩/Uref and ⟨Vr⟩/Uref are compared with measurements [176,
178] for 24% TI in the post-flame region. The legend is as in Fig. 7.10. . . . 158
7.13 Radial variation of mean normalised temperature and its rms are compared
with measurements [168, 169] for 24% TI in the post-flame region. The legend
is as in Fig. 7.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
List of figures xxv
7.14 PDF of (a) SGS Damköhler, Da∆, and (b) Karlovitz, Ka∆, numbers in the
combusting region are shown for the unstrained flamelet model on 2.2M grid
with 2% ( ) and 24% ( ) TI, and algebraic model on 1.8M grid with 24%
TI ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.15 PDFs of c˜ from LES using algebraic model on 1.8M grid ( ) at x/D = 0.6
(top) and x/D = 2.0 (bottom) and for three radial positions along the shear are
compared with β -PDFs computed using ⟨c˜⟩ and resolved variance ⟨σ2c,res⟩=
⟨c˜2⟩−⟨c˜⟩2 ( ), and using ⟨c˜⟩ and mean SGS variance, ⟨σ2c,sgs⟩ ( ). . . . . 161
7.16 Radial profiles of CH4 mass fraction, obtained using the algebraic model with
1.8M grid ( ) and the unstrained flamelet model on 2.2M grid using Eq. (7.2)
( ), are compared with experimental data [168, 169] (symbols) for (a) six
axial locations within the recirculation region and (b) three locations in the
post-flame region. Results obtained using Eq. (7.4) ( ) are also shown. . . 162
7.17 Radial profiles of O2 mass fraction are compared with experimental data [168,
169] for six axial locations within the recirculation region (top) and three
locations in the post-flame region (bottom). The legend is as in Fig. 7.16. . . 164
7.18 Radial profiles of CO2 mass fraction are compared with experimental data [168,
169] for six axial locations within the recirculation region (top) and three
locations in the post-flame region (bottom). The legend is as in Fig. 7.16. . . 165
7.19 Radial profiles of H2O mass fraction are compared with experimental data [168,
169] for six axial locations within the recirculation region (top) and three
locations in the post-flame region (bottom). The legend is as in Fig. 7.16. . . 166
7.20 Radial profiles of CO mass fraction are compared with experimental data [168,
169] for six axial locations within the recirculation region (top) and three
locations in the post-flame region (bottom). The legend is as in Fig. 7.16. . . 167
7.21 Radial profiles of H2 mass fraction are compared with experimental data [168,
169] for six axial locations within the recirculation region (top) and three
locations in the post-flame region (bottom). The legend is as in Fig. 7.16. . . 168
7.22 Radial profiles of OH mass fraction are compared with experimental data [168,
169] for six axial locations within the recirculation region (top) and three
locations in the post-flame region (bottom). The legend is as in Fig. 7.16. . . 169
7.23 Mass fractions variation of major and minor species using different progress
variable definitions for a stoichiometric methane-air flamelet. . . . . . . . . 170
7.24 Enlargement of OH mass fraction in Fig. 7.23 on the burnt side using different
definitions of c: c= 1−YCH4/Y uCH4 ( ), c=YH2O/Y bH2O ( ) and c= (YCO2 +
YCO)/(Y bCO2 +Y
b
CO) ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
xxvi List of figures
7.25 Radial profiles of CH4 mass fraction, obtained using the unstrained flamelets
model with 2.2M grid using Eq. (7.2) for three different definition of c, c =
1−YCH4/Y uCH4 ( ), c =YH2O/Y bH2O ( ) and c = (YCO2 +YCO)/(Y bCO2 +Y bCO)
( ), are compared with experimental data [168, 169] (symbols) for six axial
locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.26 Radial profiles of O2 mass fraction from LES, obtained using three different
definitions of the progress variable, are compared with experimental data [168,
169] for six axial locations. The legend is as for Fig. 7.25. . . . . . . . . . . 173
7.27 Radial profiles of CO2 mass fraction from LES, obtained using three different
definitions of the progress variable, are compared with experimental data [168,
169] for six axial locations. The legend is as for Fig. 7.25. . . . . . . . . . . 173
7.28 Radial profiles of H2O mass fraction from LES, obtained using three different
definitions of the progress variable, are compared with experimental data [168,
169] for six axial locations. The legend is as for Fig. 7.25. . . . . . . . . . . 174
7.29 Radial profiles of CO mass fraction from LES, obtained using three different
definitions of the progress variable, are compared with experimental data [168,
169] for six axial locations. The legend is as for Fig. 7.25. . . . . . . . . . . 174
7.30 Radial profiles of H2 mass fraction from LES, obtained using three different
definitions of the progress variable, are compared with experimental data [168,
169] for six axial locations. The legend is as for Fig. 7.25. . . . . . . . . . . 175
7.31 Radial profiles of OH mass fraction from LES, obtained using three different
definitions of the progress variable, are compared with experimental data [168,
169] for six axial locations. The legend is as for Fig. 7.25. . . . . . . . . . . 175
7.32 PDF of βc for algebraic ( ) and unstrained flamelet ( ) models and 24%
TI. The realisability limit, 2/(2Cm−1), is also marked. . . . . . . . . . . . . 176
8.1 A direct photograph (a) and a schematic (b) of the lean propane-air flame of [76].178
8.2 Numerical domain for the preliminary RANS. . . . . . . . . . . . . . . . . . 181
8.3 Profiles of (a) mean axial velocity and (b) mean temperature at the jet exit,
computed using RANS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.4 Variation of ε˜+ = ε˜δth/sL, with u′+∆ = u
′
∆/sL, for ∆/δth = 1.4. . . . . . . . . 182
8.5 Comparison of measured (symbols) and computed axial velocity along the
centreline; Eq. 3.50 (dashed line), Eq. 3.51 (continuos line) and Eq. 3.56 (dash-
dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.6 Scatter plot of normalised sub-grid SDR versus progress variable, obtained
using models of Eq. (3.50) (Lilly), Eq. (3.51) (Pope) and Eq. (3.56) (Colin et
al.) for u′∆. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
List of figures xxvii
8.7 Contours of ⟨u′∗∆ ⟩/⟨u′∗∆ ⟩max in the mid-plane. The values of ⟨u′∗∆ ⟩max are 0.15,
0.80 and 2.03m/s respectively for Lilly, Pope and Colin et al. models. . . . . 185
8.8 Contours of (a) filtered, U˜ , and (b) mean, ⟨U⟩, axial velocity, obtained using
Eq. (3.51) for SGS velocity. The mean progress variable isoline ⟨c⟩= 0.5, is
also shown (black line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.9 Comparison of measured and computed centreline variation of radial rms
velocity. The legend is as in Fig. 8.5. . . . . . . . . . . . . . . . . . . . . . . 187
8.10 Radial variations of mean and rms velocities in axial and radial directions. The
rms of tangential velocity is also shown. The legend is as in Fig. 8.5. . . . . . 188
8.11 Comparison of measured and computed Reynolds- and Favre-averaged progress
variable variation. The legend is as in Fig. 8.5. Error bars are also shown. . . 189
8.12 Resolved and SGS variance of the progress variable. The legend is as in Fig. 8.5
and the dotted line is for 23M grid. . . . . . . . . . . . . . . . . . . . . . . . 190
8.13 The computed isolines of ⟨c˜⟩= 0.5 are compared to the experimental result in
(a) along with the uncertainty for the measurements. A comparison of flame
brush thickness is shown in (b). The legend is as in Fig. 8.5. The dotted line is
for 23M grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.14 Normalised histograms of [76] for three velocity components at x = 50 mm for
various radial locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.15 Normalised histograms from LES for the three components of velocity at
x = 50mm and r = 10.2mm for various values of δsamp. The legend is as in
Fig. 8.5. The result for 23M grid (•••) is also shown for vr for comparison. . . 193
8.16 RMS and SGS velocity profiles for 5.4M (continuous line) and 23M (dotted
line) grids are compared to the experimental data (symbols) at x = 50 mm. . . 194
8.17 The PDF of (a) measured and (b) computed radial velocity component vr at
x = 50 mm. The modelled values using two Gaussians are shown in (b) for
the mean and variance of unburnt (c˜≤ 0.05) and burnt (c˜≥ 0.95) mixtures in
LES data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.18 The PDF (continuous line) obtained with samples collected from a single
numerical cell (δsamp < ∆) at x = 50 mm and r = 10.2 mm is compared to a
Gaussian model with the same mean and standard deviation of σL+σR (dashed
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
xxviii List of figures
8.19 Radial variation of measured (open circle) and computed (lines) rms of radial
velocity at x = 50 mm. Results are shown for 5.4M (left) and 23M (right) grids.
The effect of δsamp is shown for δsamp < ∆ (continuous line), δsamp = 3 mm for
5.4M or 2.4 mm for 23M grids (continuous line with squares); δsamp = 6 mm
for 5.4M or 5 mm for 23M grids (continuous line with triangles). The dashed
lines represent the corresponding curves when the v′r,∆ is included. . . . . . . 196
List of tables
4.1 Geometrical and physical parameters for the Sandia propane-air non-reacting
case in [214]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 List of simulations for the Sandia propane-air non-reacting case in [214]. . . . 65
4.3 Geometrical and physical parameters for the F2 cold case in [46]. . . . . . . . 69
4.4 Simulation parameters for the LES of F2 cold flow configuration in [46]. Cs is
the Smagorinsky constant appearing in Eq. (3.3). . . . . . . . . . . . . . . . 71
4.5 Geometrical and physical parameters for the bluff-body configuration [176–178]. 75
5.1 SGS kinetic energy models used for testing. . . . . . . . . . . . . . . . . . . 103
8.1 Values of β ′c obtained using the SGS velocity models in Eqs. (3.50), (3.51) and
(3.56). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
A.1 Simulations list for the piloted flames case from Chen et al. [46] - algebraic
model (AM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
A.2 Simulations list for the piloted flames case from Chen et al. [46] - unstrained
flamelets model (UFM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
A.3 Simulations list for the piloted flames case from Chen et al. [46] - strained
flamelets model (SFM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
A.4 Simulations list for the bluff body case [168, 169]. . . . . . . . . . . . . . . . 225
A.5 Simulations list for the low-turbulence flame case from Furukawa et al. [76]. . 226

Nomenclature
Roman Symbols
c Progress or regress variable
Cm Bray-model constant
Cp Heat capacity at constant pressure
Cs Smagorinsky constant
D Diameter, total derivative
D Diffusivity
h Total enthalpy (sensible + chemical)
∆h0f Enthalpy of formation
k Turbulent kinetic energy
kr Residual kinetic energy
ksgs Sub-grid scale kinetic energy
Lr Length of recirculation zone
Nc Scalar dissipation rate of c
p Pressure
r Radial coordinate
R0 Universal constant of ideal gases
sL Laminar flame speed
xxxii Nomenclature
sT Turbulent flame speed
Si j Strain tensor
t Time
U Axial velocity
u Velocity vector
u′ Root mean square axial velocity
u′∆ Sub-grid scale velocity
V Volume
v′r Root mean square radial velocity
v′t Root mean square tangential velocity
Vr Radial velocity
Vt Tangential velocity
W˙ Reaction rate in [s−1]
W Molecular weight
x, y, z Spatial coordinates
Yk Mass fraction of species k
Z Passive marker or normalised mixture fraction
Non-dimensional Parameters
Da Damköhler number
Ka Karlovitz number
Le Lewis number
Re Reynolds number
ReT Turbulent Reynolds number
ScT Turbulent Schmidt number
Nomenclature xxxiii
Greek Symbols
α Fractal model exponent
∆ Filter width
δ Zel’dovich flame thickness
δ 0L Laminar flame thickness
δt Flame brush
δsamp Sampling width
δth Thermal flame thickness
ω˙ Reaction rate in [Kgm−3 s−1]
ηk Kolmogorov’s scale
Λ Integral length scale
λ Thermal diffusivity
µ Dynamic viscosity
µT Dynamic residual eddy-viscosity
νT Kinematic residual eddy-viscosity
φ Equivalence ratio
Π Pressure dilatation term in ksgs transport equation
ρ Density
σ Standard deviation
σ2c Total variance of c
σ2c,res Resolved variance of c
σ2c,sgs Sub-grid variance of c
τ Heat release parameter
τRi j Residual stress tensor
xxxiv Nomenclature
τri j Anisotropic part of residual-stress tensor
τi j Stress tensor
ε Dissipation
ε˜c Sub-grid scale scalar dissipation rate of c
ξ Mixture fraction
Superscripts
Φ′ Reynolds residual of Φ
Φ′′ Favre residual of Φ
Φ+ Normalised Φ
Subscripts
∆ Sub-grid scale
b Fully burnt condition
f Fuel
L Laminar
u Fully unburnt condition
sgs Sub-grid scale
mix Mixture
Other Symbols
⟨Φ⟩ Reynolds average or Favre average of Φ
Φ Filtered Φ
Φ̂ Test-filtered Φ
Φ˜ Favre-filtered Φ
Acronyms / Abbreviations
CFD Computational fluid dynamics
Nomenclature xxxv
CFL Courant-Friedrichs-Lewy number
DNS Direct numerical simulation
LES Large eddy simulation
LHS Left hand side
PDF Probability density function
RANS Reynolds averaged Navier-Stokes
RHS Right hand side
SDR Scalar dissipation rate
SGS Sub-grid scale
TI Turbulent intensity
TKE Turbulent kinetic energy
URANS Unsteady-RANS

Chapter 1
Introduction
1.1 The role of turbulent combustion
The world’s energy demand is increasing at a rate of about 2% p.a., and this demand is estimated
to have increased to 37% by 2040 [14]. An investment of around 900 billion dollars per year in
the energy sector is needed by 2030 to meet this projected energy demand. The BP 2014 annual
report shows that 87% of the total energy demand is met by fossil fuels and that this percentage
has remained almost unaltered in the last decade, while renewable energies are growing in
place of nuclear energy, which faces an uncertain future. For this reason one can expect that
combustion will remain as the main source of energy for several decades. Reserves of fossil
fuels, specifically coal, oil and gas, amount in fact to 892 billion tonnes, 1688 million barrels
and 186 trillion cubic meters respectively, and are estimated to run out in 113, 53 and 55 years
respectively, using the current rates of supply. The recent growth in the North American oil
supply also helps to maintain a stability. The total fossil fuel energy supply is divided almost
equivalently between the fossil sources above, with low-carbon sources to take the place of coal
by 2040 [14]. The choice of a particular fuel depends on the particular requirement in terms
of total energy, power, emissions etc., with applications ranging from power generation for
stationary and mobile power plants to the transportation sector. However, among the fossil fuels,
the use of natural gas is seeing the fastest growth, also encouraged by increasing flexibility in
the global trade of liquefied natural gas. In contrast to oil, gas production is increasing almost
everywhere, and is expected to become the largest fuel source by 2030. This is also because
future coal use is constrained by measures to reduce emissions, despite it is having the biggest
reserves [14].
Due to the continuing dominance of combustion in meeting the world’s energy demand, con-
cern over emissions has recently been heightened. Emissions can be categorised in ‘greenhouse’
gases and air pollution gases. The first category involves mainly CO2, and to a lesser extent
2 Introduction
N2O and CH4, and takes this name because these gases contribute to the so called ‘greenhouse’
effect. The following paragraphs report data from the 2014 BP annual report and world energy
investment outlook [14]. On the current energy path, CO2 emissions are expected to contribute
to the predicted long-term global increase in temperature of 3.6◦, and the global objectives are
to restrict this rise to 2◦. Since CO2 is a main combustion product, it is proportional to the
amount of fuel used; thus its emissions can be reduced only with fuel economy improvements.
In Europe, average emission targets exist of 130g/Km for new passengers cars, and 175g/Km
for light commercial vehicles, to be reduced to 95g/Km and 147g/Km respectively by 2020.
In the US, greenhouse gas reduction target is of 30%, fixed for 2016. Other countries are
also active in producing emission reduction policies, including Japan and South Korea. The
development of technologically advanced and more efficient combustion devices assumes a
fundamental role in this picture. As CO2 emissions are proportional to the amount of fuel
used, increased efficiency will allow to energy demands to be met with less fuel, and thus less
emissions.
Different considerations exist for air pollution. Air pollution involves emissions of toxic,
intermediate combustion species, mainly NOx and CO, but also particulate matter, HCHO and
other hydrocarbons. Regulations exist for both the energy production and transportation sector,
and depend on the weight of the machine. The current European standard is the Euro 6, while
in the US emissions standards are controlled by the EPA, which has adopted the California
emissions standards. Other standards exist in Japan and South Korea. The production of
intermediate species is intrinsically related to the completeness, and thus the efficiency, of
the the combustion process. In this scenario increasingly efficient, environmental-friendly
combustion devices are needed to meet both energy demand and pollution regulations. Modern
engines can operate in two main different modes: non-premixed combustion, in which fuel
and oxidiser enter the combustion chamber separately, and premixed combustion, in which
reactants are homogeneously mixed before being burned. The combustion in both cases is
generally turbulent, as turbulence enhances the mixing between fuel and oxidiser, resulting in
higher efficiencies.
Due to concern for the environment, of these two combustion modes premixed combustion
has recently captured industry interest, as high efficiency and low emissions can be achieved
simultaneously. In fact, in homogenous mixtures, the nature of the reactants controls the thermo-
chemistry of the process and hence the maximum temperature attainable. Since emissions of
CO and NOx can be seen proportional to the maximum temperature attainable [233], premixed
combustion can effectively control the level of emissions. The maximum temperature of the
system can be reduced by making the reactants composition lean, i.e. with an excess of oxidiser.
However, lean premixed combustion is prone to instabilities, thus additional effort is required
1.2 Modelling of turbulent premixed combustion 3
to make a system operate in this condition. Moreover, in practical combustors the reactants are
never fully premixed because the time scales of the mixing are usually larger than reactants
resident time, and by measures taken to avoid potential flashbacks. The combustion systems
operate in between turbulent premixed and non-premixed modes, a condition which is referred
to as partially premixed combustion, posing additional challenges.
For all of these reasons, the development of modern, advanced lean premixed combustion
devices plays a key role in the whole energy sector, and its research is a priority for short and
long term goals.
1.2 Modelling of turbulent premixed combustion
Computational Fluid Dynamics (CFD) is quickly assuming the role of a complementary tool
for the design of modern, efficient and environmentally-friendly combustion devices. Direct
Numerical Simulation (DNS) for combustion is still unaffordable for complex geometries
involving practical flows and is used as an investigation tool to gather insight for development
of closures for Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES)
methodologies. These methodologies will be discussed in detail in later chapters. RANS and
unsteady-RANS (URANS) techniques are limited in capturing highly unsteady phenomena
such as extinctions and blow-off, which are of high interest to industry. With the advent of high-
performing computers, LES is becoming attractive as it can capture transient phenomena, but
the computational cost can be quite high if the closures involved with this methodology are not
robust in mimicking the relevant physics. The LES methodology and the associated modelling
are described in a number of earlier studies for non-reacting (for example [196]) and reacting
(for example [191]) flows, and it is seen here as a higher fidelity approach compared to URANS
as expressed in classical views for LES [54, 139, 140, 216, 217] rather than as a degenerate
from DNS. The LES studies on turbulent combustion have been reviewed in [85, 185].
In LES, the larger scales obtained using a low-pass filter are solved down to a cut-off scale,
∆, and the remaining sub-grid scales (SGS) are modelled. This operation is done through the
use of a spatial filter of a certain shape and width. Large and small scales influence each other
depending on filter size and SGS modelling, which are closely related [196]. Thus, estimating
the validity of SGS closure can become a difficult task. Furthermore, the LES solutions are
sensitive to grid anisotropies and boundary conditions used.
In LES of premixed combustion, the flame thickness is usually thinner than the smallest
scale solved [226], hence combustion typically occurs at SGS level and must be entirely
modelled. The relative importance of the SGS modelling is expected to diminish as the
filter size approaches zero, which typically dictates very fine grids leading to formidably
4 Introduction
expensive calculations. This is unattractive for industry applications. The importance of the
SGS modelling is thus fundamental when relatively coarser grids are employed, and robust,
accurate and computationally efficient models are required to bring LES to a design step.
The modelling used in this study relies on the assumption that the flame is thin and turbu-
lence does not modify its inner structure, thus the flame can be seen as an ensemble of laminar
one-dimensional structures (flamelets). Turbulence can be separated from thermochemistry
in this case, and the latter can be modelled using one equation for a progress variable. This
flamelets assumption, which will be discussed in next chapter, reduces the computational
time substantially, allowing the LES as a practical tool to use in the design cycle of ‘green’
combustion equipments.
1.3 Aims and objectives
Many flamelet-based approaches have been proposed in the past, and are discussed in the
next chapter. The aim of this study is to analyse and assess the strengths and weaknesses
of flamelets modelling based on presumed probability density function (PDF) for turbulent
premixed combustion, and propose sensible modifications to it if required. The presumed PDF
requires variance, which in turn requires modelling for its dissipation rate, which is explored
for LES by extending its modelling for RANS. Thus, the specific objectives of this study are
1. To implement, test and validate presumed PDF based flamelets for LES of premixed
combustion.
2. To study their applicability to various combustion regimes.
3. To identify and develop further improvement, if required. Specifically, the need to include
the stretch effects for sub-grid scale reaction rate closure for LES.
4. To derive and study a transport equation for the SGS variance of reaction progress
variable, to be defined in the next chapter, for premixed combustion.
5. To identify ways to minimise arbitrary tuneable parameters and thereby evaluate various
sub-models and their physical consistencies.
1.4 Outline
The outline of the thesis is as follows. Chapter 2 provides an overview of turbulent premixed
combustion. The main characteristics of this mode of combustion are briefly introduced, and
1.4 Outline 5
various existing modelling approaches are discussed. Particular emphasis will be given to
flamelet models. The scalar dissipation rate and its role for modelling in RANS and LES is
also discussed. Chapter 3 presents the numerical procedures and techniques used for this study.
General sub-grid modelling and discretisation techniques used for this study are discussed in the
first part of this chapter, and the numerical procedures associated with the flamelets modelling
are discussed in the second part. Chapter 4 details validations of the numerical procedures
and sensitivity analyses. Cold flow simulations in various flame configurations considered are
performed to assess the validity of model setup. The results obtained with different models and
different configurations are presented in Chapters 5 to 8. Piloted jet flames belonging to the thin
reaction zones regime of turbulent combustion are used as benchmark test cases (Chapters 5
and 6) to analyse the various characteristics and limitations of models and sub-models used in
this work. A confined bluff-body stabilised flame lying between thin and distributed reaction
zones regimes is analysed in Chapter 7, with a particular focus on the importance of the SGS
variance. Chapter 8 focuses on low-Reynolds number flames. An overall discussion of these
three flame configurations is given in Chapter 9, which is followed by conclusions and future
work.

Chapter 2
Background on premixed combustion
Premixed combustion is a situation where fuel and oxidiser are mixed at molecular level
before the chemical reactions occur. This is a physico-chemically complex phenomenon and
introduces several challenges. The flame structure in premixed combustion is maintained by a
dynamic balance between reaction and diffusion of heat and mass [34]. Chemical reactions
occur within the reaction zone at a rate determined by the supply of reactants. The interaction
of diffusion and reaction with straining affects the rate of mixing and enhances the scalar
gradients, and these effects become particularly relevant in LES [107]. The reaction-diffusion
coupling may lead to instabilities [152] and the strong heat release may produce other effects
such as buoyancy [197], posing additional challenges to the modelling. A brief review of the
progresses on the topic from literature is provided in this chapter.
2.1 Governing equations
Combustion gives additional complexity in the treatment of the governing equation in respect to
non-reacting problems. Species react and their rate of reaction must be modelled; species and
heat coefficients vary within the solution; transport coefficients depend on the species [191].
Because of this complexity, a number of assumptions is required. The fluid mixture is con-
sidered to be Newtonian and obeys perfect gas law. Neither Soret effects (species diffusion
coefficients do not depend on temperature gradients), nor Dufour effects (heat diffusivity does
not depend on gradients of species mass fraction) are considered, and Lewis number is taken to
be constant.
In premixed combustion the flame front advances into the reactants field at speed sL. The
interest in this study is for deflagration flames, in which this speed is much smaller than the
speed of sound and the pressure is almost constant across the flame [242]. Thus, effects of
pressure changes can be neglected in the energy equation. Despite the Mach number is low,
8 Background on premixed combustion
the density, ρ , is not constant, as there is strong heat release across the flame. Neglecting
volume forces and viscous heating, and in absence of external heat sources, the resulting set of
instantaneous equations, using Einstein notation, is as follows (see for example [191]):
• conservation of mass
∂ρ
∂ t
+
∂ρui
∂xi
= 0 (2.1)
where ui is the velocity component in spacial direction xi;
• conservation of momentum
∂ρu j
∂ t
+
∂ρuiu j
∂xi
=− ∂ p
∂x j
+
∂τi j
∂xi
(2.2)
where p is the pressure and
τi j = µ
(
∂ui
∂x j
+
∂u j
∂xi
)
− 2
3
µ
∂uk
∂xk
δi j (2.3)
is the viscous tensor, µ and δi j being the kinematic viscosity and the Kronecker delta
respectively;
• conservation of species
∂ρYk
∂ t
+
∂
∂xi
[ρ(ui+Vk,i)Yk] = ω˙k k = 1, ...,N (2.4)
where Yk is the mass fraction of the k-th species, N is the number of species and Vk,i is
the diffusion velocity in direction xi. The chemical source term of the k-th species, ω˙k, is
usually expressed in terms of forward and backward rate coefficients, which in turn are
expressed as function of temperature using Arrhenius equation, see for example [242]
for more details. The diffusion velocity can be modelled using the Hirschfelder et al.
approximation [97]. Thus, one can write VkXk =−Dk∇Xk, where Xk is the mole fraction
of species k, Dk is the diffusion coefficient of species k, which is related to the thermal
diffusivity, Dth, and the Lewis number of species k, Lek, as LekDk =Dth;
• conservation of total enthalpy (sum of sensible enthalpy, hs, and enthalpy of formation,
∆h0f )
∂ρh
∂ t
+
∂ρuih
∂xi
=−∂qi
∂xi
(2.5)
2.2 Essential parameters in premixed flames 9
where the heat flux in direction xi is modelled using Fourier’s law as
qi =−λ ∂T∂xi (2.6)
and λ and T are the thermal conductivity and temperature respectively. The molecular
diffusion term has been neglected in the above equation as it would cancel with other
terms in the full expression of the enthalpy equation when Lek = 1. This full expression
can be found for example in [191];
• state equation
p = ρR0T
N
∑
k=1
Yk
Wk
(2.7)
where R0 is the universal gas constant and Wk is the molecular mass of the k-th species.
Additional information on these instantaneous equations and their modelling in laminar pre-
mixed combustion can be found in [191]. The parameters of laminar premixed flames which
are relevant for this study are discussed instead in the next section.
2.2 Essential parameters in premixed flames
2.2.1 Progress variable
Species and temperature fields are not independent [191] in premixed combustion. One can
express fuel reaction rate, temperature, density and species, as function of a progress variable,
which takes a value of zero and one in fresh reactants and burnt products respectively. This
progress variable can be defined either in terms of temperature, θ = (T −Tu)/(Tb−Tu), or in
terms of a suitable mass fraction, c = (Yk−Yk,u)/(Yk,b−Yk,u), where subscripts u and b refer
to fully unburnt and burnt conditions respectively. These two definitions are equivalent if the
Lewis number is unitary [191]. Choice of species k depends on the problem [66] and typically
fuel or product mass fractions, or a linear combination of them, are used. A typical variation of
this parameter in freely propagating laminar premixed flame is shown in Fig. 2.1.
2.2.2 Flame thickness
The flame thickness can be used as a reference value for the choice of the numerical mesh
size, and in this work the typical cell size in the region of the flame is chosen to be one to
three times the laminar thermal thickness, δth. This quantity can be estimated either from
10 Background on premixed combustion
0
1
x
θ
Burnt gasFresh gas
δth
sL
Fig. 2.1 Typical behaviour of progress variable, θ , in freely propagating unstrained laminar premixed
flame.
scaling laws, as δ = λu/(ρuCpsL) =Dth,u/sL, also known as Zel’dovich thickness, or from the
solution, as δ 0L = (Tb−Tu) |∂T/∂x|−1max; or as the length in which θ varies from 0.01 to 0.99.
The first definition is commonly known to underestimate the flame thickness, and the third
gives an overestimation. The second definition is thus the most accurate and is used in this
study to compute δth when needed. However, it requires the knowledge of the solution, which
is not known, for example, at the first iteration of a Newton-like algorithm commonly used in
laminar flames computations. An estimation was proposed in [15] in terms of δ for constant
heat capacity and Prandtl number. This estimation is δ 0L ≈ 2δ (1+ τ)0.7, where τ = Tad/Tu−1
is the heat release parameter and Tad the adiabatic flame temperature.
2.2.3 Flame speed
In premixed flames the flame speed, sL, is the velocity at which the flame propagates toward
the reactants in the direction normal to the flame itself. It controls the fuel consumption rate
and thus the heat release rate. When the flame is unstretched, three different flame speeds can
be defined [193] absolute speed, sa = (∂θ/∂ t)/|∇θ |, which is the speed of the flame front
in respect to a fixed reference frame. It remains constant across the flame front if the flame
thickness also remains constant; displacement speed, sd = (Dθ/Dt)/|∇θ |, which is the speed
of the flame front in respect to the flow; and consumption speed, sc, which is the speed at
which reactants are consumed. Differently form the previous two, which are local, sc is global,
and can be expressed as sc =−(
∫ ∞
−∞ ω˙ f dn)/(Yf ,uρu), where ω˙ f is the fuel reaction rate, ρ the
density, n is the local normal, and subscript u refers to unburnt conditions.
2.3 Turbulent premixed flames 11
When the flame is stretched, the flame speed depends on stretch and a number of other
parameters, and becomes complex to compute [191]. Estimations of this quantity have been
provided in the limit of small stretch, and can be found for example in [242].
2.3 Turbulent premixed flames
2.3.1 Turbulence-combustion interaction
Turbulence and flame interaction involves a large range of flow and chemical scales. The heat
release affects the mixture viscosity and hence the Reynolds number, which influences the flame
by enhancing or quenching the reaction term [191]. Assuming the flow in statistical steady
state and fuel entirely combusted across the flame (fairly true in lean mixtures), a turbulent
flame speed, sT , can be defined and related to the fuel reaction rate, ω˙ f , as
AρuYf ,usT =−
∫
V
ω˙ f dV
where A is the cross section of the integration domain and V its volume. Premixed flame
analyses [1, 89] show that sT first increases linearly with the root mean square velocity, u′,
as sT ≈ sL(1+u′/sL), then becomes independent of the laminar flame speed, sL, and scales
as sT ∼ u′ for high levels of u′, before local extinctions leading to blow-off can occur for
further increase of turbulence. The increase of sT in the linear region is also proportional
to the flame surface, AT , as sT = sL(AT/A), where Ξ = AT/A is the wrinkling factor and it
increases with the Reynolds number [191]. Recent DNS studies [133] show that sT dependency
on stretch is valid only in case of thin flamelets, and sT rather depends on diffusivity and scalar
gradients. Another DNS study [62] has shown that sT is also strongly geometry dependent
because of the different evolution of the flame brush thickness in different configurations. One
dimensional turbulent studies of methane-air with forced equivalence ratio show instead that
the burning velocity decreases with increasing equivalence ratio oscillations because of the
induced damping of turbulent fluctuations [215]. Heat losses also affect the turbulent flame
speed [158].
Combustion also modifies turbulence. Kinematic viscosity increases in burnt gases of a
factor Tb/Tu ≈ 8. Consequently, the Reynolds number decreases of a factor Reu/Reb ≈ 40,
which can lead to a re-laminarisation of the flow. On the other hand, the flow is accelerated
by a factor of sL(Tb/Tu− 1) across the flame, leading to an increase in the vorticity (flame
generated turbulence). These two effects contrast each other. Experiments [165] have shown
that depending on the initial magnitude of the turbulence, a vortex can be enhanced (weak
12 Background on premixed combustion
turbulence), damped (intermediate magnitudes), or remain unmodified (strong turbulence).
The strain induced by the flame-turbulence interaction in premixed combustion alters the
instantaneous and statistical behaviour of the scalar gradients. In non-reactive flows, the scalar
gradients tend to align with the most compressive principal strain-rate of the turbulent strain
tensor [12, 174, 207]. Such alignment was found to increase the scalar gradients by reducing
the distance between scalar isosurfaces. For reactive flows, the most extensive principal strain
rate aligns with the flame surface normal [90, 227]. Steinberg et al. [221] found that this
behaviour is due to the superposition of dilatation and turbulence effects. This preferential
alignment was found to increase with the flame burning speed and decrease with increasing
values of the progress variable. The phenomenon was observed to be due to a rotation of
strain-rate structures as they approach the flame. Moreover, the maximum rotation was found to
decrease with the strength of the turbulent structure, increase with the strength of the flame, but
be invariant to the Karlovitz number. Finally, the dilatation does not generate scalar gradients
when turbulence is weak [39], since dilatational and turbulence terms balance each other in this
case. However, the two terms result with the same sign when turbulence is strong, and both
contribute to the generation of scalar gradients, altering the gradients alignment in respect to
non-reactive cases. These are leading order effects [225] and must be considered carefully for
premixed combustion modelling.
2.3.2 Combustion regimes
Because of its complexity, the phenomenology in turbulent premixed combustion is usually
divided in regimes [180]. Assuming homogenous isotropic turbulence and one-step irreversible
chemistry, the reduced number Da(r) = τ(r)/τc, where τ(r) is the turbulent characteristic
time at scale r and τc is a chemical time scale, can be used to identify different scenarios of
flame-vortex interaction. When Da(r)≫ 1, turbulence does not affect the flame inner structure.
When Da(r)≪ 1, turbulence is much faster than reaction and strongly modifies the flame. The
Damköhler number,
Da = Da(Λ) =
ΛsL
u′δ
,
and the Karlovitz number,
Ka =
1
Da(ηk)
=
u′kδ
ηksL
=
(
u′
sL
)3/2(Λ
δ
)−1/2
are representative of this time scale ratio respectively at integral scale, Λ, and Kolmogorov
scale, ηk, and are used to identify different regimes in premixed combustion. The Kolmogorov
2.4 Modelling for turbulent premixed combustion 13
velocity scale in the above expression is indicated with u′k. A possible classifications of
combustion regimes is [180]
• Flamelet regime, where the chemical time scale is smaller than any turbulent time scale
(Da > 1) and the flame thickness is smaller than any turbulent scale (Ka < 1). The
flame front is thin and with a quasi-laminar structure. This regime is further divided into
wrinkled flamelet regime, in which u′ < sL and turbulence is too weak to alter the flame
structure, and corrugated flamelet regime, in which u′ > sL and the smallest turbulent
eddies enter the preheat zone, forming pockets of fresh and burnt gases;
• Thickened flame regime, where Kolmogorov scales are smaller than the flame thickness
and can modify the inner structure of the flame (Ka > 1). It can be further divided in thin
reaction zones regime, in which turbulent eddies enter and modify the preheat zone but
not the reaction zone of the flame, which keeps a quasi-laminar structure; and broken (or
distributed) reaction zones regime, in which both diffusion and reaction zone of the flame
are affected by turbulence. No laminar structure is likely to be present in the distributed
reaction zones regime and the effect of strain can cause local extinctions.
A sketch of the various regimes is shown in Fig. 2.2. These regimes are only indicative as they
are based on order of magnitude analysis and frozen turbulence, but serve as good guidance.
For example, even when Ka > 1, Kolmogorov scale eddies may be too small or too short in
time to affect the flame structure, or may induce curvature which counteracts the influence
of strain [191]. Moreover, the vortex-flame interaction depends on time. Recent studies on
methane-air flames show that flame quench occur at Ka> 1000 rather than Ka≈ 100 [213, 220],
which can be due to high rate of dissipation in small eddies [191] and high burning rates [213].
A pilot effect through eddies carrying hot products was observed in [220], which may also
explain the high Ka for the flame quenching in respect of what expected form the regime
diagram.
2.4 Modelling for turbulent premixed combustion
Turbulence modelling can be divided in three main categories, Direct Numerical Simulation
(DNS), Reynolds Averaged Navier Stokes (RANS) and Large Eddy Simulation (LES).
DNS is the simplest approach, for no closure models are required, but also the computa-
tionally most expensive. The grid must be fine enough to resolve the Kolmogorov scale and/or
the flame thickness. Consequently, the grid size depends on the Reynolds number, and large
meshes are required even for simple computational domains with high turbulence level. In
turbulent combustion, since the flame moves quickly and exhibits a considerable wrinkling,
14 Background on premixed combustion
100 102 104
10−1
100
101
102
103
104
u
′ /
s
L
Λ/δ
Kaδ = 100
Ret = 1
Wrinkled flamelets
Corrugated flamelets
Distributed reaction
zones
Thin reaction
zones
Kaδ = 1
Daδ = 1
Fig. 2.2 Regimes diagram for premixed turbulent combustion [180]. The subscript δ indicates that
Karlovitz and Damköhler numbers are computed using the Zel’dovich thickness.
self-adaptive meshing techniques cannot be easily exploited [191]. Besides, detailed chemistry
would involve time scales of different orders which make the solution stiff. For these reasons,
DNS requires very large computational resources and simple estimates are given in [187]. Thus,
it is rather used as a tool to gather insights required for development of closures for RANS and
LES.
In RANS, the probabilistic mean is applied to the govering equations, and the theorem of
ergodicity [69] is used to substitute this probabilistic mean with temporal average. In this way
the temporal derivative disappears from Eqs. (2.1)-(2.4). The application of the probabilistic
mean on the non-linear terms leads to additional unresolved terms which need modelling. An
extension to unsteady problems is possible by keeping the temporal term in the governing
equations (Unsteady-RANS or URANS)1. Even with URANS, only the time varying mean
characteristics of the flow can be captured.
In LES, only the large-scale motion of the flow field is solved, with models to mimic the
unresolved scales. This is obtained by filtering the governing equations with a spatial filter. If
the filter is homogenous in space and time, it commutes with the differential operators, and the
resulting equations are formally identical to those for URANS, but with different meaning. The
unresolved term represents in fact the contribution of the residual motion, i.e. the motion at
scales smaller than the filter width. LES has a computational cost which is between RANS and
DNS. It is accepted generally that at least the 80% of the turbulent kinetic energy should be
1In this case the ergodicity does not apply and this approach has been largely argued in the literature.
2.4 Modelling for turbulent premixed combustion 15
resolved in a good LES because of the energy cascade from large to small scales [196], and
this can be used as a general thumb rule for the choice of the numerical grid in a LES.
2.4.1 LES filtering
A filtered variable φ(x, t) is obtained through a spatial convolution with a filter function [196]:
φ(x, t) =
∞∫
−∞
φ(x′, t)G(x− x′;∆)dx′ (2.8)
where the filter function, G, depends on space and on the filter width, ∆. Equation (2.8)
is written for simplicity for the one-dimensional case, and extension to three dimensions is
straightforward. Formally, the integration domain is unbounded. However, filters used in
practice approach zero very quickly, and the integral is truncated. Thus, filtering can be seen
as a weighted spatial average, where the weight distribution is determined by the filter shape,
and their magnitude by the filter width. The filter width also controls the accuracy in LES, as
it determines which scales are solved and which are modelled. Several filters exist and their
description can be found in the literature (see for example [196]). It is worth noting that:
• the filter commutes with the differential operator only if it is uniform;
• the filter is not chosen explicitly. It is defined by SGS modelling, and by grid and
numerical scheme if it is linked to the mesh spacing (see §3.1.3). In this case non-
uniform grids would imply non-uniform filter, and this effect can be partly compensated
using dynamic approaches [65];
• in general, for a variable φ , φ ̸= φ and φ ′ ̸= 0, where φ ′ = φ −φ is the Reynolds residual
of φ .
• establishing the actual filter shape is difficult because the effects of grid, numerical
schemes and SGS models are difficult to separate from one another in most practical
cases.
A common filter is the Gaussian filter, whose normalised expression for one dimension is
G(x) =
(
6
π∆2
)1/2
exp
(
−6 x
2
∆2
)
(2.9)
and extension to a three-dimensional case is straightforward [196].
16 Background on premixed combustion
2.4.2 Favre filtering
Because of heat release, the density varies in combustion. This variation is fundamentally
different form that of compressible flows, e.g. in presence of shock waves. In deflagration
problems density changes do not depend on pressure, which is almost constant across the
flame [180]. Because of variable density, the filtering operation leads to coupling terms
involving density and transported variables. To avoid this, a Favre filtering is defined as [191]
φ˜ = ρφ/ρ . The Favre-residual of φ is defined, in analogy with φ ′, as φ ′′ = φ − φ˜ , and in
general ˜˜φ ̸= φ˜ and φ˜ ′ ̸= 0.
2.4.3 Governing equations for LES
The LES governing equations are obtained by filtering Eqs. (2.1)-(2.4) with a uniform filter,
yielding (see for example [191])
∂ρ
∂ t
+
∂ρ u˜i
∂xi
= 0
∂ρ u˜i
∂ t
+
∂ρ u˜iu˜ j
∂x j
=
∂ p
∂xi
+
∂
∂x j
(
ρµ
∂ui
∂x j
)
− ∂
∂x j
τ˜Ri j
∂ρ h˜
∂ t
+
∂ρ u˜ jh˜
∂x j
=
∂
∂x j
(
ρ
µ
Pr
∂h
∂x j
)
− ∂
∂x j
τ˜h
R
j
∂ρ c˜
∂ t
+
∂ρ u˜ jc˜
∂x j
= ω˙ +
∂
∂x j
(
ρ
µ
Scc
∂c
∂x j
)
− ∂
∂x j
τ˜cRj
p =
ρR0T˜
W˜
(2.10)
where the species equation is substituted with the equation for a Favre-filtered progress variable,
c˜. Additional equations may be required depending on the particular closure model, which
will be discussed in Chapter 3. Also, terms due to radiation and pressure changes have been
neglected in the total enthalpy equation.
The filtered molecular diffusion and turbulent transport, appearing inside momentum, en-
thalpy and progress variable equations, need modelling. The filtered diffusion is approximated
as
ρDφ
∂φ
∂xi
≈ ρD˜φ ∂ φ˜∂xi (2.11)
2.4 Modelling for turbulent premixed combustion 17
where φ is the generic transported variable. The residual stress tensor τ˜Ri j = (ρ u˜iu j−ρ u˜iu˜ j) in
the momentum equation can be written as the sum of its trace and an anisotropic part [196]:
τ˜Ri j =
1
3
τ˜Rkk + τ˜
r
i j (2.12)
The trace is added to the pressure term, in analogy with what is done for the stress tensor in
Eq. (2.1), and computed implicitly in a pressure-correction algorithm. The anisotropic residual
term, τri j, is modelled in this work using either a Smagorinsky model or a residual-kinetic
energy model (see §3.1.1). The turbulent fluxes in scalar equations are instead modelled with a
gradient hypothesis, i.e.
τ˜Rj ≡ ρφ˜u j−ρφ˜ u˜ j =−
νT
ScT
∂φ
∂x j
(2.13)
where τ˜Rj is the turbulent transport of the generic scalar φ , νT is the kinematic residual viscosity
to be discussed in §3.1.1 and ScT is the turbulent Schmidt number. Gradient models do not
account for counter-gradient effect due to dilatation in combustion problems [24, 181]. A
model for RANS which captures this effect is for example [236]
u˜ic− u˜ic˜ = c˜(1− c˜)
(
τsL−2αu′
)
(2.14)
where τ , sL and u′ are computed ahead of the flame, and α = 0.5. Gradient behaviour was also
seen to increase with the turbulence level in past studies, see for example [45, 68, 131, 145].
More information on gradient and counter-gradient transport in premixed combustion can
be found in [25, 144]. However, due to partial resolution of the turbulent scales, gradient
hypothesis in LES is effective in most cases [66] as most of the transport is solved at the
large scales and only a part is to be modelled [17]. Thus, using a gradient hypothesis is
acceptable [131].
Further modelling of the Eqs. (2.10) is discussed in Chapter 3. The modelling approaches
for chemical kinetics and SGS reaction rates are reviewed briefly below.
2.4.4 Chemistry modelling
Infinitely fast chemistry has been shown to be appropriate in capturing flame length and
flame speed in many cases, but chemical kinetics must be considered for problems involving
soot formation, ignition delay, stabilisation in combustion devices and intermediate species
predictions [66]. The large range of turbulent and chemical scales, and the many species
involved in a combustion process, make the problem intractable in its details. Moreover, small
scale processes like the coupling of reaction and diffusion are non-linear. First approaches
18 Background on premixed combustion
were based on global schemes reduction, in which some species are considered to be in a
quasi-steady state or be a part of reaction in partial equilibrium. However, such approaches give
a computational work reduction that is less than expected [191]. Moreover, these assumptions
give poor predictions in ‘colder’ region of the flame, where transport processes are also
important [234]. To overcome these issues the literature moved towards automatic chemistry
reduction and tabulated chemistry. The first approach consists in expressing the species which
have a short time scale as a function of a small number of species. Of this class of methods is
the Intrinsic Low Dimensional Manifold (ILDM) [148]. Tabulated chemistry consists instead in
tabulating complex chemistry for 1D laminar premixed combustion in function of a limited set
of variables (mixture fraction, progress variable, etc.). The choice of the variables must capture
the impact of kinetics, diffusion and heat losses on flame structure [66]. The first models to use
tabulated chemistry were the flame prolongation of ILDM (FPI) [86] and the flamelet generated
manifold (FGM) [234]. In the two models above the tabulation is done a priori. An alternative
approach is the in situ adaptive tabulation (ISAT) [195], in which the tabulation is done on the
fly as this was shown to be advantageous in terms of computational effort when the number of
tabulation coordinates is larger than two or three. Tabulated chemistry were observed to yields
better predictions than ILDM at low temperatures, but their mathematical assumption is weaker
than for ILDM.
The tabulated values must be pre-integrated with the joint probability density function (PDF)
of the related quantities in order to take into account the statistical properties of small scales
of the simulated field in a LES. Alternatively, one can avoid tabulation and use transported
PDF [7, 31, 94, 103, 204, 238]. As will be described in next paragraph, a modelled equation
for a joint-PDF of composition variables must be solved in this case.
2.4.5 Modelling for turbulent-flame interaction
The LES methodology and its modelling are described in a number of earlier studies for non-
reacting and reacting flows, see for example the books by [196] and [191]. The LES studies on
turbulent combustion are reviewed by [185] and [85]. The combustion is usually a sub-grid
scale (SGS) phenomenon requiring modelling and various modelling approaches used for
premixed combustion are reviewed and summarised in earlier studies [33, 85, 191, 226]. These
approaches can be broadly categorised into two classes, namely, flamelets and non-flamelets
or geometrical and statistical [85]. The geometrical category of flamelets includes thickened
flame [47, 53], flame surface density or flame-wrinkling, see for example, [17, 36, 38, 88, 92,
115, 241], level-set or G equation [164, 184]. The statistical category of flamelets includes
approaches such as algebraic closure involving scalar dissipation rate [32, 61, 77, 130, 147]
and presumed probability density function (PDF) methodology with laminar flamelets. The
2.4 Modelling for turbulent premixed combustion 19
EBU (eddy-break-up) model also can be included in this category if the variance is assumed to
be bimodal (see next paragraph). The non-flamelets category includes transported PDF [6, 104]
and conditional moment closure (CMC) [13, 113] methodologies. Each of these methods has
its merits and drawbacks, and a brief summary is given below.
EBU model
The Eddy Break Up (EBU) model was first conceived for RANS. The idea is that the chemical
source term is proportional [149, 219] to a chemical, τc, and a turbulent mixing, τt , time scales,
i.e. ω˙ ∼ τ−1c + τ−1t . The model was then extended to LES [70, 84, 162] by replacing the
turbulent mixing time scale with a SGS time scale, τt,sgs ≈ ksgs/εsgs, yielding
ω˙ =Cebu
ρσ2c
τt,sgs
where Cebu is the model constant, ksgs is the subgrid turbulent kinetic energy and εsgs is the
subgrid dissipation. The latter can be estimated in LES as εsgs = cεk
3/2
sgs ∆, where cε is a model
constant. This model is valid for high Reynolds and Damköhler numbers and in the hypothesis
of infinitely fast chemistry. The variance of c˜, σ2c , can be assumed to be bimodal, σ2c = c˜(1− c˜)
in the hypothesis of infinitely thin flame. This model is computationally cheap and simple
to implement. Moreover, differently to RANS, the model constant can incorporate effects of
wrinkling. However, this model tends to overestimate the reaction rate in regions of strong
shear and does not take into account the thickness of the flame [191].
FSD models
The main idea of Flame Surface Density (FSD) models is to express the filtered reaction rate as
ω˙ = ρus0LΣ, where Σ= |∇c| is the FSD per volume unit in a computational cell. An algebraic
model for Σ is [17]
Σ= 4Ξ∆
√
6
π
c(1− c)
∆
(2.15)
where Ξ∆ = |∇c|/|∇c| is the SGS wrinkling factor and can be a constant [17] or modelled
using fractal analysis [43, 114]. This approach was observed to underestimate the pressure
peaks in engine simulation when Eq. (2.15) is used [111]. An alternative approach is to use a
dynamic procedure for Σ. The latter is decomposed in a solved contribution |∇θ |, modelled
with Eq. (2.15) with Ξ∆ = 1, and an unresolved part, |∇θ | − |∇θ |, which can be modelled
20 Background on premixed combustion
as [114]
|∇c|− |∇c| ≈ cs
[
c(1̂− c)
∆
− θ̂(1− ĉ )
∆̂
]
where cs is a model constant and the hat indicates a test-filter operation. Another approach
consists in solving a transport equation for Σ [17, 47]. Hawkes and Cant [92] showed that FSD
models can capture the net flame propagation rate independently of the filter size, and that
wrinkling becomes more and more relevant at sub-grid scale as the turbulence increases. In a
following work Hawkes and Cant [93] noted that the realisability condition, Σ≥ 0, may not be
always satisfied, and extra care is needed in this approach.
Thickened Flame Models
The approach in thickened flame model (TFM) is to artificially thicken the flame front [47].
The flame thickness scales as δ ∼ (λ/A)1/2, where A is the Arrhenius pre-exponential constant.
If A is scaled by a factor F , δ increases. Since the thermal diffusivity scales as λ ∼ sLδ , this
quantity is also multiplied by F in order to keep the value of sL unchanged. Thus, the reaction
layer thickness can be increased and resolved on the computational mesh in a LES. However,
the Damköhler number is decreased of a factor F [191], thus an efficiency function is introduced
to compensate for the resulting decrease of flame SGS wrinkling [47]. Thickened flame models
can also be used in conjunction with FGM [170] and presumed sub-grid PDF models [57].
A model based on thickened flame, in which flamelets are simply filtered instead of being
pre-integrated, was proposed by Wang et al. [240], in which the effect of the wrinkling is
recovered by estimating a wrinkling factor using asymptotic analyses [184]. In [17, 63, 157], a
filter larger than the mesh size is used to resolve the filtered flame structure, which is observed
to be necessary when the cell size is larger than the flame thickness to avoid mis-prediction
of flame speed due to the broadening of the flame front. Nevertheless, additional DNS and
LES investigations using filtered one-dimensional flamelets and FGM to model the sub-grid
combustion show that diffusion and heat release are well represented when the grid size is
of order of flame thickness, and deviations from DNS results are seen only in the unresolved
convection term in this case [166].
Level set approaches
These models treat the flame as a very thin interface, which is tracked using a transport
equation (G-equation) and assuming that this interface corresponds to a level of a progress
2.4 Modelling for turbulent premixed combustion 21
variable [100, 109, 218]. A model for the conditionally filtered velocity is required in this
model as explained in [184]. The approach models the turbulent burning speed as sT/sL =
1+A(u′∆/sL)
n, where A and n are model constants and u′∆ is a SGS velocity scale. Effects of
finite chemistry and non-unitary Lewis numbers may be included in the modelling through
sL [226]. However, this model introduces flame cusps and artificial viscosity is needed to
smooth the G-field [101].
LEM-LES
The idea behind this approach is to include effects of SGS reaction, diffusion and convection
by looking at 1D unsteady transport equation [110]:
∂Yk
∂ t
=− 1
ρ
∂τi
∂ s
+
ω˙k
ρ
+Fmix
where s is the local normal specified by the gradient of Yk. A similar equation is used for
the temperature field. A stochastic term, Fmix, is introduced into the equation to represent
the mixing at subgrid scales [41, 211, 232]. full 3D LES equations are used for larger scales.
Thus, reaction kinetics and molecular diffusion are modelled exactly without additional SGS
modelling. However, in addition to the high computational cost, the stochastic term in this
model, Fmix, needs a mapping procedure [110] and a PDF [211] which may be difficult to model.
PDF models
Filtered reaction rate are function, in general, of temperature, T , and composition space, Y. A
closure for the generic reaction rate of species k, ω˙k, can be expressed as [84, 162]
ω˙k =
∫
P(Y,T )ω˙k(Y,T ) dY dT (2.16)
where ω˙k is the laminar reaction rate and P is the multi-dimensional sub-grid PDF, which
accounts for the SGS statistical properties of the turbulent flame. This PDF can be presumed or
transported.
In presumed PDF approaches, the sub-grid PDF is usually described as function of a small
number of composition variables. In the more specific case of flamelets assumption, it is
described as function only of a progress variable and its moments. A transport equation for a
Favre-filtered progress variable of the form of that in Eqs. (2.10) is used to compute the first
moment in the turbulent domain, while other moments, if required, are either transported or
modelled. The reaction rate of the filtered progress variable, along with other quantities needed
22 Background on premixed combustion
in the turbulent computation, are thus computed as
φ =
∫
P(c;c, . . .)φ(c) dc
where φ is a generic quantity which needs closure. The simplest model of this type is the
Bray-Moss-Libby (BML) model [30, 137]. Similarly to RANS, the sub-grid PDF in LES is
modelled as
P(c) = αδ (c)+βδ (1− c)+ γ f (c) (2.17)
where δ is the Dirac function and f (c) is a function of c which integrates to one in the c space.
The constants α , β and γ represent respectively the probability of having only reactants (c= 0),
only products (c = 1) or intermediate species in the mixture. In the hypothesis of very large
Damköhler number the flame becomes a very thin interface separating products and reactants,
thus γ ≪ 1, and α and β can be expressed as functions of only the first moment of c, c˜, and
the heat release parameter, τ , with an error of O(1/Da). Also, the integral of Eq. (2.17) can be
computed analytically for any quantity φ . The BML model has the advantage of being simple
and computationally cheap. Moreover, the density can be treated as constant respectively for
reactants and products sides, and no closure is needed for the interface flux. However, the
assumption of high Da limits its application. An improvement of this approach uses a joint
PDF of progress variable and velocity field [136], and in this case the reaction rate of progress
variable was shown to be proportional to the scalar dissipation rate. Additional details will be
discussed in §2.5.
To account for finite chemistry effects, the sub-grid PDF can be modelled using a β -function.
This PDF, which will be discussed in more details in next chapter, is parametrised using c˜ and
its SGS variance, σ2c,sgs. The latter can be either modelled or transported, and it will be shown
in §3.3.2 that simple models based only on turbulence can be a very inaccurate representation
of σ2c,sgs. Integrations based on β -PDF in LES have been shown to overestimate the flame
speed in laminar regions when the filter size is larger than the laminar flame thickness [67].
Alternative PDF integrations have been investigated recently in [210, 231] using modified
laminar flamelets PDF and LEM-formulated PDF, showing improved performances compared
to β -PDF in change of additional complexity.
In transported PDF models [192], also known as filtered density function (FDF) mod-
els [194] in case of LES, a transport equation for the sub-grid PDF in Eq. (2.16) is computed,
and chemical mechanisms can in principle be applied without further simplifications. Transport
equations for the sub-grid PDF of a scalar variable or the sub-grid joint-PDF of a scalar variable
and the velocity field are commonly used, but extension to joint-PDFs to any other variable is
possible. Thus, this method can incorporate detailed and accurate chemical kinetics and is for
2.4 Modelling for turbulent premixed combustion 23
this reason very attractive for the turbulence-chemistry interaction modelling. However, the
molecular transport term in the PDF equation becomes an issue for Da≫ 1, as it is strongly
affected by the chemical reaction in this case and needs to be modelled. One approach is to
solve for a set of coupled stochastic differential equations using Lagrangian particle method and
Monte Carlo simulations [6], as done for example in [48, 204]. Another approach is to solve a
set of Eulerian stochastic fields [104], in which the stochastic fields correspond to an equivalent
stochastic system with same one-point PDFs rather than a particular realisation of the flow [80],
and a number of works have appeared recently for premixed combustion [31, 56, 102], showing
very good potentiality of this approach. A review of PDF methods for LES is for example
in [87].
CMC model
In conditional moment closure (CMC) transport equation for conditional mean of species mass
fractions and temperature are solved [13, 113]. These variables are conditional on chosen
scalars, usually a progress variable in premixed combustion. The key assumption is that the
fluctuations of the conditional mean are small and can be ignored at least for first order CMC,
and the conditional reaction rate can be expressed as ⟨ω˙|c⟩= ω˙(Q), where Q is the conditional
species or temperature (or a combination of them). These models need a closure for the condi-
tional SDR, which is not straightforward to provide. The CMC method was tested for RANS
simulations of premixed flames [4, 5, 151] but yet to be fully assessed for premixed combustion
LES. An application for compressible flows is described in [230]. More information on this
method can be found for example in [126].
More recently, scale similarity assumptions for modelling parameters such as flame wrin-
kling and flame surface density have been proposed and investigated using DNS [91, 244] and
LES [43, 88, 114, 115, 117, 127, 185, 241] for most of the above models. Dynamic models
based on the scalar dissipation rate have been investigated for non-premixed combustion [105],
and recently for premixed combustion [79, 129]. Although the assumption of scale similarity
is arguable for reacting quantities, these modelling approaches seem promising [66], and are
observed to work on different regimes [158].
The focus in this thesis will be on presumed PDF approaches. These methods are simple
and can account for finite rate chemistry with an enormous advantage in terms of computational
time in respect to other models. Thus, they have potential to meet the aims presented in §1.3.
24 Background on premixed combustion
This will be rigorously tested in this study on different configurations and regimes. Since
flamelets are usually assumed in these methods, this hypothesis is briefly discussed next.
2.4.6 Laminar flamelets approach: range of validity and limitations
Premixed flames are usually thin and quite robust to turbulent perturbations and variations
in thermochemistry [34]. Thus, if Ka< 1, turbulence can merely corrugate the flame, which
can be seen as an ensemble of thin, locally one-dimensional laminar structures (flamelets). A
laminar flamelet consists of a thin and highly wrinkled interface propagating like a strained
and curved laminar flame. In flamelets hypothesis, turbulence can be decoupled from the
chemistry; laminar properties can be used to evaluate the local rates of reaction and transport,
while the turbulent modelling is used to evaluate the main features of the flow. The coupling
between molecular diffusion and reaction is confined to a low-dimensional manifold: the
detailed chemistry from laminar computation is tabulated and parametrized by a small number
of variables [197].
One of the earliest models for turbulent premixed combustion is the Bray-Moss model [30].
This model was extended by Libby et al. [137] to include effects of molecular transport and
finite chemical reaction rates. They demonstrated theoretically that laminar premixed flames
can naturally parametrise the turbulent flames. In a following work, Libby and Bray [136]
raised doubts in the application in premixed combustion of models for turbulent transport
taken from non-reactive turbulence. In fact, the mean scalar dissipation rate (SDR), controlled
by the chemical term, influences directly turbulent mixing and dissipation. Hence, models
derived from non-reacting flows must be revised in flamelet context. Statistics of the turbulent
motion must be taken into account in flamelet approaches, e.g. by defining a joint PDF of
the progress variable and the velocity field. Under the assumption of high Reynolds number,
Bray et al. [28] showed that since reactions happen in some points within the flamelets, they
are proportional to the crossing frequency at which flamelets cross that point. A study on
the limits of flamelet assumption was carried out by Meneveau and Poinsot [155], using a
model for intermittent turbulence and taking into account all possible turbulence scales acting
on the flame front. By investigating on the effect of stretch and quenching in the thickened
reaction zones of the premixed combustion diagram, they found that the flamelets assumption
in premixed turbulent flames is valid in a domain larger than expected and that quenching
occurs depending on the level of sub-adiabaticity of the flame. This extended domain of
applicability was also observed in experiments [59, 60, 202]. Knudsen and Pitsch [118] have
shown that flamelets applicability on multi-regime flames is possible, and possible inaccuracies
are due finite chemistry effects rather than inappropriate regime predictions. More recently,
flamelets assumption in LES modelling have been observed to hold up to Ka ≈ 1000 [220].
2.4 Modelling for turbulent premixed combustion 25
Moreover, the flamelet assumption was shown to be reasonably valid for stratified flames
also [51, 95, 179, 190, 201, 203, 229].
Strain effects on flamelets have also been investigated in past studies. Poinsot et al. [188]
observed that the strain has no effect on the consumption speed, sc, at unity Lewis numbers
in premixed laminar flames. The displacement speed, sd , increases instead by almost an
order of magnitude due to hydrodynamic and diffusive mechanisms. The first effect is due
to the modification of the flame surface by lateral flow divergence and the second is due to
the misalignment of diffusive and hydrodynamic mechanisms. Moreover, curvature induces
different propagation speeds at different locations. Flamelets which are strongly curved toward
the fresh gases have higher displacement speed and come rapidly back to planar shapes. These
highly curved flamelets have a short life time, but might have strong effects on the flame
surface variation, raising questions on the choice of flamelets canonical configuration to choose
for modelling. Thermo-diffusive instability effects at non-unity Lewis number have been
investigated recently for turbulent premixed combustion [108]. It was observed that instability
arises after a critical level of strain, and this effect can be taken into account in a flamelet
model if the laminar flame speed is scaled appropriately. The fluid dynamic straining effects
were shown [122] to be important in RANS calculations of piloted stoichiometric methane-air
Bunsen flames of [46] using presumed PDF involving unstrained and strained flamelets. When
the straining effects were excluded, the computed flame length was observed to be very much
shorter than the measured length in [46] suggesting that the fuel is consumed quickly. There
is only one turbulence length and time scales involved in the RANS and the affects of the
spectrum of turbulence scales on the flame need to be included in the combustion modelling.
Various approaches such as efficiency function were introduced in the past [155] to include
flame stretching caused by the turbulent eddies which are lost in the averaging involved in the
RANS method. A similar approach is also used to include the effects of unresolved eddies on
the filtered flame in LES using thickened flame model [47, 53]. Alternatively, one could also
use strained flamelets to model filtered reaction in LES as has been attempted in [116]. The
past RANS study [122] used averaged scalar dissipation rate of a reaction progress variable
to parameterise strained flamelets whereas atomic hydrogen mass fraction was used in [116]
for LES. The SGS variance of progress variable, σ2c,sgs, required in presumed PDF methods, is
modelled in earlier studies typically using an expression developed for passive scalars. The
relevance of such modelling for turbulent premixed combustion is an open question on physical
grounds because chemical reactions ought to influence this variance as was remarked in [182]
and an algebraic model for σ2c,sgs including this effect is yet to be developed. These points are
of specifically high importance for the presumed sub-grid PDF approach because this PDF
has a direct dependence on σ2c,sgs and influences the filtered reaction rate value. Moreover,
26 Background on premixed combustion
this variance may assume a different importance on different configurations and combustion
regimes. Thus, a large scope of investigation is left with this methods. The aim of this study is
to contribute to reduce this gap and provide a deeper understanding of strengths and weaknesees
of flamelets modelling for LES of premixed combustion. The particular SGS modelling used in
this study, to be explained in detail in the next chapter, needs a model for the SDR of a progress
variable. This modelling and the role of SDR in premixed combustion are discussed in the next
section.
2.5 Scalar dissipation rate modelling in turbulent premixed
combustion
The instantaneous SDR of a reaction progress variable, c, with diffusivity Dc, is defined as
Nc =Dc(∇c ·∇c). Closures for mean and filtered reaction rates based on the SDR have been
proposed in the past (see for example [226] for a review) and can be categorised into algebraic
and transport equation models. The algebraic model assumes that the molecular dissipation
of the SDR balances its generation by turbulent straining, dilatation and chemical reaction
rate terms. In RANS, this equilibrium between production and destruction of SDR gradients
is strictly valid in the corrugated flamelets regime and the PDF of the progress variable can
be well approximated by a bimodal function in this case. Furthermore, in the thin reaction
zones regime, reaction rate and SDR contributions are of leading order in the scalar variance
equation even at high Karlovitz numbers. Since in premixed combustion the rate of heat from
the reaction to the preheat zone determines the flame propagation rate, the mean reaction rate is
directly proportional to the SDR, at least in the context of flamelets assumption [23]. If the
Damköhler number is high, the PDF of c is well approximated by a double delta function, and
it can be shown that [23]
σ2c = c˜(1− c˜) (2.18)
and
ω˙ =
2
2Cm−1ρN˜c (2.19)
where σ2c is the variance of c˜ and Cm is a constant. The model constant Cm in this relation is
defined as
Cm =
∫ 1
0 [ω˙ c]L f (c) dc∫ 1
0 ω˙L f (c) dc
, (2.20)
where f (c) is the burning mode probability density function (PDF) of c and the subscript ‘L’
refers to planar laminar flame [22, 23]. The integrals in Eq. (2.20) can be evaluated using
2.5 Scalar dissipation rate modelling in turbulent premixed combustion 27
the laminar flame data and thus Cm is a thermo-chemical parameter. This relation has been
verified for RANS [37, 40] and LES [61, 77, 129] paradigms in past DNS studies through a
priori testing. The use of this closure for RANS calculation overestimates the turbulent burning
rate [121]. However, a variant of this model, known as eddy-break-up model, involving a
time scale for SGS turbulent mixing rate gave reasonable results [70, 162, 173]. Also, it has
been shown that the scalar dissipation rate based closure can yield improved results compared
to other combustion modelling approaches for LES [32, 129, 147]. An algebraic model for
RANS for the SDR was first proposed by Borghi [18]. Nevertheless, he did not account for the
dilatation effects. Swaminathan and Bray [225] analysed the order of magnitude of the various
terms in the transport equation for the instantaneous SDR. In particular, they showed that the
dilatation is affected by two contributions, one representing a direct effect of density changes,
and the other an indirect effect due to dilatation on turbulence-scalar interaction. The original
model was then modified via a priori analyses and comparison with the turbulent flame speed
from experiments, based on new findings of the physics of turbulence-scalar interaction [120].
In particular, while neglecting the indirect term alters the true flame speed only when turbulence
is weak, neglecting the direct term always gives poor predictions [123]. Using a-priori DNS
analysis, Kolla and Swaminathan [122] improved the model of [167] by developing a closure
for corrugated flamelets regime, based on equilibrium at the leading order.
If Da ≫ 1, the progress variable variance can be modelled with equation (2.18). If the
flame is not thin, turbulent eddies can enter the reaction zone, inducing tangential strain-rate.
Flamelet-like structures can still be found under strain effect and at high Da, as described in
§2.4.6. Thus, flamelet models are still possible. Kolla and Swaminathan [123], using the SDR
as parameter, showed that the effect of the dynamic strain of reducing the burning rate could be
captured in the thin reaction zones regime. This approach is more advantageous than using the
strain as parameter in the flamelet formulation [20] since the SDR is defined at each location on
the flamelet and is naturally related to the reaction-convection-diffusion coupling of premixed
combustion, as noted earlier. Besides, asymptotic studies have shown that Nc is closely related
to the flamelet structure at both high and low strain values [138].
An extension to LES of the above model was proposed by Dunstan et al. [61] and analysed
through DNS [61] and LES [32, 129, 147]. Since the SDR features in the transport equation for
the progress variable variance and is directly proportional to the square of the scalar gradient
magnitude, this equation is also controlled by the SDR. Analyses were done by Bray et al. [26],
which proposed a transport model for the scalar variance by accounting for effects of dilatation
and proposing a model in case of high, but not infinite Da. Kolla and Swaminathan [122]
showed that unstrained flamelets model over-predict the mean reaction rate in RANS, and the
SDR was in that case added as third controlling variable in a strained flamelets formulation. The
28 Background on premixed combustion
importance of the scalar variance was shown for the mixture fraction in LES of non-premixed
combustion [105] and stratified flames [10]. Contrary to RANS, it will be shown in this thesis
that, because of the partial resolution of the strain at larger scales, strained flamelets models are
not essential in LES even at relatively high Karlovitz numbers, as long as the progress variable
variance is appropriately estimated. On the other hand, results from [116] show that strain
modelling cannot be neglected in LES. However, a simple algebraic model for the progress
variable variance was used in [116].
2.5.1 SDR model for LES
For RANS methodology, Bray [22, 23] showed that the averaged reaction rate of progress
variable is directly related to averaged scalar dissipation rate when the Damköhler number is
large, see Eq. (2.19). The validity of this relation for low Damköhler number flames was also
demonstrated in [37, 40] for RANS approach and its applicability to model filtered reaction
rate for LES was shown in [61, 77]. Dunstan et al. [61] performed an a priori analysis with
DNS, using single-step chemistry and filtering explicitly the data with different filter sizes.
They showed that the qualitative behaviours of ω˙ and ρN˜c/(2Cm−1) are significantly different
when the filter width, ∆, is smaller than the laminar thermal thickness, δth. This is because
the Bray model of Eq. (2.19) assumes a PDF which is bimodal, which does not hold when
∆< δth because of the partial resolution of the flame in LES. The model is expected to hold
when ∆≫ δth, since in this case the filter includes relatively large amounts of both burnt and
unburnt gases. Following this study [61], the Favre filtered scalar dissipation rate is expressed
as a sum of resolved and unresolved contributions as
ρN˜c = ρDc(∇c˜ ·∇c˜)+ρε˜c. (2.21)
The unresolved SDR of c, ε˜c, is related to SGS phenomena. Simple models for ε˜c based only
on turbulence scale such as ε˜c ≈K D˜cσ2c,sgs/∆2, whereK is usually a constant of order one,
would give an inaccurate representation of the sub-grid SDR because of the missing effects
of dilatation, reaction and flame-turbulence interaction terms which may be at leading order
in combustion [225], as also supported by DNS results [150, 175, 225]. Thus, such models
are unphysical and would require ad hoc tuning of the modelling constant. Dunstan et al. [61]
proposed a model which involves all these relevant effects. The sub-grid SDR is modelled in
this case as
ε˜c =F
[
2Kc
sL
δth
+(C3− τC4Da∆)
(
2u′∆
3∆
)]
c˜(1− c˜)
βc
, (2.22)
2.6 Summary 29
where F = 1− exp(−θ5∆+) with θ5 = 0.75 and ∆+ = ∆/δth is the normalised filter width.
The factor F ensures that N˜c → Nc when ∆+ → 0 [61]. The planar laminar flame speed is
denoted as sL. The SGS velocity scale is u′∆ which is to be modelled and the sub-grid scale
Damköhler number is Da∆ =
(
∆+/u′+∆
)
with u′+∆ = u
′
∆/sL. The heat release parameter, τ , can
be a constant or can also be defined using mixture density as τ = ρu/ρ−1 and it is observed
that the numerical results to be reported in §5.3 are not unduly influenced by this choice. If
the reactant mixture is stratified then the later definition is more suited as it can capture spatial
variation in τ .
The thermo-chemical parameter, Kc, in Eq. (2.22) is given as [120]:
Kc =
δth
sL
∫ 1
0 [ρNc∇ ·u]L f (c) dc∫ 1
0 [ρNc]L f (c) dc
(2.23)
where u is the velocity vector. This parameter is taken to be Kc = 0.79τ for the flames studied
here. The other parameters C3 and C4 in Eq. (2.22) are not free parameters and they are
related to SGS Karlovitz number defined as Ka∆ =
√
u′+∆
3
/∆+. These parameters are given by
C3 = 1.5
√
Ka∆/(1+
√
Ka∆ ) and C4 = 1.1/(1+Ka∆)0.4 [61].
The only one parameter which needs a close attention is βc in Eq. (2.22) and this parameter is
related to strong interaction among flamelet curvature, spatial variation of molecular diffusivity
and the spatial gradient of reaction rate resulting from the former two effects [39, 78]. This
effect is important in premixed combustion because of inherently strong coupling between
reaction rate and molecular diffusion in premixed flamelets. Both steady and unsteady RANS
calculations of a variety of turbulent premixed flames in various configurations using the RANS
counterpart of Eq. (2.22) suggested a reasonably robust value of 6.7 for βc [2–5, 120, 206, 228].
However, this value cannot be simply adopted for LES because of the difference in scales
involved for combustion modelling between RANS and LES. For example, Dunstan et al. [61]
showed that βc = 2.4 worked well for their a priori testing of Eq. (2.22). Furthermore, this
parameter was observed to vary with τ [77] and possibly with turbulence Reynolds number
also. Hence, specifying βc value for this study is postponed until the discussion in Chapter 5.
The implementation of this model and related sub-models will be discussed in Chapter 3.
2.6 Summary
The present work focuses on flamelet modelling for LES of turbulent premixed combustion.
As shown in this chapter, flamelet models have been largely used in the literature; however
their strengths and weaknesses are not fully understood. Previous models have been based on
linear relaxation model for the SDR and/or algebraic models for the SGS-variance of the key
30 Background on premixed combustion
scalars. These models, as will be shown in the next chapter, can be physically inconsistent and
lead to inaccurate results. In this work the SDR and the SGS-variance of progress variable are
modelled in a more physically meaningful way, respectively using Eq. (2.22) and a transport
equation for these two quantities. The potentials of flamelet modelling is revised in this light
by simulating premixed combustion in corrugated-flamelets, thin reaction zones and distributed
reaction zones regimes of premixed combustion. Two models are used for this purpose, one
based on fast chemistry assumption, Eq. (2.19), and another based on presumed PDF approach.
Their relative ability to predict the premixed flame structure is assessed. It will be shown in
particular that when the variance equation is carried in the simulation and the various SGS
closures related to combustion involve the physical aspects of premixed combustion, strain
effect is not needed in the LES modelling. This is an unexpected result for common views in
the literature and opens new avenues to investigate flamelets models.
Chapter 3
Numerical modelling
The governing equation are presented in Chapter 2 and as noted there, SGS processes need to
be modelled. Common models for residual stresses are used and the particular ones relevant
for this thesis are discussed first. The particular numerical scheme and discretization technique
are then introduced, before presenting reaction rate closures used in this study.
3.1 Residual stresses and common practices in LES
3.1.1 Eddy diffusivity models
Eddy viscosity models for LES are based on the assumption that the anisotropic residual tensor
is aligned to the filtered strain. For variable density flows this can be expressed as
τ˜ri j =−2νT
(
S˜i j− 13 S˜kkδi j
)
(3.1)
where
S˜i j =
1
2
(∂ u˜i
∂x j
+
∂ u˜ j
∂xi
)
(3.2)
is the Favre-filtered strain and δi j is the Kronecker delta. This assumption was shown to be
incorrect [154], since the principal directions of residual tensor and filtered strain are not
aligned. For this reason, eddy-viscosity models tend to overestimate the residual stress and
consequently are more dissipative than required. Different models for the residual viscosity
exist in the literature and a summary can be found for example in [196]. These models were
originally developed for incompressible flows and then adapted to variable density flows such
as combustion. A brief overview of those relevant for this work is given below in the framework
of variable density flows.
32 Numerical modelling
Zero-equation models
Of this class are the classical Smagorinsky model [217] and the localised dynamic Smagorinsky
model [141] used in this work. The classical Smagorinsky model assumes a mixing length
hypothesis and is expressed as
νT = (Cs∆)2|S˜| (3.3)
where |S˜| = (2S˜i jS˜i j)1/2 is the characteristic strain. The Smagorinsky constant, Cs, is found
assuming a Kolmogorov energy spectrum in the inertial range, and that the mean rate of energy
transfer from filtered to residual scales equals the viscous dissipation. It can be found that the
filtered strain is dominant around the cut-off scale ∆ and that the characteristic strain scales as
|S˜| ∼ ∆2/3 [196].
The dynamic Smagorinsky model assumes scale invariance and exploits the resolved scales
to compute the model coefficient Cs. The original model by Germano et al. [81] computes
one global averaged value for the whole flow field and assumes that the model constant is
filter-invariant. A localised formulation was proposed first by Goshal et al. [83]. In this work
the localised mixed formulation by Lilly [141] is used, which computes Cs locally and also does
not rely on the assumption of filter-invariance. Dynamic procedures can capture the physics in
shear flows and near walls by automatically lowering the magnitude of Cs. This constant may
assume negative values (backscatter) leading to numerical instabilities [212], thus it must be
spatially averaged [154, 183]. Dynamic models can also compensate the effect of errors in the
length scale due to non-uniform filters [65], thus they are useful when non-uniform grids are
employed.
One-equation models
Models described previously relate the residual stress tensor to the filtered strain at the same
time and in the neighbourhood of a point. A better approach would be to incorporate history and
non-local effects, by adding one equation for the residual kinetic energy, kr, defined as [196]:
kr =
1
2
(u˜iui− u˜iu˜i) ,
Let’s notice that by definition this is kinetic energy and not turbulent kinetic energy. In LES,
the filter width, ∆, can be assumed as characteristic length for kr, thus an additional equation,
i.e. for the energy dissipation, is not needed [52]. The residual viscosity is modelled in this
case as [55]
νT =Cvk
1/2
r ∆ (3.4)
3.1 Residual stresses and common practices in LES 33
where Cv ≈ 0.1 and a dynamic evaluation for Cv is described in [83]. Assuming uniform filter,
the transport equation for kr can be written as [35]
∂ρkr
∂ t
=− ∂ρkru˜ j
∂x j
− R
Cp
∂ q˜′′j
∂x j
+
∂ τ˜Ri ju˜i
∂x j
− τi jS˜i j + ∂∂x j
[
µ˜
∂kr
∂x j
]
+
∂ f j
∂x j
− εk− εc+Π+Kµ +Kd
(3.5)
where R = R0/W is the specific gas constant and q˜′′j is the residual heat flux, defined as
q˜′′j =Cp
(
ρT˜u j−ρT˜ u˜ j
)
(3.6)
The terms f j, εk, εc and Π are respectively turbulent transport, residual dissipation, dilatational
dissipation and pressure dilatation term, and they are defined as
f j =
1
2
ρ
(
u˜iuiu˜ j− u˜iuiu j
)
+
1
3
(
µu j
∂uk
∂xk
− µ˜ u˜ j ∂ u˜k∂xk
)
(3.7)
εk = µ
∂ui
∂x j
∂ui
∂x j
− µ˜ ∂ u˜i
∂x j
∂ u˜i
∂x j
(3.8)
εc =
1
3
[
µ
(
∂uk
∂xk
)2
− µ˜
(
∂ u˜k
∂xk
)2]
(3.9)
Π= p
∂uk
∂xk
− p∂ u˜k
∂xk
(3.10)
The remaining terms are
Kµ =
∂
∂x j
[
µ
∂ (12uiui)
∂x j
− µ˜ ∂
1
2 u˜iui
∂x j
]
(3.11)
Kd =
[(
ui
∂u j
∂xi
−u j ∂uk∂xk
)
∂µ
∂x j
−
(
u˜i
∂ u˜ j
∂xi
− u˜ j ∂ u˜k∂xk
)
∂ µ˜
∂x j
]
(3.12)
Models proposed in the literature for kr are mostly based on incompressible flows [83, 124],
in which terms εc, Π, Kµ and Kd are usually negligible. Models for q˜′′j , f j and εk can be
computed either statically or dynamically and their description is reminded to the literature (see
for example [35, 54, 83, 124, 196]). A number of authors [71, 98, 156, 171] has shown that
one equation models improve accuracy even in variable density flows. One equations models
are also beneficial in case an estimation for the sub-grid velocity is needed (see §3.3.3), and
34 Numerical modelling
because they allow for further investigations on statistical sub-grid properties, which can be
exploited for example in PDF approaches.
3.1.2 One equation model for LES of premixed combustion
An attempt to model the pressure dilatation has been recently presented by Chai and Mahesh
[35] for LES of compressible non-reacting flows. However, in the case of combustion, the
dilatation term is strongly influenced by the reaction rate and this influence must be taken into
account while modelling. In the assumption of low Mach number, Eq. (3.5) can be re-arranged
as
ρ
Dkr
Dt
=
∂
∂x j
(
µ˜
∂kr
∂x j
)
+ u˜i
∂τRi j
∂x j
− ∂ f j
∂x j
− εk +Π, (3.13)
This modelled equation is similar to that in [35] for compressible flow except for the dilatational
dissipation resulting from compressibility and that the residual heat flux in Eq. (3.6) is written
using state equation as q˜′′j =Cp(pu j− pu˜ j)/R and its divergence in Eq. (3.5) is subtracted from
Eq. (3.10). The pressure dilatation term is thus written as
Π≡ u˜i ∂ p∂xi −ui
∂ p
∂xi
=
[
p
∂ui
∂xi
− p∂ u˜i
∂xi
]
−
[
∂ pui
∂xi
− ∂ pu˜i
∂xi
]
(3.14)
By using an analogy to turbulent kinetic energy (TKE) equation for RANS [142], this term is
modelled as
Π≈ 0.5 ω˙ c˜τ2s2L+
µ˜T
ρ T˜ PrT
∂ T˜
∂xk
∂ p
∂xk
(3.15)
where PrT ≈ 1 is the turbulent Prandtl number and µT = ρνT is the dynamic residual viscosity.
3.1.3 Relation between LES filter, grid spacing and numerical scheme
Let’s distinguish between residual motion, which is the motion occurring below the filter-width
scale, and sub-grid scale motion, which is the motion occurring below the grid size. The LES
momentum equation in Eqs. (2.10) becomes formally identical to the laminar one, Eq. (2.2), if
one excludes the residual stress term. Thus, this term hides information about the LES filter.
Pope [196] showed that in homogenous and isotropic turbulence, the Smagorinsky model has a
unique implied filter, the Smagorinsky filter, obtained on the basis of a priori DNS analysis. In
particular, he noticed that the shape of the filter depends on the form assumed for the dissipation
spectrum. Further analyses showed that the Smagorinsky model implies a filter which is close
to a Gaussian one. These analyses used a filter size, imposed through the Smagorinsky term,
larger than the grid size to exclude possible effects of the numerical grid.
3.1 Residual stresses and common practices in LES 35
When the filter size is linked to the grid spacing, numerical grid and discretization schemes
affect the LES filter [125, 161]. This link appears explicitly in case of Smagorinsky-like models
(see for example Eq. (3.3)). Pope [196] showed that in this case the grid must be refined enough
and at least second order spatial schemes are required in order to avoid that numerical-diffusion
effects compromise the solution. More recently some authors have tried to unlink residual
model and grid spacing by using explicitly filtered LES (EFLES) models in non-reacting flows
(see for example [19, 153, 199, 235]). Although these methods are yet to be applied for reacting
flows, the focus of the present study is not on this kind of LES. In Smagorinsky-like models
one can reduce this dependence on grid spacing by choosing ∆ much larger than the grid size.
However, this would imply a significant increase of computational effort, as will be shown in
the next paragraph. This approach is unpractical for the purposes of this work, and the filter size
required in Eqs. (3.3) and (3.4) is set to the cubic root of the cell volume, following common
practice. Additional analyses to evaluate effects of grid spacing and numerical scheme on the
solution are left to future work. The final shape of the filter will be supposed to be Gaussian for
simplicity, following the DNS analysis in [196]. Because of this choice, residual motion and
SGS motion will be referred to indiscriminately from here onwards.
Choice of grid size
The choice of the grid size in a proper LES is usually associated with the resolution of at least
the 80% of TKE [196], which corresponds usually to 4 to 5 cells at least within the integral
length scale, Λ. However, if the filter size is bigger than the cell size, the amount of turbulent
kinetic energy at the residual level must depend on the filter size rather than the grid size as
noted in the previous paragraph. An estimation of the amount of residual kinetic energy in
terms of the integral length scale is as follows. Since the residual kinetic energy in this analysis
must represent the amount of energy ‘lost’ at residual scales, kr must be related to the residual
model. The residual kinetic energy can be expressed as kr = 3u′2∆ /2, where u
′
∆ is the residual
velocity scale, which can be related directly to the eddy viscosity model as [196] u′∆ =CqνT/∆,
where Cq is a constant. Modelling νT with Eq. (3.3) for the purpose of this analysis, one can
write
kr =
3
2
u′2∆ =
3
2
C′q∆
2S˜i jS˜i j.
where C′q = 2C2qC4s . Assuming a Kolmogorov cascade of the kinetic energy in the inertial range,
the mean strain could be approximated using rms velocity, u′, and integral length scale, Λ.
Thus, taking the statistical mean of above equation one obtains
⟨kr⟩ ≈ 32⟨C
′
q⟩∆2
u′2
Λ2
= ⟨C′q⟩k
(
∆
Λ
)2
36 Numerical modelling
where k is the mean TKE, and the mean of the products was approximated with the product of
the means just to obtain an order of magnitude scale. Following this, the SGS kinetic energy
scales as
⟨kr⟩ ∼ k
(
∆
Λ
)2
(3.16)
This equation does not account for grid effects and indicates that the amount of residual kinetic
energy increases with the filter size, at least when Smagorinsky-like models are used. This
result must be taken into account when using a filter size larger than the grid size.
3.1.4 Comparison with experiments
Strictly, statistics should be collected for each time and spatial location performing an ensemble
average over hundreds of experiments. This approach is impractical in both experiments and
simulations and, unless particular properties must be studied, ergodicity theorem is used to
apply time average in place of ensemble average [69]. In experiments, data are sampled with a
certain frequency and spatial accuracy. Non-intrusive, laser diagnostic techniques (i.e. LDA,
Raman and Raylegh scattering, CARS, PLIF, PIV) are capable to reach very high frequencies in
time, therefore the signal in time is usually directly comparable with that of a typical LES. The
accuracy in space is of order of millimetres, i.e. the signal is filtered in space. For this reason
results from LES should be processed appropriately before being compared with experiments.
Another issue comes for variable density flows. An averaged quantity is usually obtained
by simply time-averaging a filtered LES quantity, i.e. assuming ⟨φ⟩ ≈ ⟨φ⟩. Favre-averaged
quantities are instead obtained from Favre-filtered ones, as [191]
⟨ρφ⟩
⟨ρ⟩ ≈
⟨ρφ⟩
⟨ρ⟩ (3.17)
If the signal in the experimental technique is proportional to density, than Eq. (3.17) can be used
to compare LES and experimental results. However, the signals in techniques like Raman and
Rayleigh scattering do not depend on density, thus LES results and experiments are not directly
comparable. A relation between Reynolds- and Favre-averaged first order moments was derived
by Kolla and Swaminathan [121] in the hypothesis of a β -distribution for a normalised variable
φ bounded in the interval [0,1]. This relation is
⟨φ⟩ = ⟨φ˜⟩+ τ
(1+ τ⟨φ˜⟩)⟨σ
2
φ ⟩ (3.18)
where ⟨σ2φ ⟩= ⟨σ2φ ,sgs⟩+ ⟨σ2φ ,res⟩ is the total variance, sum of SGS and resolved contribution.
The resolved variance is obtained as ⟨σ2φ ,res⟩= ⟨(φ˜ −⟨φ˜⟩)2⟩, while the SGS variance, σ2φ ,sgs,
3.2 Discretisation 37
must be modelled or computed using a transport equation. The symbol ⟨ ⟩ is used here to
indicate either simple or mass-weighted time average respectively for φ and φ˜ , and in general
one can infer which one is appropriate from the nature of the LES variable, e.g. ⟨ρ⟩ indicates
simple average of ρ in time and ⟨c⟩ indicates mass weighted average of c˜. The explicit
symbolism, like ⟨φ˜⟩ and ⟨φ⟩ in Eq. (3.18), is used rather than simply ⟨φ⟩ to distinguish these
quantities when ⟨φ⟩ is computed using Eq. (3.18).
It is worth noting that the SGS variance contribution should always be included when
comparing second order moments to experimental results [191]. This is not always possible
as the SGS variance of a generic quantity is missing in the LES framework unless a model
is provided or a transport equation is computed. The only SGS variance which is always
computed in the LES framework used in this thesis is the SGS kinetic energy, which will always
be included in the comparison of TKE to be shown in the next chapters. For other second order
statistics the inclusion of the SGS variance will be discussed from case to case.
3.2 Discretisation
The numerical discretization used in this work is explained next before discussing the filtered
reaction rate closures, so that the implementation of these closures can also be described
unambiguously.
3.2.1 Numerical schemes
The LES equations (2.10) are solved using a commercial code, Precise-MB [8], which uses a
finite volume approach. This approach is summarised below. The generic transport equation
for a Favre-filtered variable φ˜ with source term ω˙ is written in its integral form as
d
dt
∫
V
ρφ˜ dV +
∫
∂V
ρ(u˜i ·ni)φ˜ dS =
∫
∂V
(
µ˜+
µT
ScT
)∂ φ˜
∂xi
·ni dS+
∫
V
ω˙ dV (3.19)
and is solved at each cell of the domain. The convective fluxes are approximated at each face
of the cell as ∫
S f
ρ(u˜i ·ni)φ˜ dS≈ (ρ u˜i ·ni)S f φ˜ f (3.20)
where S f is the f -th boundary face of the cell, ni is the component of the unity normal to
this surface in direction i and φ˜ f is the value of φ˜ at the center of the face. The latter can be
expressed in function of the values φ˜P and φ˜P+1 at the center of the contiguous cells using a
38 Numerical modelling
second order interpolation. Numerical instability can arise when using high order schemes
in regions of strong gradients produced by the flame. This instability can be treated either
ensuring a minimum number of cells within the flame thickness, or using a filter size bigger
than the cell size as proposed in [17], or using flux limiters (see for example [135]). In the first
case one reduces the relative strength of the convection in respect to the diffusion, which would
be small if the cell size is bigger than the flame thickness. This approach leads to large grid
size and is thus unpractical for the LES considered here. The other two approaches consist
in increasing the diffusion term instead. When the filter size is bigger than the cell size, the
additional diffusion comes from the SGS model, which is usually proportional to ∆. This
approach is also unpractical because ∆ must remain small to have a residual kinetic energy
smaller than 20% of the TKE, as discussed in §3.1.3. When a limiter is used, the additional
diffusion comes from the numerical scheme. Since the amount of this numerical diffusion can
be directly controlled, this approach is preferred. In Precise a blending factor can be used and
φ˜ f is evaluated as
φ˜ f = φ˜
(I)
f +β
(
φ˜ (II)f − φ˜ (I)f
)
(3.21)
where φ˜ (I)f and φ˜
(II)
f are the values obtained using first and second order approximation respec-
tively.
Diffusive fluxes are approximated as
∫
S f
D˜φ
∂ φ˜
∂xi
·ni dS≈
(
D˜φ
∂ φ˜
∂xi
·ni
)
f
S f (3.22)
where D˜φ = (µ˜+µT/ScT ). The gradients at the face centre are computed with a second order
scheme and then interpolated at the cell centre as for the convective flux.
The source term ω˙ cannot be put in conservative form and is approximated on the cell i of
volume Vi as ∫
Vi
ω˙ dV ≈ Vi⟨ω˙⟩Vi (3.23)
where ⟨ ⟩Vi refers to volume average on cell i. The volume average is then approximated to the
value at the cell centre (second order approximation), ⟨ω˙⟩V ≈ ω˙ i.
Finally, time discretisation is performed using a second order Euler scheme in order to have
a good correlation in time. A relaxation factor can be used for reacting cases to avoid numerical
instabilities if the time step is small enough and sub-loops are used. Five sub-iterations are
used at each time step for all variables and all simulations in this thesis. Additional details on
the finite volume approach can be found for example in [65, 135].
3.3 Filtered reaction rate modelling 39
3.2.2 Solver
The final set of discretised equations in Precise-MB [8] is in the form
Cφ = S (3.24)
where C is the coefficients matrix coming from the discretisation, φ is the vector of the
unknowns and S is the vector of the known terms (boundary conditions and source terms). The
linear system in Eq. (3.24) is solved using the TDMA algorithm [65], which needs a tridiagonal
matrix. Matrix C is thus first split as C = A−B, where A is its diagonal. The new system is
Aφ = Bφ +S (3.25)
and it is solved iteratively until a prefixed tolerance is reached. Additional details can be found
in [65].
All simulations discussed in later chapters have been run using 2.60GHz eight-core Sandy
Bridge E5-2670 processors of the Darwin cluster at the University of Cambridge. The time
step in each simulation is constant and is specified to have a maximum CFL number of 0.3
everywhere in the numerical domain. The LES results for each simulation are averaged first
over the sampling period and in azimuthal direction in case of comparison with radial variation
of mean quantities. The reacting cases to be presented in later chapters have been performed
after validating cold flow. These cold flow validations are presented in Chapter 4.
3.3 Filtered reaction rate modelling
Using Eqs. (2.11) and (2.13), the progress variable equation in the set of Eqs. (2.10) becomes
∂ρ c˜
∂ t
+
∂ρ u˜ jc˜
∂xi
=
∂
∂xi
[(
µ˜+
µT
ScT
) ∂ c˜
∂xi
]
+ ω˙ (3.26)
Since in premixed combustion diffusion and turbulent mixing are strongly coupled with the
reaction rate, this coupling must be retained in the modelling. Equations (2.11) and (2.13)
come from incompressible models, thus this coupling has to be considered in the reaction rate
model. In this work three closures for the filtered reaction rate are employed, one based on
algebraic expression and two on presumed PDF approach. These closures are discussed below.
A fourth closure, based on power-law, is also discussed in this section.
40 Numerical modelling
3.3.1 Algebraic closure
The filtered reaction rate of the progress variable is related directly to its scalar dissipation
rate using Eq. (2.19), in which the SDR is modelled with Eqs. (2.21)-(2.22). This model is
derived from a transport equation for N˜c, which involves reaction, diffusion and turbulent
transport [61, 77]. Thus, the algebraic closure accounts for the coupling between these three
terms. This model is derived under the assumption of high Reynolds number and filter width
larger than the laminar flame thickness. Because of this, the turbulent flame speed does not
converge to the laminar flame speed when the flow becomes laminar. Nevertheless, similar
models used for RANS have shown that the turbulent flame speed is well predicted in this
modelling framework [147].
This model is implemented as follows. Equation (3.23) is used to approximate the volume
integral of ω˙ . If a regress variable is used, then ω˙ is negative, and its value at the centre of cell
i can be written as
ω˙ci =−
[
C1D˜
(
∂ c˜
∂x j
∂ c˜
∂x j
)
+FC1C2c˜(1− c˜)/βc
]
i
(3.27)
where
C1 =
2
2Cm−1ρ and
C2 =2Kc
sL
δth
+(C3− τC4Da∆)
(
2
3
u′∆
∆
)
Equation (3.27) is non-linear in c˜ and involves its gradient. The source term is written as
ω˙ ≈ Spc˜+Su (3.28)
Where Sp and Su contribute to the LHS and RHS of Eq. (3.24), respectively. To ensure diagonal
dominance and stability in the linear system of Eq. (3.24), the conditions
Su ≥ 0 and
Sp ≤ 0
(3.29)
must be fulfilled. The only term which can be negative in Eq. (3.27) is
C34 = (C3− τC4Da∆)
3.3 Filtered reaction rate modelling 41
where C3 and C4 are defined in §2.5.1. In figure 3.1, terms in C2 are compared to each other for
different values of u′+∆ and two values of ∆
+. Values of sL, δth and τ are taken from laminar 1D
computation of a premixed stoichiometric methane-air flame. To avoid the infinity for u′+∆ → 0,
C2 is written as
C2 =
2
3
sL
δth
[3Kc+(C∗3 − τC∗4Da∆)] (3.30)
where C∗3 = C3u
′+
∆ /∆
+ and C∗4 = C4u
′+
∆ /∆
+. C34 becomes negative for a certain value of
u′∆ = u
∗
∆ which increases with ∆
+. Since C∗3 and τC
∗
4Da∆ vary monotonically with u
′+
∆ , the
largest negative value is for u′∆ = 0. Since 3Kc is always larger than (C
∗
3 − τC∗4Da∆) when
u′+∆ = 0, C2 is always positive. Same considerations apply for different fuels and equivalence
0 1 2 3 4 5 6
0
2
4
6
8
10
12
14
16
u′+∆
 
 
C ∗3
τ C ∗4Da∆
3Kc
(a)
0 1 2 3 4 5 6
0
2
4
6
8
10
12
14
16
u′+∆
 
 
C ∗3
τ C ∗4Da∆
3Kc
(b)
Fig. 3.1 Comparison of different terms of Eq. (3.30) for increasing values of u′+∆ = u
′
∆/sL, and for (a)
∆= 4δth and (b) ∆= 8δth.
ratios. Thus, an expression which fulfils the conditions in Eq. (3.29) is
Su =−C0D˜c
(
∂ c˜
∂xi
∂ c˜
∂xi
)
n
+FC0C2c˜2n/βc (3.31)
Sp =−FC0C2/βc (3.32)
Subscript n refers to the current time step, as terms in Eq. (3.31) are updated each sub-iteration.
The negative term in equation (3.31) does not affect the stability of system in in Eq. (3.24),
since its magnitude is generally small. It is worth noting that although many constant are
involved in Eq. (2.22), only u′∆ and βc are tuneable while all other parameters were found a
priori using DNS [61].
42 Numerical modelling
3.3.2 Presumed PDF approaches
A. Unstrained flamelet model
A closure for the filtered reaction rate using the unstrained flamelet is well known and this
closure is written as
ω˙ =
1∫
0
ω˙(ζ ) P(ζ ; c˜,σ2c,sgs) dζ = ρ
1∫
0
W˙ (ζ ) P˜(ζ ; c˜,σ2c,sgs) dζ , (3.33)
where P˜(ζ ; c˜,σ2c,sgs) is the density weighted sub-grid PDF of c and ω˙ = ρ W˙ is the flamelet
reaction rate obtained from a freely propagating unstrained planar laminar flame for a given
thermo-chemical condition. The symbol ζ is the sample space variable for reaction progress
variable c. This closure is analogous to the RANS counterpart and was originally proposed
for RANS by Bradley et al. [21] and then used by Cook and Riley [50] for LES, and it has
been used in many past LES of premixed combustion. The sub-grid PDF is obtained using
a presumed shape such as β -function, which has been used for reacting scalars in many past
works, see for example [57, 128, 132, 163, 239]. This presumed shape is given by
P˜(ζ ) =
ζ a−1(1−ζ )b−1
Ψ(a,b)
with Ψ(a,b) =
1∫
0
ζ a−1(1−ζ )b−1 dζ (3.34)
where the parameters a and b are related to c˜ and σ2c,sgs through
a = c˜
[
c˜(1− c˜)
σ2c,sgs
−1
]
, b =
(1− c˜)a
c˜
(3.35)
This density weighted sub-grid PDF is related to its unweighted counterpart through ρ P˜ = ρ P.
The Favre-filtered progress variable, c˜, is obtained using Eq. (3.26). The sub-grid variance is
obtained using its transport equation written as
ρ
Dσ2c,sgs
Dt
≈ ∇ ·
[(
ρD+ρ
νT
ScT
)
∇σ2c,sgs
]
+2ρ
( ˜˙Wc− ˜˙Wc˜)
−2ρ ε˜c+2ρ νTScT (∇c˜ ·∇c˜) (3.36)
It is quite straightforward to derive the above equation using the transport equations for c˜2 and
c˜ 2 because σ2c,sgs = c˜2− c˜ 2. The third and fourth terms of Eq. (3.36) need closures and the
3.3 Filtered reaction rate modelling 43
reaction related term is closed using
˜˙Wc = 1∫
0
W˙ ζ P˜(ζ ) dζ , (3.37)
which is consistent with the closure in Eq. (3.33). The algebraic closure from Dunstan et
al. [61] of Eq. (2.22) is used to model the SGS SDR, ε˜c, in which the SGS variance, σ2c,sgs, is
used in place of c˜(1− c˜) for consistency with the transported equation. Let’s notice that the last
three terms in Eq. (3.36) behave like a source term and are treated similarly to Eq. (3.28), as
−2ρε˜c+2ρ
( ˜˙Wc− ˜˙Wc˜)+2 µT
ScT
∂ c˜
∂x j
∂ c˜
∂x j
≈ Spσ2c,sgs+Su.
The SGS SDR, ε˜c, is always positive and σ2c,sgs appears explicitly in its expression, thus the
dissipation term is modelled implicitly. The last two terms in Eq. (3.36) are always positive
and are then treated explicitly. The condition (3.29) is thus satisfied by imposing
Su = 2(ω˙c− ω˙ c˜)+2 µTScT
∂ c˜
∂x j
∂ c˜
∂x j
Sp =−2ρ ε˜cσ2c,sgs
where the division by σ2c,sgs only appears because of formalism and it is not actually performed
in the computation. Finally, the precomputed filtered values of ˜˙W obtained using the unstrained
flamelet model are stored in a look-up table with c˜ and σ2c,sgs as controlling parameters.
It is worth noting that the integral in Eq. (3.33) can become an issue when the PDF becomes
bimodal for large values of the SGS variance, as the β -function has a very sharp variation near
the boundaries c˜ = 0 and 1, and it approaches infinity at c = 0 and c = 1 in these regions. This
very sharp variation of the PDF near the boundaries can make the integration inaccurate. A
common approach is to exclude points c = 0 and c = 1, find the new bounds ε and 1− ε by
imposing
1−ε∫
ε
P˜(c)dc = 1 (3.38)
and using a Newton-like algorithm, in which ε > 0 is the iteration parameter. In Fig. 3.2
a β -PDF is shown for c˜ = 0.01 and σ2c,sgs = 0.0089. Using 1000 points to discretise c, a
value ε ≈ 6.6× 10−7 is found to satisfy Eq. (3.38). The difference between the real PDF
and the approximated PDF is shown in the inset for region near c = 1. Unless the variable
44 Numerical modelling
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
c
P
(c
)
0.999 0.9995 1
0
200
400
c
P
(c
)
Fig. 3.2 β -PDF of c, obtained using Eq. (3.34) for c˜ = 0.01 and σ2c,sgs = 0.0089. Arrows indicate
presence of very large values. The approximated PDF, obtained using Eq. (3.38), is shown with dashed
line in the inset.
to be integrated approaches zero in this region, a standard numerical integration such as the
trapezoidal rule for Eq. (3.33) would yield a very poor estimate. For example, using c instead
of W˙ , the analytical integral of Eq. (3.33) is c˜ independently of the value of σ2c,sgs. However,
numerical integration with trapezoidal rule gives c˜≈ 0.18, which largely over-estimates the
imposed value c˜ = 0.01. In Fig. 3.3, isolines of c˜, obtained using Eq. (3.33) (with c integrated
instead of W˙ ) and (3.38), are shown. These isolines are not parallel, despite 10,000 points
are used to discretise c. This numerical issue is relevant whenever a quantity which does
c˜
g˜
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Fig. 3.3 Isolines of c˜, obtained using Eq. (3.33) (with c integrated instead of W˙ ) and (3.38) (continuous
lines), are shown on the plane c˜-g˜, where g˜ = σ2c,sgs/[c˜(1− c˜)]. Isolines obtained using Eq. (3.39) are
also shown (dashed lines).
3.3 Filtered reaction rate modelling 45
not approach zero at c = 0 and c = 1 has to be integrated, such as heat capacity, enthalpy of
formation and molecular weight used in the numerical procedure to be described in §3.4, and
would lead to large numerical problems and eventually divergence of the solution. It is worth
noting that this issue cannot be solved by simply increasing the number of discretisation points
because of the presence of the steep variation. The issue can be avoided very easily if one uses
integration by part to include CDF (cumulative distribution function), C , of P since the CDF
does not have such sharp variations. This is outlined below using a generic variable Q,
Q =
1∫
0
Q(ζ )P(ζ ) dζ =
1∫
0
Q d(C ) = Q(ζ = 1)−
1∫
0
C Q′ dζ , (3.39)
where the derivative Q′ = ∂Q/∂ζ is usually well behaved in the domain 0≤ ζ ≤ 1. Thus, the
integral in Eq. (3.39) can be evaluated accurately, as can be inferred in Fig. 3.3. This approach
implies that C is known or computed accurately, and it is computed using Matlab implicit
routines in this work.
B. Strained flamelet model
The strained flamelets approach used in this study is an extension of that proposed in [121] for
RANS methodology and the filtered reaction rate is modelled as
ω˙ =
1∫
0
ψmax∫
ψmin
ω˙ P(c,ψ; c˜,σ2c,sgs, N˜c) dψ dc (3.40)
where ψ is the sample space variable for Nc, which is the SDR representing the effect of
turbulent strain on the flamelets. The sub-grid joint PDF is written as P(c,ψ) = P(c)P(ψ|c)
using Bayes theorem. The marginal PDF of c is modelled using a β -function, Eq. (3.34),
discussed in §3.3.2, and the σ2c,sgs required is obtained through Eq. (3.36). The reaction term
in the variance equation is closed in a manner similar to Eq. (3.40) for consistency. The
conditional PDF P(ψ|c) is modelled as a log-normal PDF following [121], where it is written
as
P(ψ|c) = 1
(ψ|c)σ√2π exp
{
− 1
2σ2
[ln(ψ|c)−µ]2
}
(3.41)
where µ and σ2 denote respectively mean and variance of ln(ψ|c) and are related to the
conditional mean ⟨ψ|c⟩ and conditional variance G2N as
⟨ψ|c⟩= exp(µ+σ2/2), G2N = ⟨ψ|c⟩2[exp(σ2)−1] (3.42)
46 Numerical modelling
A nominal value of σ = 0.3 is taken in this thesis on the basis of previous works [121], and
results to be presented in next chapters are not unduly sensitive to this parameter when it is
increased to 0.5. The Favre filtered SDR required for the conditional PDF is approximated
as N˜c ≈ ε˜c because the premixed flame front is unlikely to be resolved in a typical LES and
thus D˜(∇c˜ · c˜)≪ ε˜c. The SGS dissipation rate is estimated using Eq. (2.22), in which the
SGS variance, σ2c,sgs, is used in place of c˜(1− c˜) as in the unstrained flamelets closure. The
conditional mean can be expressed as
⟨ψ|c⟩=
Nmax∫
ε
ψ(c,N)P(ψ|c; N˜)dψ ≈
amax∫
0
ψ(c,a)P(a)da (3.43)
where ψ(c,a) is the SDR in a flamelet subject to strain rate a. Following Kolla et al. [121],
ψ(c,a) is approximated as ψ(c,a) ≈ ψ(c∗,a) f (c), where c∗ corresponds to the location of
peak heat release rate. Using Eq. (3.40) with N˜c instead of ω˙ one then obtains
ε˜c ≈ N˜c ≈
1∫
0
⟨ψ|c∗⟩ f (c)P˜(c)dc (3.44)
and thus
⟨ψ|c⟩ ≈ ε˜c f (c)∫ 1
0 f (c)P˜(c)dc
(3.45)
It may be possible to use another flamelet attribute or variable, for example peak value of
atomic hydrogen mass fraction [116], to represent the straining effect. A quantity related to
scalar gradient is a natural first choice to represent turbulent strain and so the dissipation rate is
adopted for this study as suggested by Libby and Williams in their asymptotic analysis [138].
Furthermore, atomic hydrogen is produced through chemical reactions and thus its value would
depend on the progress of the reaction. Hence, using it as an independent variable to represent
the fluid dynamic straining is questionable.
The precomputed filtered values of ˜˙W obtained using the strained flamelets model are stored
in a look-up table with c˜ and σ2c,sgs as controlling parameters as for the unstrained flamelet
model, and ε˜c is used as a third parameter for the strained flamelets.
C. Role of SGS variance
The SGS variance of progress variable, σ2c,sgs, is modelled in earlier studies typically using
an expression developed for passive scalars, i.e. σ2c,sgs ≃ A ∆2(∇c˜ ·∇c˜), where the model
parameter A is usually obtained through dynamic evaluation and typically takes a value of
3.3 Filtered reaction rate modelling 47
about 0.5. The relevance of such modelling for turbulent premixed combustion is an open
question on physical grounds because chemical reactions ought to influence this variance
as was remarked in [182] and an algebraic model for σ2c,sgs including this effect is yet to be
developed. Pitsch [185] also noted that this model, as well as linear relaxation model for the
scalar dissipation rate, can lead to under-prediction of the SGS variance in reacting flows. These
points are of specifically high importance for the presumed sub-grid PDF approach because
this PDF has a direct dependence on σ2c,sgs and influences the filtered reaction rate value.
The SGS progress variable equation written earlier as Eq. (3.36) is rewritten below for
convenience as
∂ρ σ2c,sgs
∂ t︸ ︷︷ ︸
T1
+∇ ·
(
ρ u˜σ2c,sgs
)
︸ ︷︷ ︸
T2
≈ ∇ ·
[(
ρDc+ρ
νT
ScT
)
∇σ2c,sgs
]
︸ ︷︷ ︸
T31+T32
+2ρ
( ˜˙Wc− ˜˙Wc˜)︸ ︷︷ ︸
T4
−2ρ ε˜c︸ ︷︷ ︸
T5
+2ρ
νT
ScT
(∇c˜ ·∇c˜)︸ ︷︷ ︸
T6
(3.46)
The terms T1 and T2 are the unsteady and advective terms respectively. The molecular and
turbulent fluxes of the SGS variance are respectively denoted using T31 and T32. The other three
terms, T4, T5 and T6, signify the source and sink contributions from SGS chemical processes,
dissipation of the variance, and its production through interaction of SGS scalar flux and
gradient of c˜. It is important to recognise these physical processes in order to evaluate the
validity of the commonly used model for σ2c,sgs, which results by equating T5 and T6 after
approximating the dissipation rate as ε˜c ≃ νTσ2c,sgs/(A ScT ∆2). As one shall see below, this
model cannot be employed for LES of premixed flames.
The above equation is studied using an order of magnitude analysis as follows. The spatial
derivatives of filtered quantities, time derivative, and density are scaled using ∆, ∆/Uref, and
ρu respectively. The filtered velocity is scaled using a reference velocity Uref. The molecular
diffusivity is scaled using the laminar flame scales, sL and δth, and the SGS viscosity is scaled
as u′∆∆. The progress variable gradient is produced predominantly by the chemical reaction and
thus the appropriate scaling for ε˜c is sL/δth. However one can also scale this term using u′∆/∆
if the combustion effects are ignored and the SGS turbulence is presumed to be responsible for
the progress variable gradient which can occur through mixing of unbunt and burnt mixtures.
The above scalings yield
T1 ∼ O
(
ρusL
δth
;
Uref
u′∆
1
Da∆
)
T2 ∼O
(
ρusL
δth
;
Uref
u′∆
1
Da∆
)
(3.47)
48 Numerical modelling
T31 ∼ O
(
ρusL
δth
;
1
Da∆Re∆
)
T32 ∼O
(
ρusL
δth
;
1
Da∆
)
(3.48)
T4 ∼ O
(
ρusL
δth
; 1
)
T5 ∼ O
(
ρusL
δth
; 1
)
T6 ∼O
(
ρusL
δth
;
1
Da∆
)
(3.49)
where Re∆ = u′∆∆/(sLδth) is the SGS Reynolds number. If the local SGS Damköhler number
is large then the leading order balance is between T4 and T5 which implies that c˜ρ sL/δth ≃
ρ νTσ2c,sgs/∆2, suggesting that the SGS variance is proportional to Da∆. However, the Damköh-
ler number is finite for most practical simulations (see for example §6.5) and thus the other
terms in the balance equation cannot be neglected. If one takes a balance among sources
and sinks as it is typically done in non-reacting flows then T5 ≃ T4 +T6, which would give
σ2c,sgs ≃A1
(
c˜Da∆+∆2|∇c˜|2
)
, where A1 is a model parameter. This balance is unacceptable
because other terms in Eq. (3.46) will be of similar magnitude since they have same dependence
on Da∆.
As noted earlier, the commonly used model for σ2c,sgs ≃A ∆2(∇c˜ ·∇c˜) is obtained through
T5 ≃ T6 which can be justified only if
(
u′∆/∆
)
is used to scale ε˜c. This specific scaling gives
T5 ∼ O
(
ρusL
δth
;
1
Da∆
)
and ignores the important aspect of premixed combustion, i.e., the progress variable gradient
is produced predominantly by combustion, which is also recognised in an earlier study inves-
tigating SGS variance and dissipation rate of a conserved scalar [182]. This implies that the
commonly used model will underestimate σ2c,sgs, which is verified in Fig. 3.4 for unstrained
flamelet modelling of F1 flame of [46], to be investigated in Chapter 6, belonging to the thin
reaction zone of the premixed combustion diagram [180]. This figure is shown for two different
grid sizes at an arbitrarily chosen time, and it is obtained by plotting the modelled values
against σ2c,sgs computed using its transport equation. The data points are coloured using c˜ value.
If the modelled values agree with σ2c,sgs then the data point must cluster around the diagonal
line (A = 1). The data seems to cluster predominantly around the dash-dotted line which
has a slope of about 0.125 which suggests that A ≈ 8. The scatter is reduced and thus this
clustering is more apparent when the larger of the two grids is used. It is clear that the model
for the SGS variance underestimates σ2c,sgs as suggested by the above scaling analysis. This
reduced modelled variance, compared to σ2c,sgs from its transport equation, will give larger
values of ω˙ for 0.6≤ c˜≤ 0.8 when the unstrained flamelet model is used (see Fig. 6.1). This
issue will become even more critical in low-turbulence flame configurations, to be analysed
in Chapter 8. The difference between modelled and transported equation can be of different
3.3 Filtered reaction rate modelling 49
Fig. 3.4 Scatter plot of computed σ2c,sgs and its modelled value using A ∆2 |∇c˜|2 with A = 1 in the top
row. The results for the revised model, A1
(
c˜Da∆+∆2|∇c˜|2
)
with A1 = 0.15, are shown in the bottom
row. The first and second columns are for two different grid sizes, of 1.5M and 4.2M cells, for F1 flame
of [46] respectively.
orders of magnitude, as shown in Fig. 3.5 for a flame, to be investigated in Chapter 8, belonging
to the corrugated flamelet regime, which would result in a very inaccurate filtered reaction
rate. This overestimate of the filtered reaction rate resulting from the incorrect (modelled) SGS
variance suggests that one may need strained flamelets model for LES of premixed combustion.
If one uses the strained flamelets model with σ2c,sgs obtained from its transport equation then the
filtered burning rate is underestimated leading to substantial underestimates of various reactive
scalar mass fractions as will be observed in §6.4.
If one were to use the revised algebraic model given above, as shown in the bottom row
of Fig. 3.4, then the SGS variance would be severely overestimated because Da∆ varies from
O(0.1) to O(1) (see §6.5) yielding the modelled SGS variance substantially larger than its
maximum limit of 0.25. Indeed this overestimate can be seen in the bottom row of Fig. 3.4 for
both the grids if one were to use A1 = 1 instead of 0.15 used for this figure. This substantial
overestimate also suggests that the transport terms in Eq. (3.46) cannot be neglected to get
meaningful values for the SGS variance and this equation need to be included for LES of
50 Numerical modelling
Fig. 3.5 Scatter plot of computed σ2c,sgs and its modelled value using A ∆2 |∇c˜|2 with A = 1.
premixed combustion using the unstrained flamelet model. The additional computational cost
incurred by including this transport equation is small which is fully justified through correct
representation of turbulent premixed combustion physics as discussed above.
3.3.3 Estimation of sub-grid scale velocity
The velocity scale u′∆ influences ε˜c in Eq. (2.22) in two ways. The direct influence is felt
through the linear dependence appearing in Eq. (2.22) and additional effects come indirectly in
a non-linear manner through Ka∆ dependence of C3 and C4. Thus, the modelling of u′∆ needs to
be considered carefully. In this work, four different algebraic models are used to estimate u′∆
and these models are presented below.
1. Considering that (−τi jS˜i j) represents a production of small-scale kinetic energy [134,
140], the SGS-velocity can be estimated directly form the residual model implemented.
Dimensionally, the quantity ∆|S˜|, is a velocity, thus u′∆ can be considered proportional to
this value. A model based on a shear stress related closure was proposed by Lilly [140]
and it is given by
u′∆ =
νT√
3/2CL∆
, (3.50)
where νT = (Cs∆)2
√
2S˜i jS˜i j is the turbulent viscosity modelled using Smagorinsky
model. The model constant is CL = 0.094.
2. A second model based on scale-similarity for velocity is written as [196]
u′∆ =Cq
∣∣u˜i− ̂˜ui∣∣ (3.51)
3.3 Filtered reaction rate modelling 51
where ̂˜ui is the velocity field obtained using a test filter. Different estimations of Cq have
been given in the literature [11, 48, 159]. A dynamic evaluation of the model constant
was proposed by Cook [49], but the constant requires an estimation of the integral length
scale which may be difficult to provide. If a a Gaussian test filter is used, under the
assumption that the implicit filter is also Gaussian (see §3.1.3), then Cq = 1 gives the
best approximation for u′∆ [196]. This value is thus used in this thesis.
3. A third model, based on scale-similarity for kinetic energy proposed by Bardina et
al. [11], is written in its Galilean invariant form [64] as
u′∆ =C
′
q
∣∣∣̂˜uiu˜i− ̂˜uî˜ui∣∣∣1/2 , (3.52)
with C′q = 0.126.
4. A fourth model is based on a hypothesis of constant energy transfer down to ∆ scale. In
this hypothesis one can write [196]
u′3(ℓ)
ℓ
=
u′3∆
∆
= ε
Also, if one takes ε ≈ ε∆ and supposing that νT ≫ ν (true for ∆≫ ηk), then
u′3∆ = ∆ε∆ ≈ 2∆νt S˜i jS˜i j (3.53)
The Smagorinsky model for νT in Eq. (3.3) gives
S˜i jS˜i j =
ν2T
2C2s∆4
(3.54)
Substituting Eq. (3.54) into Eq. (3.53) yields
u′∆ =
νT
C2/3s ∆
(3.55)
This model allows one to relate SGS velocity fluctuation directly to the Smagorinsky
model constant, and is strictly valid only when Eq. (3.53) holds. Although the assumption
of ε ≈ ε∆ seems too restrictive, the Smagorinsky model typically overestimates νT [99,
160] and thus the above assumption is expected to be reasonable.
5. Colin et al. [47] derived another model by expanding Eq. (3.51) using Taylor series
truncated to second order and then applying the curl operator to exclude dilatation effects.
52 Numerical modelling
This expression is
u′∆ = c2∆
3 ∣∣∇2 (∇× u˜)∣∣ (3.56)
where c2 = 2 is typical value suggested by [47]. The above expression was modified by
multiplying it by (0.1∆/∆x)1/3, where ∆x is the mesh size estimated from local numerical
cell volume, to accommodate cases with the filter width larger than the numerical cell
size [240].
One can also obtain this velocity scale using a transport equation for the residual kinetic energy,
kr, through u′∆ =
√
2kr/3. This is also explored in this work by solving an ad hoc modelled
transport equation for kr, and its modelling was discussed in §3.1.2.
The assumption that the sub-grid scale velocity and strain are proportional is arguable [196].
Moreover, the strain in combustion is subjected to dilatation due to heat release. Since
models (3.50) and (3.55) are based on strain, the effect of dilatation is included in these
modelling. Nevertheless, the SGS velocity still would go to zero in a laminar flame if the SGS
model constant is computed dynamically, as this constant goes to zero in laminar regions. Even
though it is argued that dilatation effect must be excluded from the modelling of u′∆ [47], it is
not completely clear, as dilatational terms appear in the kr equation, which, by its definition, is
a residual kinetic energy and not a residual turbulent kinetic energy. Thus, models (3.51) and
(3.52), which are based on definition similar to kr and also capture dilatational effects, are also
related to the kinetic energy at smallest resolved scale, following scale similarity. Effects of
dilatation on the SGS velocity modelling are investigated in Chapter 8.
3.3.4 Dynamic formulations
In general, dynamic modelling is based on the assumption that an unresolved quantity of
interest is similar in neighbouring scales [196]. This scale similarity assumption for flame
wrinkling and flame surface density have been proposed and used in past studies [43, 88, 114,
115, 117, 185, 241]. Many earlier studies [18, 194, 237] have recognised that these quantities
are closely related to |∇c| and thus Nc. Hence, the scale similarity concepts can be equally
applied to the scalar dissipation rate field also.
A. Dynamic modelling for βc
The following description is based on the algebraic model of §3.3.1. The same description is
applied to unstrained and strained flamelets models by using σ2c,sgs in place of c˜(1− c˜) in all
equations below. Equation (2.22) can be written as ε˜c = f1 c˜(1− c˜)/βc for conciseness and
a comparison of this expression to Eq. (2.22) defines f1. The dynamic evaluation of βc is as
3.3 Filtered reaction rate modelling 53
follows. Firstly, one writes
ρ̂N˜c = ψ̂1+ ψ̂2/βc and (3.57a)
ρ̂̂˜Nc = ρ̂ ̂˜D(∇̂˜c ·∇̂˜c)+ ρ̂ f̂1̂˜c(1−̂˜c)/βc (3.57b)
with ψ1 ≡ ρD˜(∇c˜ ·∇c˜) and ψ2 ≡ ρ f1c˜(1− c˜), by using a test filter of size ∆̂= 2∆. Here, the
notations ̂˜Nc and f̂1 imply that they must involve quantities obtained using the test filter. A
Gaussian filter having ∆̂ as its width is used. Subtracting Eq. (3.57b) from Eq. (3.57a) and
recognising that ρ̂N˜c ≈ ρ̂ ̂˜Nc one can obtain an expression for βc. Using both parts of Eq. (3.57)
one obtains
βc =
ρ̂ f̂1̂˜c(1−̂˜c)
ψ̂1− ρ̂ ̂˜D(∇̂˜c ·∇̂˜c)+ ψ̂2/βc (3.58)
where the value of βc on the RHS can be evaluated using information from previous time step.
An additional averaging with the neighbouring cells is performed for numerical stability [212].
3 to 5 neighbouring numerical grid cells in each spatial direction were used for the test-filtering
and results were found to be insensitive to this choice in this range. To satisfy the realisability
conditions of N˜c ≥ 0 and 0≤ c˜≤ 1 this parameter must satisfy βc ≥ 2/(2Cm−1) [39] in case
of algebraic model. Thus, one gets
βc = max
(
2
2Cm−1 , Eq. 3.58
)
. (3.59)
It is worth noting that the frequency at which this condition is applied is low in the simulations
to be presented in next chapters, which is verified by looking a posteriori at the PDF of βc (see
Figs. 5.32 and 7.32). The condition βc > ε , with ε > 0 being a small number, is used for the
other closures, where ε is chosen appropriately to avoid numerical issues or singularity.
B. Dynamic power-law model
Another dynamic model explored for N˜c is based on a power-law. The static version of this
model is studied in [61, 77]. This choice was motivated by many earlier studies [17, 38, 42–
44, 114] attempting to model the flame wrinkling factor or the generalised flame surface density
using the power-law and these quantities are closely related to N˜c as noted earlier in this section.
This model for N˜c is written as [61, 77]
ρN˜c = ρD˜(∇c˜ ·∇c˜)
(
∆
ηi
)α
= ψ1
(
∆
ηi
)α
(3.60)
54 Numerical modelling
where α is the exponent and ηi is the inner cut-off scale, which scales as δth [38, 61, 115, 202].
Using a constant and non-zero value for α in Eq. (3.60) does not readily satisfy the expected
asymptotic behaviour of ρN˜c → ρD(∇c ·∇c) in the limit of ∆→ 0. To overcome this, a
dynamic evaluation can be followed. Following a method similar to that described for βc, one
gets
α =
ln[ψ̂1/ρ̂
̂˜D(∇̂˜c ·∇̂˜c)]
ln(∆̂/∆)
. (3.61)
in which the assumption that α does not change during test filtering operation has been made.
One can also write Eq. (3.60) as
ρN˜c = ψ1
(
1+
∆
ηi
)α1
(3.62)
using a different exponent α1 to satisfy the condition that N˜c → Nc as ∆→ 0. The dynamic
procedure with Eq. (3.62) gives
α1 =
ln[ψ̂1/ρ̂
̂˜D(∇̂˜c ·∇̂˜c)]
ln
[
(1+ ∆̂/ηi)/(1+∆/ηi)
] . (3.63)
If one were to have the filtered dissipation rates from Eqs. (3.60) and (3.62) equal then α1 ≈ α
for ∆/ηi ≫ 1. This can be shown as follows. The relation between α and α1 can be found by
comparing Eqs. (3.60) and (3.62):
α =
ln(∆/ηi)
ln(1+∆/ηi)
α1 = fI(∆/ηi)α1 (3.64)
which is the relation α and α1 must satisfy in order to obtain the same N˜c. The actual relation
is instead implied by the dynamic approach. Comparing Eqs. (3.61) and (3.63), and assuming
∆̂= k∆, with k > 1 being a constant, one gets
α =
lnk
ln
[(
1+ k∆/ηi
)
/
(
1+∆/ηi
)]α1 = fII(∆/ηi)α1 (3.65)
The variations of functions fI and fII with ∆/ηi are compared in Fig. 3.6. The negative brunch
in fI is due to the fact that Eq. (3.60) does not converge to ρD˜(∇c˜ ·∇c˜) for ∆→ 0. This
behaviour can be corrected using a bridging function [61, 77, 79], as shown in the picture. This
bridging function is used for small values of ∆/ηi which are not relevant for the LES analyses
in this thesis, and thus will not be further discussed here.
3.3 Filtered reaction rate modelling 55
0 0.5 1 1.5 2 2.5 3−30
−20
−10
0
10
∆/ηi
f
(∆
/
η
i)
fI
fI I
Fig. 3.6 Comparisons of functions fI and fII in Eqs (3.64) and (3.65) respectively. The dashed line
represents fIFb, whereFb is the bridging function discussed in [61, 77, 79].
Functions fI and fII do not match for small values of ∆/ηi. For a fixed value of ∆/ηi
and a given value of α1, taken for example from DNS data, in order to obtain the same N˜c in
Eq. (3.60), one should use α = fIα1. However, the value implied by the dynamic procedure
is α = fIIα1 > fIα1. For this reason one can expect over-predictions of N˜c in LES in respect
to the value from DNS, when ∆/ηi is small. Curves fI and fII converge to each other as
∆/ηi increases, so differences between LES and DNS are expected to be small in this case.
These differences are also expected to be smaller in real cases, because the volume average in
Eq. (3.61), not taken into account in Fig. 3.6, is expected to smooth down fII . This analysis
must be considered when assessing power-law modelling for LES with a priori DNS analyses.
In conclusion, it is immaterial if one uses Eq. (3.60) or (3.62) in LES with sufficiently large
filter widths. It is worth noting that the exponential constant is always positive in the dynamic
computation either using Eq. (3.61) or (3.63).
The validity of dynamic models in Eqs. (3.58) and (3.61) strongly depends on validity of
scale-similarity for quantities at ∆ and ∆̂. Specifically the scale-similarity must hold for ∇c˜
for Eq. (3.61) and for u′∆ and c˜ for Eq. (3.58) (the term ψ̂1 involving gradients in Eq. (3.58) is
small compared to the other terms in the sum and thus does not affect this observation on scale
similarity). As noted earlier, the above dynamic procedure is similar to that studied in [78],
however, the modelling of SGS dissipation rate, ε˜c, in Eq. (2.22) is not the same as that used
in [78].
56 Numerical modelling
Fig. 3.7 Strained flamelet configuration.
3.3.5 Laminar flame computation
The unstrained and strained laminar one-dimensional flames are computed using PREMIX [106]
and OPPDIFF [146] codes respectively, which solve the reacting incompressible one-dimensional
Navier-Stokes equations using finite differences technique and adaptive grid. The GRI 3.0
chemical kinetic mechanism is used for methane-air combustion and a mechanism described
by [224] is used instead in case of propane-air mixtures. In the unstrained flamelet configura-
tion reactants enter from one side and propagate in the opposite direction. The velocity in the
direction tangential to the flame is zero and the flame freely propagates toward the reactants, as
was shown schematically in Fig. 2.1. The flow is accelerated by the flame and thus reactants
and products move in the same direction. In the strained flamelet configuration reactants and
products enter from opposite sides at velocities Ur and Up respectively, as shown schematically
in Fig. 3.7. The flame is stretched because the tangential velocity increases rapidly moving
away from the centreline and thus the strain rate is a = ∇t ·u ̸= 0, where ∇t indicates that the
divergence involves only the tangential component of u. This strain rate is directly proportional
to the inlet velocities and inversely proportional to the distance L between the jet inlets in this
configuration [191], a∼ (|Ur|+ |Up|)/L. In Figure 3.8 the variation of the consumption speed,
sc, with a is shown along the centreline for methane-air flame at stoichiometric condition.
The different rate of strain are obtained by varying Ur and Up in OPPDIFF. The consumption
speed is a useful indicator of the overall burning rate and shows that this rate does not change
significantly for low rates of strain as one would expect, than decreases almost linearly to
suddenly drop for higher strain rates. Typical strained flamelets structures obtained using
3.3 Filtered reaction rate modelling 57
0 200 400 600 800 1000
0
0.5
1
1.5
2
a [s−1]
s
c
/
s
0 c
Fig. 3.8 Variation of consumption speed, sc (normalised with the unstrained flamelet consumption speed,
s0c), with characteristic strain rate for stoichiometric strained flamelets. The strain rate is estimated as
a≈ (|Ur|+ |Up|)/L. Three particular values of strain rate are marked with circles.
0 0.1 0.2 0.3 0.4 0.5 0.6
−10
−8
−6
−4
−2
0
2
4
x [mm]
U
[m
/
s]
a
(a)
−1 −0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
(x − x0)/δth
c
a
(b)
Fig. 3.9 Profiles of (a) axial velocity, U , and (b) progress variable, c, along the centreline for the three
values of strain rate marked in Fig. 3.8. x0 is the location at which c = 0.5.
OPPDIFF are shown in Fig. 3.9 for the three strain rates marked with circles in Fig. 3.8 . These
different velocities imply increasing values of strain on the flame. The flame distance from the
stagnation plane decreases with increasing strain rate as observed in Fig. 3.9a. Although the
velocity distributions across the three strained flamelets are significantly different from each
other, their respective progress variable profiles are similar. The main effect of the increased
strain is to decrease the reaction rate as shown in Fig. 3.10a. This corresponds to decreasing
values of scalar dissipation rates as shown in Fig. 3.10b. Thus, for each value of c the scalar
dissipation rate varies monotonically and thus this variable is very suitable to parametrise the
rate of strain in the procedure explained in §3.3.2, as already noted in [121]. Species mass
fractions are also affected by the strain. In Fig. 3.11 some major and minor species mass
fraction profiles across the flamelets are shown for different values of a. As one can expect,
minor species are more affected by strain than major species because of the significant variation
of reaction rate with the strain. The local behaviour of a species k with strain depends on the
58 Numerical modelling
0 0.2 0.4 0.6 0.8 1
0
500
1000
1500
2000
2500
3000
c
ω˙
c
[k
g
m
−
3
s−
1
]
a
(a)
0 0.2 0.4 0.6 0.8 1
0
1000
2000
3000
4000
c
N
c
[s
−
1
]
a
(b)
Fig. 3.10 Profiles of (a) reaction rate, ω˙c, and (b) scalar dissipation rate, Nc, in c space for the three
strain rates marked in Fig. 3.8.
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
c
Y
i
 
 
a
a
a
O2
CO2
2 × H2O
(a)
0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
0.04
0.05
c
Y
i
 
 
a
a a
CO
10×H2
10×OH
(b)
Fig. 3.11 Profiles of (a) major and (b) minor species mass fractions in c space for the three strain rates
marked in Fig. 3.8.
local balance between diffusion, reaction and convection. The balance equation of species k for
the one-dimensional laminar steady flamelet can be written as
dρUYk
dx
=
d
dx
(
ρDk
dYk
dx
)
+ ω˙k (3.66)
Integrating this equation from −∞ to x and supposing that the derivatives in x are zero at −∞
one obtains
Yk(x) =
ρuUrYk,u
ρU
+
Dk
U
dYk
dx
+
1
ρU
x∫
−∞
ω˙k(x′)dx′ (3.67)
3.4 Mathematical model and equations summary 59
0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
0.5
1
c
ρ
U
[k
g
s−
1
m
−
2
]
a
-1.455
-0.704
-0.174
3.955
2.260
0.565
Fig. 3.12 Mass flux profiles in c space for the three strain rates marked in Fig. 3.8. The arrows at c = 0
and c = 1, and the corresponding labels, indicate the mass fluxes at the boundaries.
The terms at the RHS in Eq. (3.67) represent from left to right the ratios of initial conditions,
diffusion and reaction with the mass flux, ρU . It is important to note that while the reaction rate
can decrease with the strain rate as observed in Fig. 3.10a, this decrease does not necessarily
correspond to a decrease of the reaction term in Eq. (3.67). The flow rate across the flame can
in fact be locally smaller for higher strain rate. This is observed in Fig. 3.12, where the flow
rate in the flame with highest strain becomes smaller than for lower strains after a certain value
of c. Thus, depending on the local balance between reaction and mass flux, the third term at
the RHS in Eq. (3.67) can increase with the strain. This can lead to unexpected behaviour in c
space, as for oxygen observed in Fig. 3.11a, depending on the local balance between the above
two terms and the diffusion term.
Although a significant variation of species with the strain is observed, one must consider
that flamelets in turbulent configurations are free to escape from regions of high strain [188],
and thus levels of strain in turbulent flames may be smaller than those observed for this laminar
configuration. Moreover, unsteady strain on the turbulent flame front may not be well captured,
and the use of steady flamelets must be considered here as a first step while more accurate
configurations will be considered in future studies. It is also worth noting that the laminar
flamelet may be harder to extinguish in this configuration [191]. Additional information for
this particular configuration can be found in [119, 121].
3.4 Mathematical model and equations summary
The numerical procedure which is followed in one LES time step is as follows. The Favre
filtered momentum equation in Eqs. (2.10) is solved first and the continuity equation is then
used to correct pressure using SIMPLEC algorithm [58]. The sub-grid stresses are modelled
60 Numerical modelling
using the localised dynamic Smagorinsky model, Eq. (3.3), unless otherwise specified. The
one equation closure of Eqs. (3.4) and (3.13) is used for some of the simulation to be presented
in next chapter for comparison purposes. The transport equations for c˜, Eq. (3.26), and SGS
variance in case of PDF closures, Eq. (3.36), are used to represent the thermochemistry using
flamelets hypothesis. For all combustion closures used in this study, a transport equation for
the Favre filtered total enthalpy, h˜, is included to account for possible sub-adiabaticity of the
mixture, and takes the form specified in Eqs. (2.10). The reaction rate in the progress variable
equation is modelled using Eqs. (3.27), (3.33) and (3.33) respectively for algebraic, unstrained
and strained flamelets models. Equations (2.19) and (3.60) are used instead in case of the
power-law closure. The temperature is computed from the computed total enthalpy h˜ through
T˜ = T0+
h˜− ∆˜h0f
C˜p
(3.68)
where T0 is the unburnt gas temperature, and ∆˜h
0
f and C˜p are the enthalpy of formation and
specific heat capacity at constant pressure of the gas mixture respectively. This specific heat
capacity is temperature dependent as described in [206]. The filtered density field is computed
using the state equation in Eqs. (2.10), in which W˜ is the molecular weight of the mixture.
The values for ∆h˜0f , C˜p and W˜ are obtained in case of presumed PDF model using equations
similar to Eq. (3.33) or (3.40), and stored in a look-up table along with ˜˙W and ˜˙Wc. In case of
algebraic and power-law models, a lookup table is constructed by filtering unstrained laminar
one-dimensional data with a Gaussian filter of width ∆r, where ∆r is the typical value of ∆ in the
expected flame region in LES. These values are parametrised with c˜ and accessed in run-time.
For cases involving mixing with entrained air, a transport equation for a filtered fluid marker,
Z˜, is also included and this equation is
∂ρZ˜
∂ t
+
∂ρ u˜ jZ˜
∂x j
=
∂
∂x j
(
ρ
µ
Sc
∂h
∂x j
)
− ∂
∂x j
τ˜ZRj (3.69)
where Sc is the Schmidt number and τ˜ZRj is the turbulent transport of Z˜. This dilution is included
using a mixing rule [122]
φ˜ = Z˜φ˜reac+(1− Z˜)φair (3.70)
where φ˜ is a generic variable representing C˜p or ∆˜h
0
f or W˜ or species. The subscript ‘reac’
denotes that these values are taken from the lookup table for the local c˜ value, while the
subscript ‘air’ denotes their values in the air stream. The values obtained using Eq. (3.70)
are used in Eqs. (3.68) and in the state equation. It is worth noting that Eq. (3.70) is used to
3.4 Mathematical model and equations summary 61
account for dilution rather then partial premixing effects, and that the filtered reaction rate in
Eq. (3.26) does not include contributions that could arise from mixture fraction gradient in
partially premixed combustion [27, 205]. Finally, the diffusion and turbulent transport terms
in all scalar equations are closed using Eqs. (2.11) and (2.13) respectively. All the equations
are discretised as explained in §3.2 and the solution procedure is iterated five times within the
SIMPLEC algorithm.
The boundary and initial conditions to be used while solving the LES equations with the
closures discussed in this chapter depend on the validation cases. Thus, these conditions are
discussed along with the respective test flames in the following chapters.

Chapter 4
Validation cases
Four cases in total are used for validation of numerical methods, codes and combustion models.
One of them is only a pure mixing case and the other three cases are reacting flows. These four
cases and their cold flow results are discussed in this chapter. These configurations are chosen
specifically to have a wide range of fluid mechanical features and conditions, and turbulence-
chemistry interaction leading to a range of combustion conditions so that the robustness of the
combustion closure can be tested.
4.1 Propane-air mixing
The SANDIA non-reacting propane round jet [214] is chosen as first case to validate turbulence
modelling and procedure. The purpose here is to correctly capture turbulence and mixing
properties as baseline for more complex reacting cases.
The main geometrical and physical parameters are shown in table 4.1 and a schematic of
the computational domain is given in figure 4.1.
4.1.1 Computational model
A slip flow condition is assigned on the outer surfaces of the cylindrical three-dimensional
domain, and a constant-pressure condition is assigned to the outlet. The domain boundaries
extend 13D in the radial direction and 70D from the nozzle exit in the axial direction. These
values are set by extracting positions at which gradients go to zero in the experimental results.
A pipe extending 11D upstream the nozzle exit is also included into the domain in order to
have fully developed turbulence at the exit. The sensitivity to grid size and wall boundary
conditions, as well as the effect of the pipe in the geometry, is investigated performing four
simulations, which are summarised in table 4.2. A dynamic Smagorinsky model is used for the
64 Validation cases
Slip flow 
O
ut
le
t 
Slip flow 
Non-slip flow 
Fig. 4.1 Sketch of the numerical domain used for the Sandia propane-air non-reacting case in [214].
sub-grid stresses in all simulations, and the WALE model [172] is used for the wall boundary
conditions when required. A flat velocity profile based on the mass-flow rate is assigned at
the inlet for cases A to C, as velocity and turbulence are expected to evolve within the pipe.
A smoothed top-hat velocity profile is used for case D, which does not include the pipe. This
velocity profile is given by [200]
⟨U⟩= U0
2
(
1+
r
2θ
)
, (4.1)
where U0 is the centreline velocity, r is the radial distance, and θ is the momentum thickness
assumed to be θ = D/20. Finally, a uniform velocity of Uair = 9.2m/s is assigned for the
coflow in all cases listed in table 4.2.
Table 4.1 Geometrical and physical parameters for the Sandia propane-air non-reacting case in [214].
Quantity Value Description
D 5.3mm Internal diameter
De 9.0mm External diameter
Ub 53m/s Bulk mean velocity of propane at the inlet
Uair 9.2m/s Air coflow velocity at inlet
U0 69m/s Centreline velocity at the inlet
m˙ f 2.3g/s Propane mass flow rate
T 294K Operating temperature
p 1013.3kPa Operating pressure
ρu 1.881kg/m3 Density of propane at the inlet
ρair 1.205kg/m3 Density of air at the inlet
4.1 Propane-air mixing 65
Table 4.2 List of simulations for the Sandia propane-air non-reacting case in [214].
CASE Grid size Pipe Wall Functions
A 800k Yes No
B 800k Yes Yes
C 3.5M Yes Yes
D 2.5M No N/A
Two structured grids of 800k and 3.5M cells are used and these grids have 27 and 52 cells
are along the diameter respectively. The numerical cells uniformly increase in size moving in
the axial and radial directions. A grid of 2.5M is also used, which differs from the 3.5M grid
only for the exclusion of the pipe from the computational geometry.
4.1.2 Results
Each simulation is run in parallel using 128 processors. Statistcs are collected for a period of
about 4 flow-through times, where the flow-through time is computed as the time a particle
spends from the inlet to reach x/D= 50 moving along the centreline. Additional 3 flow-through
times are used for transients to leave the numerical domain before statistics are collected. Mean
centreline axial velocity profiles are shown in Fig. 4.2 for the four cases of Table 4.2. Some
under-prediction is seen for all cases, probably because of the dissipative nature of eddy
diffusivity models (see §3.1.1). The largest under-prediction is for case D. Cases involving the
pipe produce an uncertainty on the mean axial velocity at the jet exit of 7% (±3m/s in respect
to U0), which is related to the different turbulence evolution inside the pipe. Nonetheless,
0 10 20 30 40 50 60
0
10
20
30
40
50
60
70
80
x/D
〈U
〉(
m
/
s)
 
 
Fig. 4.2 Mean axial velocity centreline profiles from LES, obtained using cases A ( ), B ( ), C ( )
and D (•••) of Table 4.2, are compared with experimental results [214] (symbols).
66 Validation cases
0 50 100
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
〈U 〉
r
/
D
x/D = 4
0 25 50
−3
−2
−1
0
1
2
3
〈U 〉
x/D = 15
0 15 30
−6
−4
−2
0
2
4
6
〈U 〉
x/D = 50
0 15 30
−6
−4
−2
0
2
4
6
〈U 〉
x/D = 30
 
 
(a)
−2 0 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
〈V 〉
r
/
D
x/D = 4
−2 0 2
−4
−3
−2
−1
0
1
2
3
4
〈V 〉
x/D = 15
−1 0 1
−6
−4
−2
0
2
4
6
〈V 〉
x/D = 50
−2 0 2
−4
−3
−2
−1
0
1
2
3
4
〈V 〉
x/D = 30
 
 
(b)
Fig. 4.3 Radial profiles of mean (a) axial and (b) radial velocity from LES are compared with experi-
mental results. Legend is as for Fig. 4.2.
0 100 200
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
〈uu〉
r
/
D
x/D = 4
0 50 100
−4
−3
−2
−1
0
1
2
3
4
〈uu〉
x/D = 15
0 5 10
−6
−4
−2
0
2
4
6
〈uu〉
x/D = 50
0 20 40
−6
−4
−2
0
2
4
6
〈uu〉
x/D = 30
 
 
(a)
0 50 100
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
〈vv 〉
r
/
D
x/D = 4
0 50 100
−4
−3
−2
−1
0
1
2
3
4
〈vv 〉
x/D = 15
0 5 10
−6
−4
−2
0
2
4
6
〈vv 〉
x/D = 50
0 20 40
−6
−4
−2
0
2
4
6
〈vv 〉
x/D = 30
 
 
(b)
Fig. 4.4 Radial profiles of (a) axial and (b) radial velocity variance from LES are compared with
experimental results (symbols). Legend is as for Fig. 4.2.
the evolution of mean axial velocity in radial and axial direction downstream of the jet exit is
similar for the four cases of Table 4.2, as shown in Fig. 4.3a, suggesting that these differences
at the inlet do not affect significantly the evolution downstream. Radial profiles of mean radial
velocity, ⟨V ⟩, obtained with models A to D of Table 4.2, are compared to the experimental data
in Fig. 4.3b. The computed profiles are similar to each other, and they are under-predicted in
downstream locations. An additional simulation has been performed with lateral boundaries
extended to ±26D to avoid possible reflection of numerical waves, which may affect the
solution in case the cells close to the lateral boundary are not big enough to dissipate them [82].
4.1 Propane-air mixing 67
0 10 20 30 40 50 60
0
0.5
1
1.5
2
2.5
3
3.5
4
(x − x0)/D
L
f
/
D
 
 
 
Fig. 4.5 Mixture-fraction half radius predictions from LES of case B ( ) and C ( ) of table 4.2 are
compared with experimental data [214] (symbols).
However, no effect on the radial velocity has been observed. A better approach may be to
assign velocity or pressure conditions [16] on the lateral boundaries. It is worth noting that the
velocity field is strongly dependent on turbulence and entrainment conditions at the jet exit,
which are not fully described within the experimental dataset.
Axial and radial velocity variance profiles are shown in Fig. 4.4, and these second order
moments show sensitivity to both grid size and turbulent conditions. Case B gives overall better
predictions than case D when compared with experimental data, and this becomes evident at
x/D = 50. Some under-prediction is seen downstream, which may be partially due to the fact
that SGS variances are not included in these profiles and partially to numerical diffusion. It is
worth noting that turbulence at the pipe exit is observed not to be fully developed in all cases
of Table 4.2 including the pipe in the geometry. A much longer pipe is likely to be needed to
reach this condition, which is unpractical. This suggests that synthetic turbulence may be a
better choice for cases where inlet turbulence is relevant. Overall, predictions obtained using
case B and C of Table 4.2 are reasonable, with some improved performance seen for case C.
These two cases are used for further analysis below.
As noted earlier, the effect of grid size becomes relevant for second order statistics. Com-
puted mixture-fraction half radius L f , defined as the radial location at which the mixture
fraction is equal to half its value at the centreline, are compared with experimental results in
Fig 4.5. A very good prediction is observed for case C, while the smaller grid used in case B
results to be inappropriate to represent the flow dynamics in downstream regions, probably
because of the numerical diffusion due to the numerical cell size increase in the axial direction.
On the other hand, the grid sensitivity observed for first order moments shown earlier suggests
that coarser grids may be adequate to estimate first order quantities.
68 Validation cases
As result of the above analysis, a pipe will not be included in configurations where inlet
conditions are well specified at the jet exit. Synthetic turbulence will be used when required for
reacting flow calculations.
4.2 Piloted jet burner
The Bunsen piloted flame configuration described in [46] is chosen to assess the validity of the
combustion models described in §3.3. This case involves a jet of stoichiometric methane-air
(a) (b)
Fig. 4.6 A schematic of (a) the experimental [46] and (b) computational setup of piloted stoichiometric
methane-air Bunsen flames.
mixture issuing from a pipe of diameter of D= 12 mm into quiescent air as shown schematically
in Fig. 4.6a. This jet is surrounded by a pilot with diameter of Dp = 68 mm having 1175 holds
of 1 mm diameter each, which inject a secondary non-reacting stoichiometric methane-air
mixture into the flow at 84 cm/s per hold. This pilot has a cooling water jacket and thus its
burnt mixture is sub-adiabatic for reacting flow. There are no turbulence generating devices
in the reactant flow path and thus the turbulence is shear generated. Three flames F1, F2 and
F3 having a bulk mean velocity of Ub = 65, 50 and 35 m/s respectively were investigated
in [46]. These flames have Reynolds number, based on jet exit diameter and Ub, of 52,000,
40,000 and 24,000 respectively and this Reynolds number is changed by varying only Ub in the
experiment. The boundary conditions, grid resolution, SGS stress closure, numerical schemes
4.2 Piloted jet burner 69
and methods used in this study are assessed here first by comparing computational results with
experimental measurements of velocity and turbulent kinetic energy (TKE) reported in [46]
for the F2 non-reacting flow. The main geometrical and physical properties for this case are
summarised in table 4.3.
Table 4.3 Geometrical and physical parameters for the F2 cold case in [46].
Quantity Value Description
D 12mm Internal diameter
Dp 63mm Diameter of the pilot
Ub 50m/s Bulk velocity of the jet
Uair 0.22m/s Velocity of coflow at inlet
k0 10.8m2/s2 Turbulent kinetic energy at inlet on
the centeline
T 298K Temperature of the reactants
p 1013.3kPa Operating pressure
ρmix 1.15kg/m3 Density of jet and pilot mixture
ρair 1.205kg/m3 Density of the air
Λy 2.4mm Integral length scale at the inlet in the
radial direction
Λx 4.8mm Integral length scale at the inlet in the
axial direction
τt 0.65ms Integral time scale at the inlet
4.2.1 Computational model
A three-dimensional computational domain spanning 40D in all three directions as shown in
Fig. 4.6b is considered in this analysis, which is discretised using structured multi-block grid
having non-uniform numerical cells. These cells are fine near the burner exit and grows slowly
in the downstream and radial directions. A typical grid is shown in Fig. 4.7 and two grids are
considered for this study. A coarse grid having 22 cells inside the jet diameter, D, and about 4
cells within the turbulence integral length scale, measured to be about Λy = 2.4 mm [46] near
the exit, gives a total of about 1.5 million cells for the computational volume of π(40D)3/4
considered. This grid has 404 cells in the streamwise direction along the centreline. Increasing
the cell count to 32 for D and 6 for Λy, keeping other grid parameters to be almost the same,
yields about 4.2 million cells in total. These two grids are used to assess the grid sensitivity of
the LES results reported in this study.
The jet exit velocity is specified using the radial profile reported in the experimental
study [46]. There is no velocity fluctuation at the jet exit for the simulations as the turbulence
70 Validation cases
(a)
(b) (c) (d)
Fig. 4.7 Numerical grid used for piloted flame case [46] for 1.5M case. An axial slice through the
mid-plane (z = 0) is shown in (a) for a region of 0≤ x≤ 15D and 0≤ y≤ 5D. The grid distribution in a
radial slice of size y× z = 5D×5D is shown for axial locations of (b) x = 0, (c) 4.5D and (d) 10.5D.
in reacting region is predominantly shear generated. A uniform velocity is specified for the
pilot stream on the basis of total volumetric flow rate obtained from the experimental data. A
small velocity of 0.2 m/s is assigned to the coflowing air to mimic the air entrainment. The
lateral boundaries are specified to be slip walls and the outlet is set to have zero gradient in the
streamwise direction for all the variables. The temperature of jet, pilot stream and entrained air
are taken to be 298K for the isothermal flow analysis.
The large eddy simulations are run using 96 cores for about 0.15 to 0.25 second of real
flow time. Out of these total durations, the period of data collection for averaging and analysis
corresponds to about 26 flow-through times, where the flow-through time is defined using the
computational domain length and the bulk-mean velocity, Ub, at the jet exit.
4.2 Piloted jet burner 71
4.2.2 Cold flow results
The sensitivity to coflow velocity, SGS stress model and Schmidt number is assessed first
using the 4.2M grid. These cases are summarised in Table 4.4. Mean axial velocity and
Table 4.4 Simulation parameters for the LES of F2 cold flow configuration in [46]. Cs is the
Smagorinsky constant appearing in Eq. (3.3).
CASE Uair [m/s] ρmix [kg/m3] ScT C2s
B 0.22 1.15 0.7 Dynamic
C 0.10 1.15 0.7 Dynamic
SC 0.22 1.15 Dynamic Dynamic
SM1 0.22 1.15 0.7 0.030
SM2 0.22 1.15 0.7 0.025
TKE centreline profiles, obtained for the cases of Table 4.4, are compared with experimental
results in Fig. 4.8. These quantities are normalised respectively with the jet bulk velocity
0 5 10 15 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
〈x/D〉
〈U
〉/
U
b
 
 
(a)
0 5 10 15 20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
〈x/D〉
〈k
〉/
U
2 b
 
 
(b)
Fig. 4.8 Centerline profiles of (a) mean normalised axial velocity, ⟨U⟩/Ub, and (b) turbulent kinetic
energy, ⟨k⟩/U2b , computed using LES on 4.2M grid for cases B ( ), C ( ), SM1 (+), SM2 (×)
and SC ( ) of Table 4.2 are compared with experimental results (circles) for the F2 cold flow
configuration [46].
Ub = 50m/s and its square, U2b . The TKE, ⟨k⟩, is obtained as ⟨k⟩ = 0.5∑⟨uiui⟩+ 1.5⟨u′2∆ ⟩,
where ui = U˜i−⟨Ui⟩. The values of u′∆ are obtained using the Lilly’s model in Eq. (3.50) for
the cold flow analysed here. The energy transfer rate based model in Eq. (3.55) was also used in
the analysis but no significant difference was observed for the cold flows. Small over-prediction
72 Validation cases
of mean axial velocity in the first few jet diameters are due to the absence of inlet turbulence.
This over-prediction corresponds to under-prediction of TKE in the same region, as can be
seen in Fig. 4.8b. After about 3D, the TKE increases because of the shear, suggesting that
the absence of inlet turbulence affects only the first few diameters. Cases B, C give similar
predictions, indicating that the effect of coflow velocity is weak. The effect of using a dynamic
Schmidt number, case SC, on the scalar equations (only the passive marker used to track of
the air entrainment in this case) is also small as one would expect. Case SM1 suggests that a
Smagorinsky constant C2s = 0.03 is too large. Decreasing this constant to a value of C
2
s = 0.025,
case SM2, produces a mean axial velocity similar to case B, with a higher level of TKE in
the region between 4D and 10D, indicating that the choice of a dynamic formulation is more
appropriate as one would expect. Because of the analysis above, case SC is kept for further
analysis below and for reacting cases in the later chapters.
Figure 4.9 compares the computed and measured radial variations of averaged streamwise
velocity and TKE for five axial positions. These quantities are normalised using Ub = 50 m/s
0
0.5
1
1.5
〈U
〉/
U
b
 
 
0 0.5 1 1.5 20
2
4
6
8
10
r/D
〈k
〉/
k
0
0.5 1 1.5 2
r/D
0.5 1 1.5 2
r/D
0.5 1 1.5 2
r/D
0.5 1 1.5 2
r/D
x/D = 2.5 4.5 6.5 8.5 10.5
Fig. 4.9 Comparison of measured [46] (symbols) and computed (lines) normalised mean axial velocity
and turbulent kinetic energy in cold flow of flame F2. The radial variations are shown for 5 axial
locations and two numerical grids, 1.5M (solid line) and 4.2M (dotted line).
and k0 = 10.8 m2/s2. Results obtained using 4.2M and 1.5M grids are compared to each other
to assess the sensitivity to the grid size. The averaged streamwise velocity computed in this
study agrees very well with measured values as one observes in Fig. 4.9 and the differences
between 1.5M and 4.2M grid results are negligibly small not only for ⟨U⟩/Ub but also for
normalised ⟨k⟩. There are some differences between the computed and measured values of ⟨k⟩
for r ≤ 0.55D. This difference decreases as one moves in the downstream direction. As seen in
Fig. 4.9, the turbulence levels represented by ⟨k⟩ in regions of turbulent combustion (as one
shall see in Chapter 5 this would be r > 0.6D) are captured quite well. Thus, no turbulence
is specified for the jet fluid. However, feeding some inlet turbulence in commensurate with
4.3 Bluff body burner 73
2
8
4
m
m
Shear and
mixing layer
U0 CH4 + air
Postflame
y
x
Recirculation
zone
45◦
D stem
Ure f
79 × 79 mm
D
(a)
Slip flow 
No-slip flow 
Uair
U0
284
mm
17.5
D =
778
mm
(b)
Fig. 4.10 A sketch of (a) the experimental burner configuration and (b) its numerical model.
experimental values did improve the comparison of ⟨k⟩ for x = 2.5D but no significant changes
were observed for other downstream locations because the turbulence is dominated by the
shear generation mechanism in this flow configuration. Thus, ignoring a small level of inlet
turbulence is acceptable. The 1.5M grid chosen here satisfies the 80% criterion of Pope [196]
for TKE. Thus, it is clear that this 1.5M grid is adequate to capture the flow dynamics in the jets
simulated here and hence this grid is used for flame calculations to be discussed in Chapter 5
as this choice is consistent with the views discussed earlier in the Introduction. Nevertheless,
4.2M grid will be also used for further testing and verification for reacting cases, when required.
4.3 Bluff body burner
The configuration of this burner investigated in [176–178] is shown in Fig. 4.10 along with its
computational model. The flow enters a combustion chamber of dimension of 79×79×284 (in
mm) at 294 K. The bluff-body has a base-diameter of D = 44.45 mm, stem-diameter of
Dstem = 12.7 mm and an apex angle of θ = 45◦. It is placed at the centre of a square duct
which also forms the combustion chamber in the downstream of the bluff-body as shown in
Fig. 4.10. A turbulence generator having 3.46 mm diameter holes is placed at about 58 mm
upstream of the bluff-body base. The flow entering the combustor section has a bulk-mean
velocity of Uref = 15 m/s and turbulence intensity of 2% or 24% [176–178].
74 Validation cases
(a) (b) (c)
Fig. 4.11 Computational mesh in the mid-plane (z = 0) around the bluff body for (a) 1.8M and (b) 2.2M
grids. An enlargement of the 2.2M grid is also shown in (c).
4.3.1 Computational model
The computational domain begins at 58 mm upstream of the base of conical bluff-body as
shown in Fig. 4.10b and this computational boundary is specified as inflowing boundary using
measured mass flow rate. The intricate geometry of the turbulence generating device used in the
experiments is excluded. An additional region of 4.5D×4.5D×17.5D is added downstream as
shown in Fig. 4.10b for numerical stability by avoiding the effects of numerical wave reflection
on the combustion region if this additional domain was not present. The combustor walls are
specified to be adiabatic and no-slip whereas the walls of the additional section are treated to
be slip walls. The outlet at the end of the whole computational domain is specified to have
zero gradients for all the variables. A flat velocity profile having U0 = 11.5 m/s is specified at
the inlet boundary based on the measured mass flow rate giving Uref = 15 m/s at the entrance
of the combustion chamber. A small velocity of Uair = 0.2 m/s is specified as illustrated
in Fig. 4.10b for the additional computational domain to mimic the air entrainment at this
location. Turbulence at the inlet of the computational domain is specified using the digital filter
technique [112] and its level at the inlet is empirically adjusted so that the turbulence intensity
is about 2% or 24% at the combustor entry.
The domain is discretised using two block structured meshes having 1.8M and 2.2M
numerical cells and these two meshes have about 47 cells for D. Both of these grids satisfy
the 80% criterion of Pope [196] for the turbulent kinetic energy. The larger grid has refined
cells near the bluff-body to capture the boundary layer development along the body as shown
in Fig. 4.11b. The wall refinement can be characterised in terms of the dimensionless wall
distance, y+ = yu∗/ν , where y is the local normal to the wall, ν is the kinematic viscosity
4.3 Bluff body burner 75
Table 4.5 Geometrical and physical parameters for the bluff-body configuration [176–178].
Quantity Value Description
D 44.45mm Diameter of the bluff-body
Dstem 12.7mm Diameter of the stem
h 79mm height of the combustor section
L 284mm Length of the combustor chamber
θ 45◦ Apex angle
Uref 15m/s Reference velocity
U0 11.5m/s Domain inlet velocity
Uair 0.2m/s Velocity of coflow
T 294K Inlet temperature
and u∗ is the friction velocity, defined as u∗ =
√
τw/ρ , with τw being the wall shear stress.
This quantity was estimated a priori using experimental data [168, 169] and re-computed a
posteriori using LES data. Roughly 2 cells are encompassed in the viscous sublayer in the
case of 2.2M grid, where the viscous sublayer was considered to be the region y+ ≤ 5. In case
of 1.8M grid, the cell size near the wall is y+ ≈ 9 at the side of the bluff body and y+ ≈ 90
at the base, where the velocity gradients are small because of the effect of the recirculation
zone. Since these values are outside the viscous sublayer, wall functions are employed when
the 1.8M grid is used. All parameters are summarised in Table 4.5. The simulations have been
run using 80 processors for a period of ten flow-through times, where the flow-through time
is computed as twice the length of the combustor divided by the reference velocity, Uref. The
first half of this period allows the transient from the initialisation to escape the computational
domain and the later half is used to collect statistics.
4.3.2 Cold flow results
The characteristics such as the length of the recirculation zone behind the bluff-body is provided
within the experimental dataset and is used to assess the adequacy of the numerical grid and
appropriateness of the various boundary conditions used. This quantity is extremely important
as it affects significantly the evolution of the flow field. Figure 4.12 compares the computed and
measured axial velocity along the centreline when the approach turbulence intensity (TI) is 2%
and 24%. No synthetic turbulence is specified at the inlet for the 2% case since the turbulence
evolves as the inflow approaches the combustor entry. Synthetic turbulence is imposed using
axial rms velocity and lateral length scale from experiments using the digital filter technique
described in [112] in case of 24% TI. The recirculation zone length, Lr, is underestimated in
the calculations using 1.8M grid and the 2.2M grid yields the correct length for both conditions
76 Validation cases
0 1 2 3 4
−1
−0.5
0
0.5
1
1.5
2
2.5
x/D
〈U
〉/
U
r
e
f
(a)
0 1 2 3 4
−1
−0.5
0
0.5
1
1.5
2
2.5
x/D
〈U
〉/
U
r
e
f
(b)
Fig. 4.12 The variation of computed (lines) and measured [176] (symbols) ⟨U⟩/Uref along the centreline
for cold flow with (a) 2% and (b) 24% TI. The results are shown for both 1.8M ( ) and 2.2M ( )
grid. The sensitivity to wall functions is shown for 1.8M grid ( ).
with 2% and 24% turbulence intensity. The sensitivity of Lr to the use of wall functions is also
investigated and this is found to be small as shown in Fig. 4.12.
Overall, the comparison shown in this figure is very good for 2.2M grid suggesting that the
numerical model used here approximates the burner flow and its attributes very well, and that
the inlet turbulence technique used here [112] is appropriate.
4.4 Non piloted jet burner
The jet burner configuration studied by Furukawa et al. [76] is investigated as fourth and final
case in this thesis. This configuration involves a cylindrical pipe of diameter D = 26mm from
which a lean propane-air mixture spreads into quiescent air. A pilot was not required to stabilise
the flame in the experiments because this configuration involves low velocities (the centreline
velocity at the pipe exit is as low as 4.5 m/s in the cold flow) and thus the flame is stabilised
on the rim of the pipe. This flame lies in the corrugated flamelets regime of the combustion
diagram and thus its investigation is of interest to have a broad perspective of the combustion
closures in different regimes. This flame is also prone to instabilities and flow transition, giving
additional challenges for combustion modelling. Unfortunately, no cold flow measurements are
available for this burner configuration. Thus, its detailed description is reminded to the reacting
flow investigation in Chapter 8.
4.4 Non piloted jet burner 77
The isothermal flow results shown in this chapter suggest that the computational models,
grids and boundary conditions used for the various configurations are good and represent the
burner flow conditions well. The results for the reacting flows in these burners are discussed in
the following chapters.

Chapter 5
Piloted flames - Algebraic closure
In this chapter the piloted flame configuration described in §4.2 is investigated. The relative
performance of models and sub-models described in §3.3 is assessed by comparing LES results
with experimental data [46], and an extensive analysis is conducted on properties of flamelets
modelling and SDR based modelling. The algebraic closure of §3.3.1 is first explored in §5.3
using a static value for βc in Eq. (2.22). The dynamic procedures described in §3.3.4, their
advantages and limitations are assessed in §5.4.
All the simulations discussed in this chapter are listed in Table A.1 along with other
simulations used to test the sensitivity to various parameters.
5.1 Experimental Cases
The piloted Bunsen flames investigated experimentally by Chen et al. [46] are considered for this
study. A jet of stoichiometric methane-air mixture issues from a pipe of diameter of D= 12 mm
into quiescent air as shown schematically in Fig. 4.6. This jet is surrounded by laminar pilot
flames of stoichiometric methane-air mixture and the pilot diameter is Dp = 68 mm. This pilot
has a cooling water jacket and thus its burnt mixture is sub-adiabatic. There are no turbulence
generating devices in the reactant flow path and thus the turbulence is shear generated. Three
flames F1, F2 and F3 having a bulk mean velocity of Ub = 65, 50 and 35 m/s respectively
were investigated in [46]. These flames have Reynolds number, based on jet exit diameter
and Ub, of 52,000, 40,000 and 24,000 respectively and this Reynolds number is changed by
varying only Ub in the experiment. It is worth to note that, because of the large pilot, the air
entrainment has an effect of dilution on the burnt gases in these flames, which can thus be
supposed to operate in a premixed mode. In this regard, an analysis by Wang et al. [240] has
shown that the probability to find values of equivalence ratio smaller than 0.8 is below 1% for
the stoichiometric piloted flame with largest Reynolds number in [46]. Similar conclusions
80 Piloted flames - Algebraic closure
100 102 104
10−1
100
101
102
103
104
u
′ /
s
L
Λ/δ
F2
F3
Corrugated
Wrinkled
Distributed reaction 
           zone
Kaδ = 1
F1
Kaδ = 100 Thin reaction 
       zone
Ret= 1
Fig. 5.1 Combustion regime diagram showing conditions of flames F1, F2 and F3.
were made by Kolla et al. [122] in terms of mixture fraction for the same configuration using
RANS. Additional terms in Eq. (3.26) to account for partial premixing are thus not considered
here and the effect of dilution on the thermophysical properties (heat capacity, enthalpy of
formation and molecular weight) is taken into account as described in §3.4.
The radial variations of averaged temperature, fluid velocity, turbulent kinetic energy,
various species mass fractions at axial locations ranging from x/D= 2.5 to 10.5 were measured
and reported in [46]. These are useful to evaluate the overall model performance and in this
respect, these flames were also considered by a number of past RANS [5, 96, 122, 143, 198,
208, 209, 222, 223] and LES [53, 56, 130, 186, 240, 243] studies using different combustion
modelling. The availability of these results provides strong impetus for using these flames to
address the objectives of this study.
The conditions of turbulent combustion in these flames, F1, F2 and F3, are shown in a
combustion regime diagram [180] as Fig. 5.1. All of these flames are in the thin reaction zones
regime [180] and the conditions of F1 to F3 are calculated using the centreline value of the
turbulence RMS velocity, u′, and the lateral integral length scale, Λ, of 2.4 mm measured [46]
near the nozzle exit. The Zeldovich thickness is calculated as δ = νu/sL for the stoichiometric
methane-air mixture at room temperature, where νu is the kinematic viscosity of the reactant
mixture. The turbulence Reynolds number is defined as ReT = u′Λ/νu. The flames F1 to F3
are simulated using LES methodology employing the models discussed in section 3.3.1 and
details of these simulations are discussed next.
A value of about 0.83, based on a DNS analysis [77] is used for the Bray constant Cm
in Eq. 2.19 in cases where this algebraic closure is used. Values of laminar flame speed,
5.2 Specific numerical detail 81
thermal thickness and heat release parameter are assigned consistently from stoichiometric
one-dimensional computations as sL ≈ 0.41m/s, δth ≈ 0.42mm and τ ≈ 6.48. The heat release
parameter can also be defined using mixture density as τ = ρu/ρ−1 and it is observed that
the numerical results to be reported in section 5.3 are not unduly influenced by this choice.
If the reactant mixture is stratified then this definition is more suited as it can capture spatial
variation in τ . The four closures in Eqs. (3.50) to (3.52) and (3.55) are used to model the SGS
velocity scale u′∆ required for ε˜c closure in Eq. (2.22) and the results of this testing is discussed
in §5.3. The closure for u′∆ in Eq. (3.56) is not used here for the following reason. This model
was derived specifically for use in conjunction with thickened flame model and its efficiency
function requiring turbulence level of unburnt mixture that would have existed locally in the
absence of heat release rate. However, it would be inappropriate to exclude the dilatational
effects on u′∆ required in Eq. (2.22) because the SGS velocity in this equation is related to the
SGS kinetic energy, which must include such effects. The local gradients of c will be influenced
by combustion even in the absence of turbulence and the local values of ∇c result from a fine
balance among turbulent transport, heat release, advection and molecular diffusion. All of these
processes will be influenced by the heat release effects and the local velocity will be affected
by the acceleration induced by dilatation resulting from the flame. The model of Eq. (3.56) will
be specifically used for comparisons in cases where the thermal dilation is relevant in Chapter 8
in order to assess this effect on the SGS velocity modelling.
The static value of the model parameter βc is chosen empirically for this study and this is the
only tuneable parameter in the algebraic reaction rate closure given in Eqs. (2.19) and (2.22).
The values used in this study for the three flames are guided by an earlier analysis [77]. The
value of βc can be evaluated dynamically as discussed in §3.3.4 and this will be investigated
in §5.4. Overall, the algebraic closure for the filtered reaction rate involving the filtered scalar
dissipation rate works acceptably well and of course this closure is based on high (local)
Damköhler number. The influences of local finite rate chemistry can be included in this
approach using either unstrained or strained flamelets and this would be explored in Chapter 6.
5.2 Specific numerical detail
The time step used to keep the CFL below 0.3 in the simulations to be discussed later is of 3µs
(2µs), 4µs (3µs) and 6µs (4µs) respectively for flames F1, F2 and F3 using 1.5M (4.2M)
grid. The sub-grid stresses are modelled using dynamic Smagorinsky model [81, 141] and
the SGS scalar fluxes are computed using gradient hypothesis with dynamic Schmidt number
approach [141]. The test-filter size for all dynamic procedures used in this study is ∆̂ ≈ 2∆
82 Piloted flames - Algebraic closure
following common practices with filter width estimated as ∆= (Vi)1/3, where Vi is the volume
of computational cell i.
The exit velocity for the jet is specified using its measured radial profile [46]. There is no
fluctuation specified for this velocity since the turbulence in reacting region is shear generated,
as discussed in §4.2. The pilot stream temperature is unspecified in the experimental study and
values ranging from 1785 to 2248 K were used in past studies [5, 53, 56, 96, 122, 143, 186,
198, 208, 209, 222, 223, 240, 243] suggesting that the heat loss to the pilot burner varies from
0 [53] to 34% [56, 143]. A value of 17% was used in [96] and 20% heat loss was assumed
in [186, 240]. For this study, it is taken to be 16% following [5, 122, 129], which gives 1950 K
for the pilot temperature. These past studies showed that the pilot temperature uncertainty
influences the temperature field only close to the jet exit (x ≤ 3D), which is also confirmed
here by changing this temperature over a range of about 200 K using for both algebraic and
unstrained flamelet models. Other initial and boundary conditions are as those specified for the
cold flow in table 4.3.
Out of many possibilities to define the instantaneous c, the fuel mass fraction, Yf , is used
here to define the progress variable as c = 1−Yf /Yf ,u so that it takes a value of 0 and 1 in the
unburnt and burnt mixtures respectively. The fuel mass fraction in unburnt mixture is denoted
as Yf ,u. This definition of c avoids spurious flame that will appear numerically in the mixing
layer between the pilot and coflow if it is defined using temperature T or any other species
mass fraction. The filtered progress variable, c˜, has a value of 0 in the jet exit and 1 for the pilot
and coflowing streams. The passive fluid marker, Z˜, needed for Eq. (3.70) is specified to be 1 at
the jet and pilot exits, and 0 for the air inlet. The total enthalpy at these boundaries is specified
to be consistent with the respective temperature and mixture composition. Other computational
details such as grids and boundary conditions are discussed in §4.2. As noted in §4.2.1, these
simulations are conducted using 96 cores for a period of about 32 flow-through times, defined
using the computational domain length and the respective Ub. This is substantially larger than
what has been used in past studies of these flames. These computations took about 12 h on the
wall clock for 1.5M grid and 24 h for the 4.2M grid in the Darwin cluster at Cambridge (see
§3.2.2). A period of 16 flow-through times is used for collecting statistics, after allowing the
initial transient to leave the domain in the first 16 flow through times.
5.3 Results for static βc
The algebraic reaction rate closure in Eqs. (2.19) to (2.22) and its efficacy to capture flame
brush structure is explored in this section. This algebraic reaction rate closure needs a model
for SGS velocity scale, u′∆. The models of Eqs. (3.50) to (3.52) and (3.55) have been tested in
5.3 Results for static βc 83
F1 (1.5M)
y/D
x
/
D
 
 
−10 −5 0 5 100
5
10
15
20
25
30
35
40
(a)
y/D
x
/
D
F3 (1.5M)
 
 
−10 −5 0 5 100
5
10
15
20
25
30
35
40
(b)
y/D
x
/
D
F2 (1.5M)
 
 
−10 −5 0 5 100
5
10
15
20
25
30
35
40
(c)
F2 (4.2M)
y/D
x
/
D
 
 
−10 −5 0 5 100
5
10
15
20
25
30
35
40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(d)
Fig. 5.2 The spatial variation of normalised filtered temperature, T˜+, field in z = 0 plane for the three
flames, obtained using algebraic model for ω˙ with βc = 7.5. The result from 4.2M grid is shown only
for F2 to depict the grid influence.
an a priori analysis using DNS data in [130]. The restults of this analysis show some sensitivity
of the filtered dissipation rate, ε˜c, to the u′∆ model. The sensitivity of various statistics obtained
from LES results to the modelling of u′∆ is discussed in the following section along with a
discussion on grid sensitivity of LES statistics obtained using the algebraic closure.
5.3.1 General flame features
The spatial variations of temperature at an arbitrarily chosen time are shown in Fig. 5.2 for the
three flames, F1, F2 and F3. The temperature is normalised as T˜+ = (T˜ −Tu)/(Tb−Tu) and it
varies from 0 in the reactant and 1 in product mixtures. The subscripts u and b respectively
denote unburnt and burnt mixtures. This normalised temperature field is shown in the mid z
plane from simulations using 1.5M grid for all three flames. As one would expect, the presence
of shear driven Kelvin-Helmholtz instability is seen in the shear-layer region between the jet
and pilot stream and this becomes more obvious in F1 because of higher Reynolds number. The
potential core region, denoted by the middle region with T˜+ ≈ 0, is of similar length in all the
flames. When the refined grid with 4.2M cells is used for the flame F2, this length is reduced
slightly and the fine scale structures emerging from the Kelvin-Helmholtz instability become
more apparent. Also, the flame length is reduced substantially. Furthermore, the normalised
temperature field seems to be more intermittent compared to that for 1.5M grid. The filter width
implied by the numerical grid volume in 4.2M case is ∆+ ≤ 1, for which the filtered reaction
rate closure in Eq. (2.19) is not fully valid as has been discussed in earlier studies [61, 77, 147].
This yields locally lower temperatures in case of the 4.2M grid and this effect is seen clearly
84 Piloted flames - Algebraic closure
F1 (1.5M)
r/D
x
/
D
 
 
−10 −5 0 5 100
5
10
15
20
25
30
35
40
r/D
x
/
D
−10 −5 0 5 100
5
10
15
20
25
30
35
40
F2 (4.2M) F2 (1.5M)
r/D
x
/
D
F3 (1.5M)
 
 
−10 −5 0 5 100
5
10
15
20
25
30
35
40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 5.3 The spatial variation of averaged and normalised temperature, ⟨T˜+⟩, in three flames, obtained
using algebraic model for ω˙ with βc = 7.5.
in the averaged normalised temperature, T˜+, field shown in Fig. 5.3. The variations of ⟨T˜+⟩
obtained using 1.5M grid are similar to those observed in many earlier numerical studies of
these flames using various combustion modelling approaches.
The normalised minimum filter sizes, ∆+min, implied by the minimum cell volume are
respectively 1.3 and 0.8 for 1.5M and 4.2M grids. These sizes become respectively 2.8 and
1.7 if one uses the diagonal distance of the smallest numerical cell. Since the grid is spatially
non-uniform the implied filter width varies spatially and a typical distribution obtained using
the local cell volume is shown in Fig. 5.4 as a histogram for the two numerical grids used in
this study. This histogram is constructed using the numerical grid information gathered from
the whole and two different subparts of cylindrical computational domains having 0≤ x≤ 20D
and 0 ≤ r ≤ 10D, and 0 ≤ x ≤ 11D and 0 ≤ r ≤ 2D, where r =
√
y2+ z2. For the smallest
subdomain considered, the most likely width for the implied filter is about 2δth and 0.9δth for
1.5M and 4.2M grids. This information is interesting as the reactions happen close to the jet exit
and thus this subdomain is representative of the region of the reactions. When the subdomain
size is increased in the axial and radial directions the most likely and averaged filter widths are
increased substantially, especially for 1.5M grid. The most likely value of ∆+ is about 4 and
the average value is about 5 to 6. There is a further increase in these values when the whole
domain is considered for this analysis. The increase in the count around ∆+ ≈ 14 and about 10
respectively in Figs. 5.4a and 5.4b is due to the larger grid sizes near the lateral and outflow
boundaries (see Figs. 4.6 and 4.7) of the computational domain. The above values of ∆+ are
substantially larger than typical values used in past LES studies of these experimental flames.
5.3 Results for static βc 85
0 5 10 150
1
2
3 x 10
5
∆+
O
cc
ur
re
nc
es
(a)
0 5 10 150
2
4
6
8 x 10
5
∆+
O
cc
ur
re
nc
es
(b)
Fig. 5.4 Histogram of normalised filter width, ∆+, for (a) 1.5M and (b) 4.2M numerical grids. Different
domain sizes have been used: whole computational domain ( ), 0≤ x≤ 20D, 0≤ r ≤ 10D ( ), and
0≤ x≤ 11D, 0≤ r ≤ 2D ( ).
5.3.2 Sensitivities to SGS velocity modelling
As noted in §3.3.3, the SGS velocity influences the filtered reaction rate directly through the
sub-grid scale scalar dissipation rate, ε˜c, modelled by Eq. (2.22). The influence of u′∆ modelling
on the filtered reaction rate computed in the LES of flame F2 is shown in Figure 5.5 for four
models used in this study. Countours obtained using the ksgs equation, to be discussed in
more details in §5.3.4, are also shown. The reaction rate is normalised using ρu sL/δth and the
colourmap is shown for ω˙+ > 0.01 for an arbitrarily chosen time. The algebraic models in
Eqs. (3.50) to (3.52) give relatively thinner fronts compared to that obtained using Eq. (3.55)
or the ksgs equation. The flame length is more or less the same for all the models and there
is a slightly longer flame for the scale-similarity model in Eq. (3.51). The Lilly’s model in
Eq. (3.50) gives the shortest flame. Also, the peak reaction rate near the burner exit is seen
only for Eq. (3.51) and ksgs transport equation. The peak reaction rate predicted by the other
models is relatively smaller. The averaged normalised reaction rate shown in Figure 5.6 is
almost similar for the various SGS velocity models. The difference is seen only in the region
close to the jet exit (x≤ 2.5D). The differences observed in Figures 5.5 and 5.6 suggest some
sensitivity to u′∆ modelling. It is not easy to measure the reaction rate in experiments and thus it
may not be easy to assess the accuracy of these SGS velocity models. However, one can make
this assessment for overall performance of these models using statistics gathered from LES and
measurements as has been done next.
86 Piloted flames - Algebraic closure
y
/
D
x/D
 
 
−
2
0
2
510152025 (
a)
E
q.
(3
.5
0)
y
/
D
x/D
 
 
−
2
0
2
510152025 (
b)
E
q.
(3
.5
1)
y
/
D
x/D
 
 
−
2
0
2
510152025 (
c)
E
q.
(3
.5
2)
y
/
D
x/D
 
 
−
2
0
2
510152025 (
d)
E
q.
(3
.5
5)
y
/
D
x/D
 
 
−
2
0
2
510152025
00.
1
0.
2
0.
3
0.
4
0.
5
(e
)
E
q.
(3
.1
3)
Fi
g.
5.
5
C
ol
ou
rm
ap
of
no
rm
al
is
ed
fil
te
re
d
re
ac
tio
n
ra
te
,ω˙
+
,f
or
fla
m
e
F2
,c
om
pu
te
d
fo
rv
ar
io
us
u′ ∆
m
od
el
s.
5.3 Results for static βc 87
y
/
D
x/D
 
 
−
2
0
2
510152025 (
a)
E
q.
(3
.5
0)
y
/
D
x/D
 
 
−
2
0
2
510152025 (
b)
E
q.
(3
.5
1)
y
/
D
x/D
 
 
−
2
0
2
510152025 (
c)
E
q.
(3
.5
2)
y
/
D
x/D
 
 
−
2
0
2
510152025 (
d)
E
q.
(3
.5
5)
y
/
D
x/D
 
 
−
2
0
2
510152025
00.
1
0.
2
0.
3
0.
4
(e
)
E
q.
(3
.1
3)
Fi
g.
5.
6
C
ol
ou
rm
ap
of
no
rm
al
is
ed
m
ea
n
re
ac
tio
n
ra
te
,⟨
ω˙
+
⟩,
fo
rfl
am
e
F2
,c
om
pu
te
d
fo
rv
ar
io
us
u′ ∆
m
od
el
s.
88 Piloted flames - Algebraic closure
The flame F2 simulated using 4.2M grid is chosen for this assessment because the results
for 1.5M grid is better than what is shown here. Figures 5.7 and 5.8 respectively show the
sensitivity of fluid dynamic and scalar quantities to u′∆ modelling. The radial variation of
0.5
1
1.5
0
5
10
0.5
1
1.5
0.5
1
1.5
〈U
〉/
U
b
0
5
10
〈k
〉/
k
0
0.5
1
1.5
0
5
10
0 0.5 1 1.50
1
2
r/D
0 0.5 1 1.50
5
10
r/D
0
5
10
4.5
6.5
8.5
10.5
x/D = 2.5
Fig. 5.7 Influence of SGS velocity modelling on averaged streamwise velocity, ⟨U⟩, and turbulent
kinetic energy, ⟨k⟩, obtained using algebraic model for ω˙ with βc = 7.5. Symbols (◦) are experimental
results [46], lines are from simulations using Eq. 3.50 ( ), Eq. 3.51 ( ), Eq. 3.52 (•••) and Eq. 3.55
( ).
averaged streamwise velocity, ⟨U⟩, normalised using Ub is shown in Fig. 5.7 for five streamwise
locations. The averaged normalised turbulent kinetic energy, ⟨k⟩/k0, is also shown in this figure.
The contribution of SGS kinetic energy to ⟨k⟩ is obtained using the same u′∆ model consistently
5.3 Results for static βc 89
0.5
1
 
 
2
4
6
0.5
1
2
4
6
0.5
1
〈T
+
〉
2
4
6
〈Y
C
H
4
〉%
0.5
1  
2
4
6
0 0.5 1 1.50
0.5
1
r/D
0 0.5 1 1.50
2
4
6
r/D
8.5
6.5
4.5
x/D = 2.5
10.5
Fig. 5.8 Influence of SGS velocity modelling on averaged fuel mass fraction and normalised temperature,
obtained using algebraic model for ω˙ with βc = 7.5. The legend is as in Fig. 5.7.
for four different cases discussed in section 3.3.3. The variation of ⟨U⟩/Ub is observed to be
very weakly sensitive to this SGS velocity modelling as seen in Fig. 5.7. However, a weak
sensitivity is seen for ⟨k⟩/k0 as one moves downstream. All the models overestimate ⟨k⟩ for
r ≤ 0.8D when x≥ 8.5D and their estimates for the outer region, r > D, compare reasonably
well with the measured values.
The flame related quantities, fuel mass fraction and normalised temperature, are shown
in Fig. 5.8. The estimates of ⟨YCH4⟩ by the scale-similarity model in Eq. (3.51) are closer to
90 Piloted flames - Algebraic closure
the measured values. Although a similar behaviour is seen for the normalised temperature
there are some large underestimates for r > 0.7D, which is related to the performance of
filtered reaction rate closure in Eq. (2.19) on 4.2M grid supporting the observation made in
Figs. 5.2d and 5.3b. More elaborate discussion on this point will be provided later while
discussing Fig. 5.12. Based on these assessments, the scale-similarity model in Eq. (3.51)
seems appropriate for LES of reacting flows, at least for the conditions investigated in this study.
The SGS stress related models in Eqs. (3.50) and the model of Bardina et al. in Eq. (3.52) are
less satisfactory compared to the scale-similarity model in Eq. (3.51) and energy transfer based
model in Eq. (3.55). It is to be noted that SGS stress based model considers only the deviatoric
part of the SGS stress tensor and not the trace of this tensor while the SGS kinetic energy per
unit volume is related to this trace. For these reasons, Eqs. (3.51) is selected for the following
analyses.
5.3.3 Analysis of flame brush structure
De & Acharya [53] compared RANS and LES results for the experimental flames from a
number of past studies and thus their results are used here for comparison. They used 1.88M
and 5.91M cells in a domain of size 20D and 4D in the axial, x, and radial, r, directions
respectively, and their results using a modified thickened flame approach on the 1.88M grid
is used here. The RANS-strained flamelets study of Kolla and Swaminathan [122] used an
axisymmetric domain of x = 70D and r = 45D with about few hundred thousand cells. The
LES-thickened flame study of Wang et al. [240] used unstructured grids ranging from about
3.2M to 8.5M cells in a domain of x = 120D and r = 40D. Their results from the 8.5M grid is
used for comparison here. The recent LES-PDF study using Eulerian stochastic fields approach
by Dodoulas & Navarro-Martinez [56] used a computational domain of x = 15D and r = 5D
employing about 510k and 225k grids in the physical space and their results on their largest
grid, and with 16 stochastic fields, is used here. Further detail on the combustion modelling
and numerical methods used in these studies can be found in these references and the results
available there are used for comparison here.
The radial variation of ⟨U⟩/Ub for various streamwise locations are shown in Fig. 5.9 along
with experimental measurements. Only two representative results from past studies are shown
here because those results are very similar to one another. The current results are shown for the
1.5M grid and the 4.2M grid result is shown only for F2 flame because the relative behaviour is
very similar for the other two flames. The difference in ⟨U⟩/Ub obtained using the 1.5M and
4.2M grids is negligibly small and the values obtained using the 1.5M grid compare well with
the measurements also. These results agree well with those from past studies, however, a close
examination of Fig. 5.9 suggests that the agreement with the experimental data is better for the
5.3 Results for static βc 91
0.5
1
1.5
F1 F2
 
 
0.5
1
1.5
F3
0.5
1
1.5
〈U
〉/
U
b
0.5
1
1.5
0 0.5 1 1.50
0.5
1
1.5
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
x/D = 2.5
4.5
6.5
10.5
8.5
Fig. 5.9 Comparison of radial variation of normalised axial velocity, ⟨U⟩/Ub, in experiments [46] (sym-
bols) and simulations ( ) using Eqs. (2.19), (2.22) (with βc = 7.5) and (3.51) on 1.5M grid. Results
obtained using 4.2M grid (•••) are shown only for F2. Previous RANS [122] ( ) and LES [240] ( )
results are also shown for comparison.
results of [240], which used 8.5M unstructured grid. Nevertheless, the difference among the
numerical results and the data is observed to be small.
The radial variation of ⟨k⟩/k0 at various streamwise locations is shown in Fig. 5.10 for the
three flames. The statistics from LES are improved compared to previous RANS results as one
would expect. The agreement between the computational results of this study and measured
values improves as the turbulence Reynolds number increases (going from F3 to F1). This is
due to the fact that the underlying assumption of high Reynolds number for the algebraic model
of Eq. (2.19) is better verified for flame F1 and this observation is consistent with other scalars
to be discussed later. The variations of ⟨k⟩/k0 computed using the 4.2M grid shown for the
flame F2 agree quite well with measured values in the near field and the agreement becomes
poor in the inner region of the jet at downstream locations. However, the results from 1.5M and
92 Piloted flames - Algebraic closure
0
5
10
F1 F2
 
 
0
5
10
F3
0
5
10
〈k
〉/
k
0
0
5
10
0 0.5 1 1.50
5
10
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
8.5
4.5
x/D = 2.5
10.5
6.5
Fig. 5.10 Comparison of measured [46] (◦ ◦ ◦) and computed ( ) normalised mean TKE, ⟨k⟩/k0,
variation with r/D, obtained using algebraic model for ω˙ with βc = 7.5. The computational results
are for the 1.5M grid. Results of the 4.2M grid (•••) are shown only for F2. Previous results [53] (△),
[122] ( ) and [240] ( ) are shown for comparison.
4.2M grids agree well with measured values for x≥ 0.8D. Generally, second order statistics
such as TKE have increased sensitivity to mesh resolution compared to first order statistics
as implied in Fig. 19 of Wang et al. [240]. Thus, one would expect to see a better agreement
for the 4.2M case of flame F2 in Fig. 5.10 and the large difference seen here is because, the
combustion submodel for the filtered reaction rate is at its limit for this grid as noted earlier.
This SGS combustion model underestimates the reaction rate, effects of heat release rate and
their interactions with fluid dynamics as noted earlier for the 4.2M grid case (see the discussion
for Figs. 5.2, 5.3, 5.7 and 5.8).
5.3 Results for static βc 93
2
4
6
F1 F2
 
 
2
4
6
F3
2
4
6
〈Y
C
H
4
〉%
2
4
6
0 0.5 10
2
4
6
r/D
0.5 1
r/D
0.5 1
r/D
 
 
x/D = 2.5
4.5
6.5
8.5
10.5
Fig. 5.11 The mean fuel mass fraction computed on the 1.5M grid ( ), obtained using algebraic
model for ω˙ with βc = 7.5, is compared to the measurements (◦ ◦ ◦). Previous results [53] (△), [56]
(□), [122] ( ) and [240] ( ) are shown for comparison.
A. Comparison of reactive scalars:
The radial variation of averaged fuel mass fraction is compared to its measured values in
Fig. 5.11 for all the three flames. The results of previous numerical studies are also included in
this figure. The thickened flame-LES results of De & Acharya [53] agree very well with the
measurements for x ≤ 6.5D for the high Da flame F3 and this computational results slowly
move away from the experimental data as seen in this figure for the flame F1. However, the
results of Wang et al. [240] using dynamic thickened flame approach shows a good agreement
uniformly. The agreement between the measured and computed values using the LES-PDF
method [56] is not uniformly good. The agreement is relatively better for x≥ 8.5 for the flames
F1 and F2, and for the flame F3 the agreement seems to improve from x = 6.5D as seen in
Fig. 5.11. It had been suggested that including the differential diffusion effects could improve
94 Piloted flames - Algebraic closure
0.5
1
F1 F2
 
 
0.5
1
F3
0.5
1
〈T
+
〉
0.5
1
0 0.5 1 1.50
0.5
1
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
4.5
x/D = 2.5
8.5
6.5
10.5
Fig. 5.12 The mean normalised temperature computed on the 1.5M grid ( ), obtained using algebraic
model for ω˙ with βc = 7.5, is compared to the measurements (◦ ◦ ◦). Previous results [53] (△), [56]
(□), [122] ( ) and [240] ( ) are shown for comparison.
these estimates for the LES-PDF method. The mean fuel mass fraction computed in this study
using the flamelet based algebraic model agrees very well with the measurements as shown in
Fig. 5.11.
A similar comparison is shown in Fig. 5.12 for the normalised averaged temperature, ⟨T+⟩.
The values shown in Figs. 5.8 and 5.12 are calculated using Eq. (3.18) for the normalised
temperature. The sub-grid variance, σ2T,sgs, needed in Eq. (3.18), is unavailable in the algebraic
modelling framework since one need to solve a transport equation for σ2T,sgs to meaningfully
include the influences of reaction rate, dissipation rate, and turbulence, as discussed in §3.3.2.
Although one should use ⟨T+⟩ for comparison with measurements, no significant difference
between ⟨T+⟩ and ⟨T˜+⟩ was observed for the flame conditions considered in this study. The
flame F3 has the highest Damköhler number in the set and the computed values agree well
with the data as shown in Fig. 5.12. A quite good agreement among various numerical studies
5.3 Results for static βc 95
is also observed for this flame. A large overestimate seen for the flame F1 for x ≤ 4.5D is
because of the uncertainty in the pilot temperature as it is not reported in [46]. Furthermore, the
flame in the near field of the jet is expected to have quasi-laminar unsteady structure since the
turbulence in these regions is low. The results in Fig. 5.12 suggest that the compounded effects
of these two factors (uncertainty and quasi-laminar unsteady flames) become more apparent as
the turbulent Reynolds number increases (compare the difference between the computational
and experimental values at x = 2.5D for all the three flames). The values of ⟨T+⟩ computed in
this study agree well with the measurements at downstream locations as shown in Fig. 5.12.
The under-prediction seen for r > 0.6D at x/D = 10.5 is likely to be due to the missing effect
of SGS fluctuations, which could not be included in Eq. (3.18) for temperature because of the
reason explained earlier. A detailed analysis on the effect of the SGS fluctuations on scalars will
be provided in Chapter 7. In general, a good agreement is observed for the statistics computed
in this study with measurements.
The radial variations of averaged mass fractions of major, O2, H2O and CO2, and minor,
OH, species and intermediates, H2 and CO, at various axial locations are shown respectively
in Figs. 5.13 – 5.18, since the ability of the SGS combustion submodel to capture the flame
brush structure is also of interest. The current statistics are also compared to past results
available in the literature. A 2-step chemistry with empirical rate expression was used by De &
Acharya [53] and 1-step chemistry was used by Wang et al. [240]. The LES-PDF study in [56]
used a 4-step, 15-step and GRI3.0 mechanisms and the results of 15-step mechanism are used
here. Because of these differences in the chemistry, a uniform comparison cannot be made for
the various species measured in [46].
The mass fractions shown in these figures are obtained using Eq. (3.70) to include the
dilution effect, which is elucidated using the oxygen mass fraction at axial positions of x/D =
2.5, 8.5 and 10.5 for the flame F1. In the near field (x/D = 2.5), the influence of this effect is
almost negligible and including this effect captures the oxygen mass fraction increase in the
outer region, r > 0.8D, due to dilution with the air. The oxygen mass fraction computed using
a two-step chemistry and thickened flame model by De & Acharya [53] is relatively larger
compared to those obtained in the current study and those from the LES-PDF calculation in [56].
This difference increases with downstream distance. Wang et al. [240] did not report O2 and
CO2 mass fractions and thus they are not shown in Figs. 5.13 and 5.15. The current results are
improved over the previous RANS results as one would expect for the flames F1 and F2. This
improvement is observed for H2O, H2 and OH. The averaged mass fraction of water vapour
computed in this study agrees well with measured values and those reported in earlier LES
studies. The underestimate seen for CO2 mass fractions in Fig. 5.15 is consistent with many
earlier studies [53, 56, 122] employing different combustion modelling approach. However,
96 Piloted flames - Algebraic closure
0
10
20
F1 F2
 
 
0
10
20
F3
0
10
20
〈Y
O
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
4.5
6.5
8.5
10.5
x/D = 2.5
Fig. 5.13 Comparison of mean mass fraction of O2: experimental data (◦ ◦ ◦), results obtained using
algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□), and [122] ( ). The dash-dotted line ( )
for F1 at x/D = 2.5, 8.5 and 10.5 shows the value obtained without using Eq. (3.70).
the values obtained using the LES-PDF method are relatively closer to the measurements for
downstream locations. Although CO2 is transported in the calculations reported in [53] there
is a large underestimate inside the flame brush for x ≥ 6.5D for the flame F1. For high Da
flame F3, the computed CO2 mass fractions agree well with the measured values. The OH
mass fractions obtained using the LES-PDF [56] and the current LES-flamelets methods agree
quite well with one another and also with measurements for the flame F1. These agreements
are relatively poor for upstream locations and for the largest Da flame F3 as shown in Fig. 5.16.
Nevertheless, the general trend is captured well by both of these two modelling approaches.
The mean mass fraction of H2 computed in this study agrees well with the measured values
as shown in Figs. 5.17. None of the previous LES studies reported this result and thus they are
not included in this figure for comparison. The slight overestimate at x/D = 2.5 is because of
5.3 Results for static βc 97
0
10
20
F1 F2
 
 
0
10
20
F3
0
10
20
〈Y
H
2
O
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
4.5
6.5
8.5
x/D = 2.5
10.5
Fig. 5.14 Comparison of mean mass fraction of H2O: experimental data (◦ ◦ ◦), results obtained using
algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□), [122] ( ) and [240] ( ).
the uncertainty in the pilot boundary condition and this uncertainty does not seem to be of any
importance as one moves downstream.
The agreement for ⟨YCO⟩ shown in Fig. 5.18 is not good and there is a large overestimate for
all the three flames. Flamelets based models are known to generally overestimate the CO mass
fraction. However, a proper reason for the overestimate observed here is unclear at this time
and this overestimate is consistent with the underestimate observed for the CO2 mass fraction.
A similar behaviour is seen in figure 5.18 for many previous studies using different combustion
modelling. However, the CO values computed by De & Acharya [53] seem to agree quite well
with the measurements but this seems to be inconsistent with the CO2 values (see Fig. 5.15)
computed by them.
98 Piloted flames - Algebraic closure
0
10
20
F1 F2
 
 
0
10
20
F3
0
10
20
〈Y
C
O
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
x/D = 2.5
4.5
8.5
10.5
6.5
Fig. 5.15 Comparison of mean mass fraction of CO2: experimental data (◦ ◦ ◦), results obtained using
algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□), and [122] ( ).
B. Flame brush thickness comparison:
Although there is a reasonable comparison between the computed and measured values of
various mass fractions and temperature, it is instructive to compare the computed and measured
flame brush thicknesses. This result is shown in Fig. 5.19. The flame brush thickness is calcu-
lated as δt = 1/
(
∂ ⟨T+⟩/∂ r
)
max
in the experimental study [46]. Thus, one can independently
calculate this value for various axial locations using the experimental data in Fig. 5.12. The
values of δt calculated by fitting a spline curve for the measured variation of ⟨T+⟩ with r in
Fig. 5.12 are compared to the δt reported in Fig. 16 of Chen et al. [46]. This comparison
is shown in the top row of Fig. 5.19 for all the three flames. The values computed using
⟨T+⟩ variation are shown as diamonds and the squares represent the values, δt,Y , obtained
using measured ⟨Y CH4⟩ variation with r. The flame brush thicknesses obtained using either
5.3 Results for static βc 99
20
40
F1 F2
 
 
20
40
F3
20
40
7
5
×
〈Y
O
H
〉%
20
40
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
0 0.5 1 1.50
20
40
r/D
6.5
4.5
8.5
10.5
x/D = 2.5
Fig. 5.16 Comparison of mean mass fraction of OH: experimental data (◦ ◦ ◦), results obtained using
algebraic model for ω˙ with βc = 7.5 ( ), [56] (□) and [122] ( ).
temperature or fuel mass fraction variations are expected to be the same for adiabatic unity
Lewis number flames and the stoichiometric methane-air flame has the Lewis number close to
unity. This expectation holds very well for the flame F3 and also at downstream positions for
the flame F2 as shown in Fig. 5.19. However, the agreement is not so good for the flame F1
and the difference between δt and δt,Y seen in Fig. 5.19 for F1 is because of non-adiabaticity,
which is verified systematically by computing δt and δt,Y values using LES results. The effect
of heat loss is pronounced for x≤ 7D in the flame F2 and it is non-negligible over the region
of measurement locations for the flame F1. These results seem to suggest that this effect also
depends on the jet exit Reynolds number. The agreement between δt values computed here as
noted above and reported by Chen et al. [46] is very good for the flames F2 and F3, but it is not
so for the flame F1. These difference for the flame F1 raises some uncertainty on the flame
100 Piloted flames - Algebraic closure
0
10
20
F1 F2
 
 
0
10
20
F3
0
10
20
1
0
0
×
〈Y
H
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
x/D = 2.5
6.5
8.5
10.5
4.5
Fig. 5.17 Comparison of mean mass fraction of H2: experimental data (◦ ◦ ◦), results obtained using
algebraic model for ω˙ with βc = 7.5 ( ), and [122] ( ).
brush thickness values. One must keep this in mind while comparing the flame brush thickness
obtained from LES results, or in fact from numerical simulations in general for the flame F1.
The bottom row of Fig. 5.19 compares the values of δt computed from the LES results
with those obtained in earlier numerical studies and using the measured radial variation of
⟨T+⟩. This choice is made for the experimental values to be consistent with the temperature
comparison discussed earlier. For the F2 and F3 flames, the agreement is good. However, there
are some relatively larger differences for the flame F1. The two LES simulations, this study
and by Wang et al. [240], yield almost the same flame brush thickness for x ≤ 6D. For the
downstream positions, x≥ 9D, the flame brush thickness computed in this study seem to be in
good agreement with the experimental values of the flame F1. The values of δt obtained using
a modelled transport equation for ksgs are also shown for comparison and these values suggest
that initially the flame brushes are thicker compared to those obtained using the scale-similarity
5.3 Results for static βc 101
20
40
F1 F2
 
 
20
40
F3
20
40
1
0
×
〈Y
C
O
〉%
20
40
0 0.5 1 1.50
20
40
r/D
0.5 1 1.5
r/D
0.5 1 1.5
r/D
 
 
6.5
10.5
8.5
4.5
x/D = 2.5
Fig. 5.18 Comparison of mean mass fraction of CO2: experimental data (◦ ◦ ◦), results obtained using
algebraic model for ω˙ with βc = 7.5 ( ), [53] (△), [56] (□), and [122] ( ).
model. However, one must not read too much into these difference for the ksgs model because
of the uncertainties of the sub-models used in the ksgs transport equation. Overall, the flame
brush structure captured using the algebraic closure for the filtered reaction rate investigated in
this study is acceptable and satisfactory.
5.3.4 Analysis of SGS kinetic energy equation
Additional simulations have been performed using the additional transport equation for the
SGS kinetic energy of Eq. (3.13), referred here as ksgs. This equation is described in §3.1.2 and
it is used in this analysis to provide a closure for the SGS velocity scale, as u′∆ =
√
2ksgs/3.
The sub-grid stress closure is that of Eq. (3.4) in this case, where Cv is computed dynamically
for each computational cell using scale similarity. The turbulent transport is closed using a
gradient hypothesis and assuming unity turbulent Schmidt number, while the dissipation of
102 Piloted flames - Algebraic closure
5 10 15
x/D
F2
 
 
0 5 10 150
5
10
15
20
x/D
δ t
F1
 
 
5 10 15
x/D
F3
 
 
5 10 15
x/D
F2
 
 
0 5 10 150
5
10
15
20
x/D
δ t
F1
 
 
5 10 15
x/D
F3
 
 
Fig. 5.19 Flame brush thicknesses, δt (mm), computed here using the radial variation of averaged
temperature (⋄) and fuel mass fraction (□) reported in [46] are compared to δt values reported (◦)
in [46] (top row). The values of δt obtained from LES using algebraic model for ω˙ with βc = 7.5 and
SGS velocity model in Eq. (3.51) (+) and ksgs transport equation (×) on 1.5M grid are compared with
experimental data [46] (⋄) and past RANS [122] (•) and LES [240] (△) studies in the bottom row.
ksgs is computed as εk =Cερk
3/2
sgs /∆. Finally, the pressure dilatation term Π is computed using
Eq. (3.15). This model is used here specifically to include the dilatation resulting from heat
release effects. The modelling of Π and εk is included systematically as listed in Table 5.1 to
study their influence on LES. The case K1 excludes the pressure-related term, Π, and a static
value of 0.916 is used for Cε , whereas the case K2 includes Π. The value for Cε is determined
5.3 Results for static βc 103
Table 5.1 SGS kinetic energy models used for testing.
Case Π-model Cε
K1 No 0.916
K2 Yes 0.916
K3 Yes dynamic
dynamically in the case K3. Typical results obtained using the modelled ksgs equation is shown
in Fig. 5.20 for the flame F2 along with the results obtained using Eq. (3.51). The influence of
0
0.5
1
1.5
〈U
〉/
U
b
x/D = 2.5 x/D = 10.5
0
5
10
15
〈k
〉/
k
0
0
2
4
6
〈Y
C
H
4
〉%
0.5 1 1.5
r/D
0.5 1 1.5
r/D
x/D = 6.5
 
 
0 0.5 1 1.50
0.5
1
r/D
〈T
+
〉
Fig. 5.20 Radial profiles of mean normalised velocity, ⟨U⟩/Ub, normalised TKE, ⟨k⟩/k0, methane mass
fraction, ⟨CH4⟩ and normalised temperature, ⟨T+⟩, obtained using algebraic model for ω˙ with βc = 7.5
and models K1 ( ), K2 ( ) and K3 (•••) of table 5.1, are compared with LES results obtained using
Eqs. (2.22) and (3.51) ( ) and with experimental data [46] (symbols) for flame F2 and 1.5M grid.
104 Piloted flames - Algebraic closure
ksgs based u′∆ modelling on ⟨U⟩/Ub is negligible and the results obtained using these models
are very close to those obtained using Eq. (3.51) as one observes in the top row of Fig. 5.20.
A similar behaviour is observed for ⟨k⟩/k0 and the averaged fuel mass fraction shown in the
second and third rows of Fig. 5.20. Although there are some small differences in the near field
for the normalised averaged temperature, the differences become negligible for downstream
locations as seen in the bottom row of Fig. 5.20. From these results, one concludes that the
results of ksgs based modelling of u′∆ are very similar to those of scale-similarity model in
Eq. (3.51) and thus it does not seem to justify the additional computational effort required for
ksgs transport equation. On the other hand, these comparisons may suggest that the effect of the
pressure dilatation is not big for these flames, and thus including this effect into modelling of
u′∆ is not critical as the contribution of shear generated turbulence is much stronger compared to
flame generated turbulence effects, as one shall see in Chapter 8. Also, one must acknowledge
that the various sub-modelling for ksgs equation used here are ad hoc and more investigation is
required to develop new or to improve these models.
5.4 Assessment of dynamic procedure for βc
The results in the previous section are obtained with particular values of βc chosen empirically
after several tests, see Table A.1. As noted in §2.5.1, this parameter depends on heat release
parameter and Reynolds number and so it would be ideal to evaluate it dynamically in LES.
The procedure for this evaluation was discussed in §3.3.4, and it is assessed here. A similar
dynamic procedure has been studied recently [79, 129] for flames with different values of
Lewis and turbulent Reynolds numbers using DNS data.
Since βc is computed dynamically, the only tunable parameter in the combustion model of
Eq. (2.22) is for the SGS velocity scale, u′∆. The analysis in §5.3 has shown that a simple model
based on scale-similarity of velocity field as in Eq. (3.51) works well. The same analysis has
demonstrated further that the model in Eq. (2.22) is not unduly sensitive to u′∆ model and thus
this scale-similarity model is used for the analysis in this section. Detailed discussion in this
section suggests that the dynamic procedure works well and physical insights and reasonings
are provided to explain the observed behaviour.
5.4.1 Flame structure
Typical behaviour of dynamic models are elucidated using only flames F1 and F3. Figure 5.21
shows the spatial variation of mean progress variable field, ⟨c˜⟩, in the flame F1 obtained using
both static and dynamic approaches for βc and the results are shown for both 1.5M and 4.2M
5.4 Assessment of dynamic procedure for βc 105
grid. The results for the static, βc = 7.5, value is the same as that reported in §5.3. There
r/D
x
/
D
F 1
 
 
−5 0 50
5
10
15
20
25
Static 
4.2M
Dynamic 
4.2M
(a)
r/D
x
/
D
F 1
 
 
−5 0 50
5
10
15
20
25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Static 1.5M
Dynamic 
1.5M
(b)
Fig. 5.21 The spatial variation of mean progress variable field for flame F1 obtained using algebraic
model for ω˙ with static and dynamic approaches for βc. This comparison is shown for both 1.5M and
4.2M grids.
seems to be very small difference in the mean progress variable field obtained using either
static or dynamic approach for βc on a given numerical grid. However, a close study of this
figure suggests that the flame brush is thinner for the dynamic βc model leading to a slightly
shorter flame brush as seen in Fig. 5.21. If one compares the results of 4.2M and 1.5M grids
then the flame brush computed using the 1.5M grid is about 2 to 3D longer compared to that
obtained using 4.2M grid. This behaviour is consistent with what is observed in Fig. 5.3 when
discussing of the static formulation for βc. The filtered reaction rate closure in Eq. (2.19) is not
fully valid when ∆+ < 1 [61, 77, 147], which is the case of the 4.2M grid in the region of the
strong reaction close to the jet exit, and this is reflected also in the dynamic computation.
The radial variations of averaged axial velocity, normalised using the respective bulk mean
velocity at the jet exit, computed using the dynamic and static βc values are compared to the
experimental measurements [46] in Fig. 5.22 for F1 and F3 flames. The results from both
106 Piloted flames - Algebraic closure
0.5
1
1.5
F1
0.5
1
1.5
F3
0.5
1
1.5
〈U
〉/
U
b
0.5 1 1.5
r/D
0 0.5 1 1.50
0.5
1
1.5
r/D
0.5
1
1.5
6.5
10.5
x/D = 2.5
4.5
8.5
Fig. 5.22 Comparison of measured [46] (symbols) and computed (lines) radial variation of normalised
axial velocity, ⟨U⟩/Ub, obtained using algebraic model for ω˙ with dynamic ( , •••) and static ( ,
•••) βc models for flames F1 and F3. The results are shown for 1.5M (solid lines) and 4.2M (dotted)
grids.
1.5M and 4.2M cases are shown for few axial locations investigated in the experimental study
of Chen et al. [46]. It is hard to distinguish the various curves for the flame F1, having the
largest turbulence Reynolds number, suggesting that the computed averaged axial velocity is
insensitive to the choice of βc modelling and numerical grid. The insensitivity to numerical
5.4 Assessment of dynamic procedure for βc 107
grid can also be seen for the flame F3, which suggests that the chosen grids are good. A small
difference is seen in ⟨U⟩/Ub for the flame F3 as in Fig. 5.22 and this difference grows to be
about 20% at axial location of x = 8.5D and radial position of about 0.8D. This is because the
value of βc obtained dynamically is smaller than the static value used for these locations. As
one shall see later, the combustion is almost completed before this radial position for this axial
location and the difference in βc values is because of some small differences in the gradients of
c˜. The variations and comparisons shown in Fig. 5.22 are typical in many earlier numerical
studies of these piloted Bunsen flames and thus the statistics shown here are acceptable.
The computed averaged TKE normalised using its centreline value, k0, near the jet exit is
compared to the measurements for few axial locations in F1 and F3 flames in Fig. 5.23. The
values of k0, 12.7 for F1 and 3.82 m2/s2 for F3, reported in [46] are used for this study. The
comparisons seen in this figure are typical of many earlier studies. However, the agreement
with the experimental data can be improved further than shown here by using finer grids as has
been demonstrated in [240], but we restrict ourselves to relatively smaller grids to study the
efficacies of the dynamic βc modelling for the filtered reaction rate closure. This is because the
closure in Eq. (2.22) works well when ∆+≫ 1 as discussed earlier, and ∆+ is already smaller
than unity for 4.2M grid used here. Thus, refining the numerical grid further will yield ∆+≪ 1,
which would invalidate the closure in Eq. (2.22) as also noted in [61, 77, 147]. Moreover,
refining the grid further would not be in line with a broader objective of this project as noted in
the introductory chapter of this thesis. A close study of the results in Fig. 5.23 suggests that
the peak values of ⟨k⟩/k0 obtained using both static and dynamic βc are similar to one another
and also to measured values. However, the radial location of this peak value obtained using
the dynamic approach agrees quite well with the measurement and the agreement, in general,
at downstream positions is improved when 4.2M grid is used for the flame F1 as one would
expect.
The radial variations of averaged fuel mass fraction and normalised temperature T+ = (T˜ −
Tu)/(Tb−Tu) are compared with experimental results [46] in Figs. 5.24 and 5.25 respectively.
The variation of averaged fuel mass fraction obtained using the power-law based model is also
shown in Fig. 5.24 for both flames computed using 4.2M grid. The variation of averaged fuel
mass fraction computed using this model suggests that this model is not representing the flame
behaviour and it is rather representing a convective-diffusive scalar behaviour. This is because
the scale-similarity for the gradient of c˜ does not hold even on the finest, 4.2M, grid used in this
study because ∆min+ is of order unity. This gradient is non-zero only inside the resolved flame
suggesting that there must be at least three points within δth for this scale-similarity to hold
implying that the test filter width must be smaller than 0.5δth. This gives severe underestimate
of α in Eq. (3.60) and thus ω˙ leading to the overestimate of averaged fuel mass fraction values
108 Piloted flames - Algebraic closure
5
10
15
F1
0
5
10
15
F3
0
5
10
〈k
〉/
k
0
0
5
10
0 0.5 1 1.50
5
10
r/D
0.5 1 1.5
r/D
4.5
8.5
10.5
6.5
x/D = 2.5
Fig. 5.23 Comparison of measured [46] (symbols) and computed (lines) radial variation of normalised
mean turbulent kinetic energy, ⟨k⟩/k0. The legend is as in Fig. 5.22.
seen in Fig. 5.24. The above grid requirement condition demands that the current grid should
be refined at least by a factor of 27 (3 in each direction). This gives many tens of millions
of grid cells, increasing the computational cost enormously, perhaps impractical for industry
applications. The dynamic evaluation of βc in Eq. (3.58) involves scale-similarity for u′∆ and c˜
(the test filter of ψ appears as sum at denominator in Eq. (3.58) and its magnitude is negligible
5.4 Assessment of dynamic procedure for βc 109
2
4
6
F1
F3
2
4
6
0 0.5 1 1.50
2
4
6
r/D
0.5 1 1.5
r/D
2
4
6
〈Y
C
H
4
〉(
%
)
2
4
6
10.5
6.5
4.5
x/D = 2.5
8.5
Fig. 5.24 Comparison of measured [46] (symbols) and computed (lines) radial variation of mean fuel
mass fraction. The legend is as in Fig. 5.22. The dash-dotted line is for the power-law based model in
Eqs. (3.60) and (3.61) for 4.2M grid.
in respect to the other terms) and thus, this assumption of scale similarity can be met with
relatively coarser grid. For these reasons, the power-law based model is not considered further
in this study.
The computed statistics for fuel mass fraction from static and dynamic formulation of βc
are similar on both 1.5 and 4.2M grids. The more pronounced under-prediction close to the jet
110 Piloted flames - Algebraic closure
0.5
1
F1
0.5
1
F3
0.5
1
〈T
+
〉
0.5
1
0 0.5 1 1.50
0.5
1
r/D
0.5 1 1.5
r/D
8.5
x/D = 2.5
4.5
6.5
10.5
Fig. 5.25 Measured [46] (symbols) and computed (lines) radial variations of normalised mean tempera-
ture are compared above. The legend is as in Fig. 5.22.
exit in case of dynamic βc is due to the fact that the flame in this region is quasi-laminar [46]
and the dynamic formulation relies on the existence of a clear range of scales, i.e. it is weak
in low-turbulence condition. The large difference in the statistics obtained using the dynamic
model on 1.5M and 4.2M grids observed for r ≤ 0.3D, especially at the downstream locations
for flame F1, cannot be commented because of the absence of experimental data in these radial
5.4 Assessment of dynamic procedure for βc 111
positions. Overall, results for CH4 mass fraction are good and comparable with previous studies
(see for example [53, 56, 122, 240]) despite the relatively small grids used here.
The variations of averaged normalised temperature shown in Fig. 5.25 are consistent with
the results for fuel mass fraction. The dynamic evaluation of βc seems to overestimate the fuel
consumption rate compared to the static case leading to an overestimate of peak temperature.
The level of this overestimate decreases as one moves downstream, implying that the perfor-
mance of the dynamic approach improves for both flames F1 and F3. The large overestimate
of mean temperature near the jet exit results from uncertainties in pilot stream temperature as
noted in §5.3 and in earlier studies [122, 240]. However, the averaged temperature computed
using the current sub-grid reaction rate closure is improved as noted in §5.3 and some additional
improvement from the dynamic approach is observed for downstream positions as seen in
Fig. 5.25.
The radial variations of averaged mass fractions of various scalars are shown in Figs. 5.26 to
5.31. The effects of entrained air is included in the computations of these species mass fraction
values as described in § 3.4. This effect is responsible for the increasing values of oxygen mass
fraction for r > 0.8D in Fig. 5.26 and this has been shown clearly in §5.3. The influence of
dynamic evaluation of βc on these various scalar mass fractions is found to be consistent with
observations noted for averaged fuel mass fraction and temperature shown in Figs. 5.24 and
5.25. The mass fractions of H2O and H2 are well estimated by the method used in this study.
The mass fraction of CO2 is underestimated and the level of this underestimate is larger for
the flame F1. There is a substantial overestimate in the mass fractions of CO and OH for both
static and dynamic evaluation procedure for βc. The computed increasing trend of OH mass
fraction with r improves for the flame F1 at downstream positions for both static and dynamic
computation of βc and some better prediction is seen in the latter case for flame F3, as seen
in Fig. 5.29. The large overestimate of CO mass fractions seen here is consistent with many
past studies. As noted earlier, the difference in various statistics obtained using the dynamic
and static evaluation of βc is observed to be small for the following reason. The static value
of this parameter is obtained for these two flames empirically by conducting many LES as
noted in §5.3 until an overall satisfactory behaviour is noted. The dynamic procedure described
here avoids this empirical testing and the average value of βc obtained in this procedure is
comparable to the static value as one shall see next.
5.4.2 The PDF of βc
The PDF of βc obtained by post-processing the LES data is shown in Fig. 5.32 for both flames,
F1 and F3. This PDF is constructed using βc values collected from the computational domain
by imposing the limits 0.05≤ c˜≤ 0.95 and ω˙ > 0.05⟨ω˙⟩max. These limits are used to avoid
112 Piloted flames - Algebraic closure
0
10
20
F1
0
10
20
F3
0
10
20
〈Y
O
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
8.5
x/D = 2.5
4.5
6.5
10.5
Fig. 5.26 Comparison of measured [46] (symbols) and computed (lines) mean O2 mass fractions. The
legend is as in Fig. 5.22.
regions with very low reaction rates so that the values of βc used for this PDF are physically
meaningful. The lower limit of βc is imposed by the realisability condition in Eq. (3.59). The
upper limit for F3 is nearly twice that for F1 and the PDF has a long tail for large βc values for
both of these flames. These PDFs are similar to those shown in the DNS analysis in [129]. It
is worth noting that the mean values ⟨βc⟩ given for the flames F1 and D6 are almost the same
because the Damköhler and turbulence Reynolds numbers are similar for these two flames.
5.4 Assessment of dynamic procedure for βc 113
0
10
20
F1
0
10
20
F3
0
10
20
〈Y
H
2
O
〉%
0
10
20
0.5 1 1.5
r/D
0 0.5 1 1.50
10
20
r/D
x/D = 2.5
6.5
8.5
10.5
4.5
Fig. 5.27 Comparison of measured [46] (symbols) and computed (lines) mean H2O mass fractions. The
legend is as in Fig. 5.22.
This value of ⟨βc⟩ ≃ 7.7 is very close to the static value of βc = 7.5 used for the flame F1 and
thus it not surprising to observe small differences, as noted above, in various statistics obtained
using the static and dynamic evaluation of βc. The value of ⟨βc⟩ = 10.5 given in Fig. 5.32
for the flame F3 is larger than for F1 because of higher Da and lower Ret values for F3, and
this value is about 40% larger than the static value used for the F3 flame. This good match
in the βc values for the static and dynamic procedure for flame F1 is not coincidental. The
114 Piloted flames - Algebraic closure
0
10
20
F1
0
10
20
F3
0
10
20
〈Y
C
O
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
x/D = 2.5
6.5
8.5
10.5
4.5
Fig. 5.28 Comparison of measured [46] (symbols) and computed (lines) mean CO2 mass fractions. The
legend is as in Fig. 5.22.
static value was obtained with empirical testing as discussed in §5.3. The dynamic procedure
described here eliminates this empiricism while retaining the accuracy with a small increase in
the computational cost.
The efficacy of the combustion modelling approach discussed in this paper is tested further
by calculating the thicknesses of the simulated flame brushes and comparing them with mea-
sured values. The results are shown in Fig. 5.33 for both flames F1 and F3. The flame brush
5.4 Assessment of dynamic procedure for βc 115
20
40
F1
20
40
F3
20
40
7
5
×
〈Y
O
H
〉%
20
40
0 0.5 1 1.50
20
40
r/D
0.5 1 1.5
r/D
x/D = 2.5
10.5
8.5
4.5
6.5
Fig. 5.29 Comparison of measured [46] (symbols) and computed (lines) mean OH mass fractions. The
legend is as in Fig. 5.22.
thicknesses computed have been compared to the reconstructed experimental values [46] as
discussed for Fig. 5.19 in §5.3.3. For the flame F3, the comparison shown here is reasonable.
The combustion closure is at its limit for the 4.2M grid when ∆+ < 1 and this results in thicker
flame brushes for both F1 and F3 flames. In other words, the closure in Eq. (2.19) is strictly
valid for ∆+ substantially larger than unity as explained in [61, 77, 147]. For the 1.5M grid
case, the values of δt obtained using the static βc model agrees very well with the measured
116 Piloted flames - Algebraic closure
0
10
20
F1
0
10
20
F3
0
10
20
1
0
0
×
〈Y
H
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
10.5
x/D = 2.5
8.5
4.5
6.5
Fig. 5.30 Comparison of measured [46] (symbols) and computed (lines) mean H2 mass fractions. The
legend is as in Fig. 5.22.
values for the flame F3 as shown in Fig. 5.33. The thicker flame brush for 4.2M grid case
is because of the combustion closure is at its limit for this grid as noted few times earlier.
Although the difference between the measured and computed δt values seems large in Fig. 5.33,
the maximum difference observed is about 4 mm while the maximum difference observed
for the comparisons shown in the top row Fig. 5.19 is also about 4 mm (some experimental
5.4 Assessment of dynamic procedure for βc 117
20
40
F1
20
40
F3
20
40
1
0
×
〈Y
C
O
〉%
20
40
0 0.5 1 1.50
20
40
r/D
0.5 1 1.5
r/D
10.5
4.5
6.5
x/D = 2.5
8.5
Fig. 5.31 Comparison of measured [46] (symbols) and computed (lines) mean CO mass fractions. The
legend is as in Fig. 5.22.
inconsistencies). Thus, whether this difference between computed and measured values of δt is
big or small, is an open question.
118 Piloted flames - Algebraic closure
0 5 10 150
0.2
0.4
0.6
F1
βc
P
(β
c
)
2
2Cm−1
〈βc〉 = 7.7
(a)
0 10 20 30
0
0.05
0.1
0.15
F3
βc
P
(β
c
)
〈βc〉 = 10.5
2
2Cm−1
(b)
Fig. 5.32 The PDF of βc obtained from LES of flames (a) F1 and (b) F3 using algebraic model for ω˙
with dynamic computation of βc with 1.5M grid. The average value and the lower limit imposed by
Eq. (3.59) are also shown.
0 5 10 150
5
10
15
20
x/D
δ t
F1
 
 
5 10 15
x/D
F3
 
 
Exp.
4.2M, βc - stat.
1.5M, βc - stat.
4.2M, βc - dyn.
1.5M, βc - dyn.
Fig. 5.33 The recomputed (⋄) and LES values of flame brush thickness, δt , obtained using algebraic
model for ω˙ with dynamic computation of βc, are compared for various cases.
5.5 Summary 119
5.5 Summary
The algebraic closure used for the simulation studied in this chapter inherently assumes that
the local Damköhler number is very large. Thus, the influence of finite rate chemistry effects is
not included. These effects are important from a practical perspective and can be included by
using unstrained and strained flamelets. This is discussed in the next chapter. The dynamic
evolution of βc eliminates the empiricism and thus it will be used for further investigation in
the following chapters, unless otherwise noted.

Chapter 6
Piloted flames - Presumed PDF closures
The unstrained and strained flamelets closures for filtered reaction rate described in §3.3.2
are studied in this chapter. The dynamic scales of turbulence containing about 80% to 90%
of turbulent kinetic energy are resolved in a typical LES. Hence, the influences of resolved
turbulent eddies on the flamelets are captured fully in the LES and of course one would need a
model to represent the effects of unresolved sub-grid eddies on the flamelets and these eddies
may have less than 10% of turbulent kinetic energy depending on the numerical grid employed
for LES. Thus, these scales may not have enough energy to impart considerable influence on the
flamelets to alter their structure and burning characteristics. This is only a conjecture here for
LES and careful analyses are required to make an assessment. Similar views have been debated
in the past while investigating premixed combustion regimes using DNS methodology [189] and
has been suggested that the small-scale eddies do not have adequate energy to impart significant
perturbations on unstrained laminar flame front structure. In the light of these observations,
whether one needs strained flamelets for typical large eddy simulations of premixed combustion
or not is an open question at least for configurations not involving local extinctions or blow off.
This question is answered to some extent in this chapter.
All the simulations obtained using unstrained and strained closures and shown in this
chapter are listed in tables A.2 and A.3 along with other simulations used to test the sensitivity
of various parameters in these modelling approaches.
6.1 Specific numerical detail
The relative performances of the strained and unstrained flamelet closures are assessed by com-
paring numerical results from LES to experimental measurements. The numerical procedure
for these closures is described in §3.3.2 and §3.4. The SGS velocity, u′∆, needed in Eq. (2.22),
is modelled using Eq. (3.51) on the basis of the analysis of §5.3.2. The dynamic procedure for
122 Piloted flames - Presumed PDF closures
σ
2 c
,s
g
s
Unstrained
 
 
c˜
0 0.5 1
0
0.1
0.2
σ
2 c
,s
g
s
εc = 5 [s
−1]
 
 
c˜
0 0.5 1
0
0.1
0.2
0
0.5
1
c˜
σ
2 c
,s
g
s
εc = 100 [s
−1]
 
 
0 0.5 1
0
0.1
0.2
c˜
σ
2 c
,s
g
s
εc = 500 [s
−1]
 
 
0 0.5 1
0
0.1
0.2
0
2
4
0
5
10
15
0
0.5
1
Fig. 6.1 Typical variation of normalised filtered reaction rate in unstrained and strained flamelets.
the combustion parameter βc described in §3.3.4 is used to compute this parameter based on
the results obtained in the previous section. However, comparisons with a static formulation
are also discussed, if required. Grids, boundary conditions and other numerical details are
discussed in §4.2, §5.1 and §5.2. The additional boundary for the SGS variance, σ2c,sgs, see
Eq. (3.36), is specified to be zero for all the inlets as it is generated predominantly by chemical
reactions inside the domain.
6.2 Filtered reaction rate
Typical variations of filtered reaction rate, ω˙ , normalised by δth/(ρusL), are shown in Fig. 6.1
for the unstrained and strained flamelets as a function of the controlling parameters (see §3.3.2).
This result is shown for stoichiometric methane-air flame, and similar variations are observed
for other stoichiometry values. The results are shown for three values of ε˜c for the strained
flamelets model. The maximum value of ω˙ occurs around c˜ ≃ 0.8 for very small values of
σ2c,sgs. For values around this c˜, as the variance increases the filtered reaction rate decreases,
which is because the burning part of the sub-grid PDF of c decreases as σ2c,sgs increases, which
is easy to verify using the β -function employed for this study. For 0.2≤ c˜≤ 0.5, the sub-grid
PDF broadens as σ2c,sgs increases which implies that the burning part of the flamelet are seen
occasionally leading to relatively larger reaction rate as seen in Fig. 6.1. The straining decreases
6.3 Grid sensitivity 123
the peak reaction rate substantially as one would expect. Except for this change, the variation
of filtered reaction rate is similar for the unstrained and strained flamelets. These tabulated
values are used in LES to simulate the piloted flames [46] studied in the previous chapter and
the results are discussed in the following sections.
6.3 Grid sensitivity
The sensitivity of the computational results to the numerical grid is discussed first. The radial
variation of mean axial velocity, TKE, fuel mass fraction and normalised temperature is shown
in Figs. 6.2-6.5 respectively. The results are shown for various streamwise positions for the
flames F1 and F3. Typical results are shown only for the F1 and F3 flames here because they
have the extreme values of Da, Ka and ReT in the range investigated experimentally in [46].
The mean velocity is normalised using the respective bulk-mean velocity at the jet exit and
the mean TKE is normalised using the values k0 = 12.7 m2/s2 for the F1 flame and 3.82 for
the F3 flame reported in [46]. The temperature is normalised as T+ = (T −Tu)/(Tb−Tu),
where T is the filtered temperature calculated using T˜ obtained from h˜ using Eq. (3.18), and
the fuel mass fraction is constructed from the computed c˜. The effect of the SGS variance
could not be included in ⟨T+⟩ because this quantity is not available in the LES framework
used here. These figures compare all of these quantities to their experimental values (open
circles) and those reported in earlier studies. For the discussion in this section one shall focus
on the present computational results obtained using 1.5M and 4.2M grids in reference to the
experimental values to understand the grid sensitivity and other results shown in these figures
will be discussed in later sections.
The mean velocity variations in the flames F1 and F3 show that the 1.5M grid is adequate
to capture the experimental variations. The results from the 4.2M grid shown only for the flame
F1 clearly demonstrates that the mean velocity is not strongly sensitive to the numerical grids
used here, which has been also observed for the algebraic closure investigated in the previous
sections. Similar behaviour is also observed for the strained flamelets. The TKE results shown
in Fig. 6.3 also suggest similar conclusion. However, the computed TKE are relatively large
for the strained flamelets and show some sensitivity to the grid as seen in Fig. 6.3. Despite
this, the general variation of the TKE does not seem to be influenced by the numerical grid.
The reason for the sensitivity observed for the strained flamelets model will become clearer
when the results of unstrained and strained flamelets are discussed in the next subsection. The
sensitivity (not shown) of mean mass fraction of methane and normalised temperature to the
numerical grid is similar to that shown for ⟨U⟩. These results suggest that the current numerical
124 Piloted flames - Presumed PDF closures
0.5
1
1.5
F1
0.5
1
1.5
F3
0.5
1
1.5
〈U
〉/
U
b
0.5 1 1.5
r/D
0 0.5 1 1.50
0.5
1
1.5
r/D
0.5
1
1.5
6.5
8.5
10.5
4.5
x/D = 2.5
Fig. 6.2 Comparison of measured [46] (symbols) and computed (lines) radial variation of ⟨U⟩/Ub for
F1 and F3 flames. Unstrained flamelet result is shown for 1.5M ( ) and 4.2M (•••) grids, and strained
flamelets result is shown for 1.5M ( ) and 4.2M ( ) grids. The results of 4.2M grid is shown only
for the F1 flame.
results are weakly sensitive to the numerical grids employed for this study and the second order
quantity such as TKE shows a sensitivity for the strained flamelets. The results of 1.5M grid
are used for the subsequent analysis below because of the weak grid sensitivity observed for
the mean quantities.
6.4 Unstrained vs Strained Flamelets 125
5
10
15
F1
0
5
10
15
F3
0
5
10
〈k
〉/
k
0
0
5
10
0 0.5 1 1.50
5
10
r/D
0.5 1 1.5
r/D
6.5
10.5
8.5
x/D = 2.5
4.5
Fig. 6.3 Comparison of measured [46] (symbols) and computed (lines) radial variation of ⟨k⟩/k0 for F1
and F3 flames. The legend is as in Fig. 6.2.
6.4 Unstrained vs Strained Flamelets
The fluid dynamic straining effects were shown [122] to be important in RANS calculations
of the Bunsen flames of [46] using unstrained and strained flamelets. When the straining
effects were excluded, the computed flame length was observed to be very much shorter than
the measured length in [46] suggesting that the fuel is consumed quickly. There is only one
turbulence length and time scales involved in the RANS and the effects of the spectrum of
126 Piloted flames - Presumed PDF closures
2
4
6
F1
F3
0.5 1 1.5
r/D
2
4
6
〈Y
C
H
4
〉(
%
)
2
4
6
2
4
6
0 0.5 1 1.50
2
4
6
r/D
6.5
4.5
x/D = 2.5
8.5
10.5
Fig. 6.4 Comparison of measured [46] (◦ ◦ ◦) and computed mean fuel mass fraction. The computational
results are shown for the 1.5M grid using unstrained ( ) and strained ( ) flamelets. Previous results
[53] (△), [56] (□), [122] ( ) and [240] (•••) are shown for comparison.
turbulence scales on the flame need to be included in the combustion modelling. On the
other hand, a range of scales is involved in LES, allowing the strain to be partially resolved
implicitly at super-grid scales. Various approaches have been investigated in the past to include
flame stretching using RANS and LES paradigm as discussed in §2.4.6. It was noted in
this section that the sub-grid variance of the progress variable is modelled in earlier studies
typically using an expression developed for passive scalars, and these expression is likely to
6.4 Unstrained vs Strained Flamelets 127
0.5
1
F1
0.5
1
0.5
1
〈T
+
〉
0.5
1
0 0.5 1 1.50
0.5
1
r/D
0.5 1 1.5
r/D
F3
10.5
6.5
4.5
8.5
x/D = 2.5
Fig. 6.5 The normalised mean temperature computed on the 1.5M grid using unstrained ( ) and
strained ( ) flamelets is compared to the measurements [46] (◦ ◦ ◦). Previous results [53] (△),
[56] (□), [122] ( ) and [240] (•••) are shown for comparison.
severely under-estimate this variance across the flame, as shown in the analysis of §3.3.2. These
under-prediction of SGS variance would result in an over-prediction of reaction rate in an
unstrained flamelets formulation, inducing one to think that strained flamelets are required.
To avoid such underestimates, a transport equation for the SGS variance of progress variable,
see Eq. (3.36), must be used. Unstrained and strained flamelet models described in §3.3.2
are thus compared in this study to understand their relative behaviour in LES in light of the
128 Piloted flames - Presumed PDF closures
above observation, and to assess if strain is effectively needed in flamelets modelling for LES
in general. This assessment is done in this section by comparing radial profiles of velocity,
temperature, TKE and species mass fractions obtained using unstrained and strained flamelet
closures to experimental results [46].
The results shown for normalised mean axial velocity, ⟨U⟩/Ub, in Fig. 6.2 show that the
heat release induced acceleration in the flame F3 having the highest Da is underestimated by
the strained flamelets model, compared to the unstrained flamelet model, for the upstream
positions as seen in this figure for x/D = 2.5 and 4.5. This implies that the local burning rates
may be under estimated by the strained flamelets model and it seems to be the case as one shall
see later while discussing Figs. 6.6 and 6.7. However, this difference becomes negligible as one
moves in the streamwise direction. Similar observations are made for the highest ReT flame F1
which has the lowest Da.
The normalised mean TKE obtained using the two flamelets is compared in Fig. 6.3 along
with experimental data. The trend obtained using these two flamelets are similar for the flame
F3 and there is some over estimation of the TKE in flame region, 0.55≤ r/D≤ 0.7. The values
obtained using the strained flamelets are generally larger than those obtained for the unstrained
flamelet model and the TKE values obtained using the later model compares favourably with the
experimental values for the flame F1 as shown in Fig. 6.3. A closer study of this figure suggests
that the shear layer, where the filtered flame is expected to be residing, is shifted slightly
towards the centreline which is suggested by the shift in the peak value of the mean TKE. The
production of TKE by the fluid dynamic shear is likely to be aided by the radial acceleration
of the fluid resulting from the heat release effects. Because of these effects the mean TKE is
overestimated for r/D ≤ 0.6 as observed in Fig. 6.3. Nevertheless, the comparisons shown
here for the unstrained flamelets are very similar to those observed in earlier studies using
various SGS [53, 56, 186, 240, 243] and RANS [5, 96, 122, 130, 143, 198, 208, 209, 222, 223]
combustion modelling approaches.
It is worth to investigate spatial variation of reaction rate before discussing the results in
Figs. 6.4 and 6.5 in detail. This spatial variation for the unstrained and strained flamelets
are compared in Figs. 6.6 and 6.7 for the flames F3 and F1 respectively. These results are
shown for two different times and log(100ω˙+) for ω˙+ > 0.01, where ω˙+ = ω˙δth/ρusL is the
normalised filtered reaction rate. The maximum value observed in the computations for ω˙+
is about 2.63 and the logarithm is used here to depict the spatial variations clearly because
the combustion and thus the chemical reactions are small scale phenomena. The laminar-like
structure with negligible wrinkling is observed in the F3 flame having the highest Da when
the unstrained flamelet model is used. If the strain effects are included then the influence of
shear layer roll-up resulting from Kelvin-Helmholtz instability on the filtered flame becomes
6.4 Unstrained vs Strained Flamelets 129
(a) (b) (c) (d)
Fig. 6.6 Spatial variation of log(100ω˙+) in flame F3 obtained at an arbitrarily chosen time, t1, using (a)
unstrained and (b) strained flamelets with 1.5M grid. These results at 20 ms later are shown in (c) and
(d). The contours are shown for ω˙+ > 0.01.
apparent as in Fig. 6.6 for the two arbitrary time instances chosen. Also, the peak reaction rate
occurs near the jet exit and this value is reduced considerably (see Fig. 6.1) as one would expect
when the straining effects are included. The overall flame length is more or less the same and
there is island formation as in Fig. 6.6 for the two combustion models used in this study.
Although the peak reaction rate is reduced for the strained flamelets model, the overall
mean burning rate is maintained by the increase in flame area resulting from the increased level
of flame wrinkling compared to the unstrained flamelet model. The overall mean burning rate
and thus the mean heat release rate given by m˙ f∆Hc, where ∆Hc is the heat of combustion, must
be the same for both the model because the fuel flow rate, m˙ f , is identical and there are no local
extinction. Despite this, a distinct difference is observed for the flame F1 having the highest ReT
and lowest Da as seen in Fig. 6.7. The peak reaction rate (red colour) is spatially intermittent in
Fig. 6.7a and an indication of its temporal intermittency can be seen by comparing Figs. 6.7a
and 6.7c, which are expected in high ReT flames. The most apparent difference between the
130 Piloted flames - Presumed PDF closures
(a) (b) (c) (d)
Fig. 6.7 Spatial variation of log(100ω˙+) in flame F1 obtained at an arbitrarily chosen time, t1, using (a)
unstrained and (b) strained flamelets with 1.5M grid. These results at 20 ms later are shown in (c) and
(d). The contours are shown for ω˙+ > 0.01.
filtered reaction rates computed using the unstrained and strained flamelet models is seen in
the region x/D≥ 10 for the flame F1. The strained flamelets model confines the reaction rate
to thin layers all the way to x/D ≈ 15 whereas the reaction rate is distributed more or less
uniformly over larger region for x/D≥ 11 when the unstrained flamelet model is used. A small
change in the size of this region is observed if one compares Figs. 6.7a with 6.7c. It is not
easy to comment which one of the behaviour shown in Figs. 6.6 and 6.7 for the unstrained
and strained flamelet models is the correct behaviour because the experimental study [46] did
not report results on reaction rate. Furthermore, it is not quite easy to quantitatively measure
the reaction rate but it is possible to image surrogates of reaction rate qualitatively using laser
diagnostics for OH, CH2O and CH species. In the absence of such information, one can
compare and analyse statistics gathered from the experimental and, past and current numerical
studies to further assess the relative behaviour of strained and unstrained flamelets models. The
comparisons discussed above for the mean velocity suggests that both unstrained and strained
6.4 Unstrained vs Strained Flamelets 131
0
10
20
F1
0
10
20
F3
0
10
20
〈Y
H
2
O
〉%
0
10
20
0.5 1 1.5
r/D
0 0.5 1 1.50
10
20
r/D
x/D = 2.5
6.5
10.5
8.5
4.5
Fig. 6.8 Comparison of measured [46] and computed H2O mass fraction. The legend is as in Fig. 6.4.
flamelets are good but the TKE variation suggests that the unstrained flamelets model may be
preferred. One must also investigate the behaviour of scalar mass fractions which is conducted
next.
6.4.1 Major species comparison
The radial variations of mean fuel mass fraction and normalised temperature are compared
to the experimental measurements and past numerical results in Fig. 6.4. The results are
132 Piloted flames - Presumed PDF closures
0
10
20
F1
0
10
20
F3
0
10
20
〈Y
O
2
〉%
0 0.5 1 1.50
10
20
r/D
0
10
20
0.5 1 1.5
r/D
6.5
8.5
10.5
4.5
x/D = 2.5
Fig. 6.9 Comparison of measured [46] and computed O2 mass fraction. The legend is as in Fig. 6.4.
shown for various streamwise positions in the F1 and F3 flames. The averaged fuel mass
fraction computed using the unstrained flamelet model is lower than the measured values
for 0.5 ≤ r/D ≤ 0.7 at x/D = 2.5 for both the F1 and F3 flames as shown in Fig. 6.4. The
estimates obtained using the strained flamelets are relatively better for this location. However,
the unstrained flamelet model values are closer to the measured values compared to those from
the strained flamelet model as one moves in the downstream direction. Indeed, the strained
flamelet model overestimates the averaged fuel mass fraction and the level of this overestimate
increases with downstream distance for both the F1 and F3 flames as observed in Fig. 6.4.
6.4 Unstrained vs Strained Flamelets 133
0
10
20
F1
0
10
20
F3
0
10
20
〈Y
C
O
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
x/D = 2.5
6.5
8.5
4.5
10.5
Fig. 6.10 Comparison of measured [46] and computed CO2 mass fraction. The legend is as in Fig. 6.4.
This is consistent with the reduced reaction rate observed in Figs. 6.6 and 6.7. The results in
Fig. 6.4 suggest that the unstrained flamelet model predictions are comparable to those obtained
using other combustion modelling approaches such as thickened flames [53, 240] and Eulerian
stochastic fields method [56].
The normalised mean temperature variation with normalised radius is shown in Fig. 6.5
for the flames F1 and F3 at various axial positions. No significant difference was observed
between ⟨T+⟩ and ⟨T˜+⟩, and thus the mean temperature is denoted using ⟨T+⟩ in the following
discussion. A significant overestimate at x/D= 2.5 is because of the uncertainty in the boundary
134 Piloted flames - Presumed PDF closures
condition for the pilot stream as observed also for the algebraic closure results in the previous
sections and in past studies. This overestimate decreases as one moves downstream for both
F1 and F3 flames. The relative behaviour of the unstrained and strained flamelets models is
consistent with that for the fuel mass fraction. Although the strained flamelets model gives a
good comparison for x/D = 6.5 for the F1 flame, it underestimates the peak temperature for
downstream positions and the level of under-prediction increases gradually with downstream
distance as seen in Fig. 6.5, whereas the unstrained flamelet model prediction improves with the
distance and is comparable to previous results and measured values. The difference between
these two flamelet models is small for the flame F3 with the highest Da.
Figure 6.8 compares the computed values of mean mass fraction of H2O to the measure-
ments [46] and results of previous studies. The results of both unstrained and strained flamelet
models are shown in this figure and the strained flamelets model under-predicts the mean mass
fraction of water vapour, which is consistent with the fuel mass fraction variation discussed
earlier. The values obtained using the unstrained flamelet model are close to the measurements
for all the axial positions in both F1 and F3 flames. Also, one can observe in Fig. 6.8 that the
unstrained flamelet predictions are improved compared to some of the earlier studies employing
nearly 3 to 4 times the grid used in the current study. The comparisons for other major species
such as O2 and CO2 are similar to those shown here for water vapour and fuel and are shown
in Figs. 6.9 and 6.10.
6.4.2 Minor species comparison
Figures 6.11-6.13 respectively compare the radial variations of minor species, OH, H2 and CO,
mass fractions with the results of past studies and measurements. The results of past studies
are shown in these figures only if they were reported in those studies. There is a severe under
prediction of OH and H2 mass fractions by the strained flamelets model whereas the unstrained
flamelet model yields good prediction of these minor species for both F1 and F3 flames. The
OH mass fraction comparison shown in Fig. 6.11 is much better than the comparison shown in
Figs. 5.16 and 5.29 for the algebraic reaction rate closure based on high Da combustion. This
improved comparison is because the unstrained flamelet model allows one to include local
finite rate chemistry effects on the filtered reaction rate. Computed CO mass fraction profiles
from unstrained flamelet closure are largely overestimated and is overestimation was also seen
in Figs. 5.18 and 5.31 for the algebraic model and in previous studies [56, 122]. However,
this over-prediction seems to be related to under-predictions seen for CO2 in Fig. 6.10. Good
predictions of CO mass fraction are observed when the strained flamelets closure is employed
in case of flame F1. Nevertheless, the same quality is not observed for flame F3.
6.4 Unstrained vs Strained Flamelets 135
20
40
F1
20
40
F3
20
40
7
5
×
〈Y
O
H
〉%
0 0.5 1 1.50
20
40
r/D
0.5 1 1.5
r/D
20
40
x/D = 2.5
10.5
8.5
4.5
6.5
Fig. 6.11 Comparison of measured [46] and computed OH mass fraction. The legend is as in Fig. 6.4.
The results presented so far suggest that the unstrained flamelet model performs consistently
better compared to the strained flamelets, which is interesting to note in the light of the
observation made in [122] suggesting that the strained flamelets are required for correct
prediction of these piloted Bunsen flames using RANS methodology. Furthermore, the strained
flamelets model is observed to behave differently depending on the flame (F1 or F3) and the
axial position, suggesting a dependence from the local Reynolds number which is not observed
in case of the unstrained flamelets closure.
136 Piloted flames - Presumed PDF closures
0
10
20
F1
0
10
20
F3
0
10
20
1
0
0
×
〈Y
H
2
〉%
0
10
20
0 0.5 1 1.50
10
20
r/D
0.5 1 1.5
r/D
10.5
8.5
x/D = 2.5
4.5
6.5
Fig. 6.12 Comparison of measured [46] and computed H2 mass fraction. The legend is as in Fig. 6.4.
The under-predictions seen here for the strained flamelets model are because of under
prediction of reaction rate observed in Figs. 6.6 and 6.7. When the flame front is stretched its
reaction rate is expected to drop and the stretching is caused by straining, bending (curvature)
and contortion of the flame front by the turbulent eddies. The majority of these energy
containing eddies are captured in typical LES as that reported here and thus their influence
on the filtered flame is included inherently through c˜, σ2c,sgs and enthalpy transport equations
and their interaction with the resolved flow fields. So, including stretching effects through
strained flamelets overestimates the influences of stretch. However, the stretching caused by
6.4 Unstrained vs Strained Flamelets 137
20
40
F1
20
40
F3
20
40
1
0
×
〈Y
C
O
〉%
20
40
0 0.5 1 1.50
20
40
r/D
0.5 1 1.5
r/D
10.5
x/D = 2.5
6.5
8.5
4.5
Fig. 6.13 Comparison of measured [46] and computed H2 mass fraction. The legend is as in Fig. 6.4.
the unresolved SGS turbulent motions may not be small if the SGS kinetic energy is large. The
numerical grid used here suggests that this kinetic energy is small and the comparison shown
in Fig. 6.3 is good specifically for the flame F1 computed using the unstrained flamelet model.
Thus, including the flame stretching effects of SGS turbulence through strained flamelets over
estimates the stretching effects leading to a substantial reduction in the burning rate, which is
reflected in all the scalar profiles investigated in this study. Thus, one may not need to explicitly
account for flame stretching by SGS turbulence in a typical LES, which is expected to have
small kinetic energy in sub-grid scales.
138 Piloted flames - Presumed PDF closures
−2 −1 0 1 2
0
0.5
1
1.5
2
log10 Da∆
P
(l
o
g 1
0
D
a
∆
)
−2 −1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
log10 Ka∆
P
(l
o
g 1
0
K
a
∆
)
−2 −1 0 1 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
log10 Da∆
P
(l
o
g 1
0
D
a
∆
)
−2 −1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
log10 Ka∆
P
(l
o
g 1
0
K
a
∆
)
Fig. 6.14 PDFs of Da∆ and Ka∆ from flames F1 (top row) and F3 (bottom row). The results are shown
for unstrained ( ) and strained ( ) flamelets.
It is also important to note that the SGS variance of the progress variable is computed using
its transport equation rather than through a commonly used model, σ2c,sgs ≃ A ∆2(∇c˜ ·∇c˜)
to include correct and meaningful interactions among turbulence, scalar mixing and reactive
fields, as discussed in §3.3.2. If one uses this algebraic model then ω˙ is overestimated severely
for unstrained flamelets (see Fig. 6.1), leading to a conclusion that one would need strained
flamelets as noted in the beginning of this section. Thus, care is required for LES to maintain
the consistencies among various sub models.
6.5 Analysis of Da∆ and Ka∆
Figure 6.14 suggests that the typical values of Da∆ range from about 0.1 to 1.8 in the F1 flame
and 0.16 to 5 in the F3 flame. These ranges are not strongly influenced by the stretching effects
included in the strained flamelets model, except that the most probable value is reduced slightly.
6.5 Analysis of Da∆ and Ka∆ 139
0 5 10 150  
0.1
0.2
βc
P
(β
c
)
0 5 10 15 20 25 300  
0.05
0.1
βc
P
(β
c
)
Fig. 6.15 The PDF of βc from flame (a) F1 and (b) F3 obtained using unstrained ( ) and strained ( )
flamelets with 1.5M grid.
The relative behaviour between the unstrained and strained flamelet models does not seem to
vary between F1 and F3 flames, which is consistent with the flame statistics discussed earlier.
The variation of Ka∆ PDF is similar and consistent with the PDF of Da∆. These two PDFs are
more or less symmetric about the most probable value suggesting that they may be lognormal.
However, Ka∆ PDF shows a slightly longer negative tail because of the non-liner dependence
on the small scale quantities.
The scale-dependent βc in Eq. (2.22) is an important parameter in the SGS combustion
modelling approach used in this study. This parameter is obtained dynamically using the
scale-similarity methodology as discussed in section 3.3.4 and it varies spatio-temporally. The
PDF of βc obtained by post-processing the LES data is shown in Fig. 6.15 for both F1 and F3
flames. These PDFs are constructed using βc values collected from the entire computational
domain by imposing the limits 0.05 ≤ c˜ ≤ 0.95 and ω˙ > 0.05⟨ω˙⟩max to avoid regions with
very low reaction rates so that the values of βc used for this PDF are physically meaningful,
which was verified in §5.4.2 and in [129] in conjunction with the algebraic closure of §3.3.1.
The PDFs shown here for the unstrained and strained flamelets are similar to that in Fig. 5.32,
except for a lower limit used in that study. The lower limit was set to satisfy a realisability
condition of N˜c ≥ 0, which was obtained by analysing the transport equation for the mean
scalar dissipation rate [39]. For the reaction rate closures used in this study, the realisability
condition for the filtered dissipation rate, N˜c ≥ 0, is met if C3 ≥ τC4Da∆, see Eq. (2.22), which
is satisfied automatically by the functional forms of C3 and C4 described in §3.3.1.
The PDF of βc is shown for both unstrained and strained flamelets for the 1.5M grid
simulations. These results for the 4.2M grid are very similar to those shown here. There are no
difference between the unstrained and strained flamelets and thus the flame stretching effects
140 Piloted flames - Presumed PDF closures
introduced through the strained flamelets does not seem to influence the physical processes
described by the βc parameter.
6.6 Comparison of static and dynamic βc
The static formulation for the combustion parameter βc is also employed to have a direct
comparison with the dynamic formulation in the context of unstrained flamelets. The differences
in case of the strained flamelets closure are smaller than for the unstrained one and thus are not
shown here. A value βc = 7.5 is chosen based on the analysis of §5.3, which is also meaningful
in light of the PDFs shown in Fig. (6.15). Mean profiles of fuel mass fraction obtained using
static and dynamic formulation for βc are compared to experimental data [46] in Fig. 6.16. The
mean fuel mass fraction profiles obtained using static βc compare to experimental data better
than for the dynamic formulation at x/D = 2.5 and 4.5. This is because the separation of scales
needed for the scale-similarity assumption in the dynamic computation of βc is not completely
fulfilled in the quasi-laminar region close to the jet exit, resulting in an under-prediction of
βc for this region. A similar relative behaviour was observed in §5.4.1 (Fig. 5.24) when
comparing static and dynamic formulation of βc for the algebraic closure. It is worth noting
that the static value of βc is chosen empirically and is optimal for the flame configuration.
The differences in results between static and dynamic βc formulation become smaller when
one moves downstream, where the turbulence level is sufficiently large. In case of flame F1,
results obtained using the dynamic formulation for βc look better than those obtained with the
static formulation at downstream locations, where the effect of air entrainment is high, and for
r < 0.5D. This suggests that the dynamic parameter may absorb effect of air entrainment. This
is also indicated by the fact that similar improvements are not seen in case of flame F3, where
the air entrainment is much lower than for F1.
6.7 Summary
The analysis presented in this chapter shows that there is substantial difference between the
results obtained using the unstrained and strained flamelets closures. The stretching of SGS
turbulence is parameterised using the SGS dissipation rate in the strained flamelets closure
used in this study. Since the numerical grids employed for the simulations resolve most of
the turbulent kinetic energy the SGS kinetic energy is expected to be small implying that the
SGS eddies may be weak to impart any stretching effects on the flame front. This is similar
to the views expressed by Poinsot et al. [189] and Roberts et al. [202] while investigating
turbulent combustion regimes. Thus, using the SGS dissipation rate or any parameter of
6.7 Summary 141
2
4
6
F1
F3
0.5 1 1.5
r/D
2
4
6
〈Y
C
H
4
〉(
%
)
2
4
6
2
4
6
0 0.5 1 1.50
2
4
6
r/D
6.5
4.5
x/D = 2.5
8.5
10.5
Fig. 6.16 Comparison of measured [46] (◦ ◦ ◦) and computed mean fuel mass fraction. The compu-
tational results are shown for the 1.5M grid using unstrained flamelet model with dynamic ( ) and
static ( ) formulation for βc.
strained flamelets established in counterflow geometries to include the affects of SGS stretching
is likely to overcompensate the SGS stretch effects compared to the influences of resolved
dynamic eddies on the unstrained flamelet. Hence, the burning rate is underestimated by
the strained flamelets leading to over prediction of fuel mass fraction and under-estimates of
various reactive scalar mass fractions as observed in this study. The influences of resolved
dynamic scales are inherently included in the unstrained flamelet closure through the various
142 Piloted flames - Presumed PDF closures
transport equations and their interactions. The SGS variance equation plays an important role
in this aspect and using an algebraic closure for σ2c,sgs is inadequate, which is demonstrated
using order of magnitude analysis of the SGS variance equation in §3.3.2. The commonly
used model, σ2c,sgs ≃ A ∆2(∇c˜ ·∇c˜), severely underestimates the SGS variance because it
excludes the combustion effects. This underestimated variance will give large reaction rate
for the unstrained flamelet closure (see Fig. 6.1) implying a need for the strained flamelets, as
observed for example in [116]. The results presented in this chapter suggests that the unstrained
flamelet model is preferred at least for the combustion conditions investigated in this study
and one must pay close attention to the modelling of SGS variance to retain the subtleties of
premixed combustion physics consistently across the various SGS closures for LES of premixed
combustion when flamelets closure, in general, is used with presumed PDF. Applying this
modelling framework to other flame configurations such as bluff-body stabilised flames is of
interest and will be explored in the next chapter.
The effect of finite rate chemistry is also of interest and it can be observed if one compares
the results obtained in this chapter using unstrained flamelets closure with those for algebraic
closure in previous chapter. From this comparison, it seems that a significant difference is
seen only for minor species mass fractions for the piloted flame configuration analysed here.
It would be ideal to evaluate this relative model performance on a different flame spanning a
range of combustion regimes. For this reason the unstrained flamelet closures is also compared
to the algebraic closure in the bluff body stabilised configuration which is described in the next
chapter.
Chapter 7
Bluff body stabilised flame
The bluff-body stabilised flame configuration studied experimentally [168, 169, 176–178] is
presented in this chapter. Bluff-body configurations are commonly used in engines and many
combustion devices to provide flame anchoring mechanism. The recirculation zone behind
the bluff-body provides this mechanism by increasing the residence time of the burned gas
aiding combustion to sustain itself [191]. Despite a simple geometry, the physical processes
involved here are complex encompassing strong interplay among turbulence, combustion and
heat transfer. The performance of these type of burners, in general, is influenced by size and
shape of and turbulence in the recirculation zone, which in turn is affected by a number of
factors such as inlet turbulence, equivalence ratio and geometry. Thus, this configuration is
ideal to further explore capabilities of flamelet based modelling.
Results obtained using the algebraic model and the unstrained flamelets model described
in §3.3.1 and 3.3.2 are compared to experimental data to assess the relative performance of
these two reaction rate closures and the sensitivity to a number of parameters, such as inlet
turbulence, grid size and flamelet used, is studied. The SGS velocity, u′∆, needed for the SGS
scalar dissipation rate in Eq. (2.22) is modelled using Eq. (3.51) as this model gave good results
in the analysis of §5.3. The parameter βc is computed dynamically using Eq. (3.58), as this
procedure was observed to provide accurate results when turbulence level is relatively high, as
seen in §5.4 and Chapter 6.
Particular emphasis is given to the investigation of sub-grid fluctuations, which are observed
to play a key role in the model performances when turbulence intensity is high. This is
addressed by comparing results obtained using two different conditions of turbulent intensity
(TI) with related experimental measurements.
144 Bluff body stabilised flame
0 0.5 1 1.5 20
1
2
3
4
5
6
∆+
P
(∆
+
)
(a)
0 0.5 1 1.5 20
1
2
3
4
5
∆+
P
(∆
+
)
(b)
Fig. 7.1 Histograms of ∆+ = ∆/δth for (a) 1.8M and (b) 2.2M grids are shown for the combustor region
0≤ x/D≤ 6 and a fraction of it for r/D< 0.5.
7.1 Experimental case
The experimental burner uses a methane-air mixture at φ = 0.586. The burner configuration, its
geometrical details and boundary conditions are as specified in §4.3 and Table 4.5. A sketch of
geometry and boundary conditions is shown in Fig. 4.10. The domain is discretised using two
meshes of 1.8M and 2.2M cells each and the difference in size between the two meshes is due
to wall refinement on the bluff body surfaces for 2.2M grid. A sketch of these two grids close
to the bluff body is shown in Fig. 4.11. Turbulence conditions at the inlet of the computational
domain are specified using the digital filter technique [112] and the two fluctuations levels
u′/Uref = 2% and 24%, as for the cold flow. The integral length scales in the three directions
have been assigned according to the size of the holes (3.46 mm) of the turbulence generator
device. The progress variable is defined as c= 1−YCH4/Y uCH4, where Y uCH4 is the methane mass
fraction in the reactant mixture, and is assigned to be 0 at the inlet of the computational domain.
Other choices have also been explored and will be discussed in §7.5.1.
Since the algebraic closure of §3.3.1 requires filter sizes larger than the laminar flame thick-
ness as observed in §5.3 and §5.4, an a priori assessment is done for the two grids. Histograms
of the normalised filter size are shown in Fig. 7.1 for both grids, and the filter size is defined
as ∆ = V 1/3i , where Vi is the volume of the cell. These histograms are shown both for the
entire combustor and for a smaller region, 0≤ r/D≤ 0.5, to illustrate the effect of the cell size
increase in the radial direction. Both grids have a peak at ∆+ ≈ 1. This value is acceptable to
fulfil the modelling requirements of both algebraic and unstrained flamelets models [61, 129].
Nevertheless, because of wall refinement, local values can go below 1 for the 2.2M grid, and
7.2 Flame and recirculation region 145
this restricts the use of the algebraic reaction rate closure requiring ∆+ > 1 [61]. These grids
are relatively small compared to what is commonly used for LES of combusting flows if one
considers that about 720k cells are used for the region downstream the combustor, but they
are adequate to yield accurate results for the modelling framework used in this study. This
was already demonstrated for a piloted jet flame configuration in §5.3 and §5.4, and will be
reconfirmed in this study for a bluff-body flame.
The full list of simulations done on this configurations is shown in Table A.4. and the most
relevant for this study are discussed here. All these simulations have been run for a total of 10
flow-through times as specified for the cold flow, corresponding to about 24h on a wall clock.
The CFL number is kept below 0.3 everywhere in the flow using a constant time step of 10µs
and 7.5µs respectively for the 1.8M and 2.2M grids.
7.2 Flame and recirculation region
The reacting flow field was divided in three regions in [178]: (i) a first region (R1) between
the flame holder and x/Lr = 0.12, where the flame is thin and combustion time scale is much
smaller than the turbulence time scale. The assumption of flamelets holds in this region; (ii) the
recirculation region (R2), 0.12< x/Lr < 1, where coherent structures start to grow, thickening
the flame. The regime was stated to be in the thin reaction zones of the premixed regime
diagram; (iii) a region (R3) further downstream of the rear stagnation point, where the flame is
thick and turbulent structures strongly influence the flame. The flame condition in this region
was suggested to be in the distributed reaction zones regime.
Experimental results are provided for regions (i) and (ii) for both cases with 2% and 24%
TI, and the results for region (iii) are available only for the case with 24% TI. These results are
compared with the LES statistics.
7.2.1 2% approach turbulence
LDA and CARS techniques were respectively used by Pan et al. [176, 178] to measure the
fluid velocity and temperature respectively. These techniques were reported to have 1% and
±30 K uncertainty respectively. The fast chemistry assumption is valid close to the bluff-body
base where the flame is anchored. In this region, the flame was observed to be thin in the
experiments [178]. The flame brush becomes thick as one moves downstream and it has become
almost 12 times thicker at an axial location close to the rear stagnation point of the recirculation
zone [178]. A similar behaviour is observed in the computational results also as shown in
146 Bluff body stabilised flame
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
2
2.5
3
y/D
x
/
D
 
 
0
0.5
1
1.5
2
2.5
R1
R2
R3
(a)
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
2
2.5
3
r/D
x
/
D
 
 
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
(b)
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
2
2.5
3
y/D
x
/
D
 
 
1
1.5
2
2.5
3
(c)
Fig. 7.2 Stream lines of (a) filtered and (b) averaged velocity fields along with contours of (filtered and
averaged) log10(1000 ω˙
+
), where ω˙+ = ω˙δth/(ρusL). The results are on the mid-plane for unstrained
flamelets model and 2% TI. The contour of log10(1000Da∆) is shown in (c). The three regime regions,
R1, R2 and R3, are separated in the figures using dashed lines.
Fig. 7.2. The variations of filtered and averaged reaction rates are shown as colour maps in
Fig. 7.2 for the mid-plane obtained from the simulation employing unstrained flamelet model.
Figure 7.2a shows the filtered reaction rate along with the streamlines. The variation of averaged
reaction rate and the streamlines of averaged velocity are shown in Fig. 7.2b. The variation of
SGS Damköhler number, Da∆, in the corresponding plane is shown in Fig. 7.2c. The filtered
reaction zone is very thin close to the bluff body but its width increases quickly as one moves
downstream. The length of the recirculation zone is about 2D as seen in Fig. 7.2b and the
averaged reaction zone thickness grows in the downstream direction. Also, one observes that
the averaged reaction zone is already very thick at the end of the recirculation zone and thus the
reaction zone structure is likely to be significantly influenced by turbulence from this location
onwards and effects of finite rate chemistry become non-negligible. In particular, the algebraic
model relies on the high Damköhler number assumption which is expected not to hold well in
this region. Some insight can be gleaned by studying the SGS Damköhler number variation
shown in Fig. 7.2c. The relatively low values of Da∆ near the edges of bluff-body are due to
intermittency in the axial velocity fluctuation and the flame in this region is expected to be
quasi-laminar. The values of Da∆ ∼ 0.01 are because of spatio-temporal intermittency effects
and a value of order unity is not observed in regions of flame suggesting that the SGS fluid
dynamic time scale is typically shorter than the chemical time scale, δth/sL. This suggests that
7.2 Flame and recirculation region 147
0 1 2 3 4
−1
−0.5
0
0.5
1
1.5
2
2.5
x/D
〈U
〉/
U
r
e
f
(a)
0 2 4 60.2
0.4
0.6
0.8
1
1.2
x/D
〈T
+
〉
(b)
0 2 4 60
0.1
0.2
0.3
0.4
x/D
√ 〈σ
2 T
〉
(c)
Fig. 7.3 Centreline variation of (a) ⟨U⟩/Uref, (b) ⟨T+⟩, and (c)
√
σ2T for 2% TI. The results with
algebraic model on 1.8M ( ) and 2.2M ( ) grids and with unstrained flamelets model on 1.8M ( )
and 2.2M ( ) grids are compared with experimental data [176, 178] (symbols).
the finite rate chemistry effects are important in these flames. This observation must be kept
in mind while comparing LES statistics obtained using the algebraic and unstrained flamelet
closures with the measurements.
First and second order statistics from LES are compared next to experimental results. Both
1.8M and 2.2M grids are used for both algebraic and unstrained flamelets closures for further
assessment of grid sensitivity. Wall functions with WALE model [172] are used in case of
1.8M grid. Figure 7.3a compares the averaged axial velocity, ⟨U⟩, obtained from LES, with
measurements. This velocity is normalised using Uref and its variation along the centreline is
shown in this figure. The experimental data are taken from [176]. The algebraic and unstrained
flamelets models yield similar statistics for x/D≤ 1.1 and the underestimate of ⟨U⟩ by these
two models for x < D is due to adiabatic wall used in the simulation. This wall treatment
avoids heat losses and thus the flow accelerates more in the computation. As one moves in
the downstream direction, the results of these two models differ significantly. The length of
the recirculation zone, which is critical for the overall dynamics of the flow, is well estimated
by the unstrained flamelets model compared to the algebraic model. It was suggested in the
experimental study [176] that the flame at locations downstream of the rear stagnation point
(x∼ 2D) is thickened and combustion is of distributed reactions regime type. Pan et al. [178]
argued that the flame thickness near the rear stagnation point axial location is about 12 times
its size near the flame holder, as noted earlier. This flame is about 5 times ticker than δth and
might pose a challenge to the flamelets approach. Significant overestimates are observed in
Fig. 7.3a for the algebraic model in this region independently of the grid used. However, the
unstrained flamelets model captures the experimental observation suggesting that this model is
sufficiently accurate for this combustion.
148 Bluff body stabilised flame
A typical comparison for the centreline variation of normalised mean temperature, ⟨T+⟩=
(⟨T ⟩−T0)/(Tad−T0), computed here and measured values is shown in Fig. 7.3b. This tempera-
ture is underestimated by the algebraic model whereas the unstrained flamelets model estimates
are very good apart for some overestimates nearer to the bluff-body due to adiabatic condition
used for the simulations as noted earlier. The better behavior for 1.8M grid in case of algebraic
model in Figs. 7.3a and 7.3b is due to the fact that the condition ∆+ > 1 [61] is not met close to
the bluff body for 2.2M grid and this affects significantly the reaction rates as already noted in
§5.3 and §5.4. The difference in these statistics obtained for the unstrained model using these
two grids is very small implying that this model is more robust and accurate than the algebraic
model. The centreline variation of rms (root-mean-square) of normalised temperature is shown
in Fig. 7.3c. The rms value is estimated here using
⟨σ2T ⟩= ⟨T˜+2−⟨T˜+⟩2⟩+ ⟨σ2T,sgs⟩= ⟨σ2T,res⟩+ ⟨σ2T,sgs⟩ (7.1)
where σ2T,sgs indicates the SGS contribution of the normalised temperature, which is approx-
imated as ⟨σ2T,sgs⟩ ≈ ⟨σ2c,sgs⟩ because the Lewis number is close to unity for this flame and
a temperature based progress variable would have the same boundary conditions as c˜ in the
LES domain. The above result is shown only for the unstrained flamelets model since σ2c,sgs is
unavailable for the algebraic model. Higher mean temperature was seen to correspond to lower
rms and vice-versa in the experiments [178] and this behaviour is captured by the unstrained
flamelets model. The underestimate of
√
⟨σ2T ⟩ for x < D is related to the overestimate of
mean temperature in this region resulting from adiabatic conditions used. Overall, the rms
temperature variation along the centreline is captured reasonably well and this result shows
negligible sensitivity to the numerical grid.
Further insight can be gained by studying radial variation of normalised mean and rms
temperature shown in Fig. 7.4 for three axial locations. The latter is shown only for the
unstrained flamelets model for the reasons noted earlier. The results are shown for both
1.8M and 2.2M grids. The algebraic model estimates the mean temperature variation well for
x = 0.1D and 0.8D, where high Damköhler number combustion is expected to hold. Since this
model works well for ∆+ > 1, the statistics obtained using 1.8M grid are closer to the measured
values for these two locations. The statistically significant value of ∆+ does not go below about
1.0 in the domain for the 1.8M grid, while it can become less than 0.8 and indeed below 0.5
in regions close to the bluff body when the 2.2M grid, involving wall refinement, is used. A
severe underestimate is observed for x = 2D as shown in Fig. 7.4. The radial variation of mean
temperature obtained using the unstrained flamelet model compares well with the measured
values for the downstream location and an overestimate of about 10% observed for x = 0.1D
and 0.8D for 0≤ r ≤ 0.5D is probably because of the adiabatic wall used for the bluff-body
7.2 Flame and recirculation region 149
0.2 0.4 0.6
r/D
0.2 0.4 0.6
r/D
0  0.2 0.4 0.60  
0.2
0.4
r/D
√ 〈σ
2 T
〉
 
 
0.6
0.8
1  
〈T
+
〉
2.0x/D = 0.1
x/Lr = 0.05
0.8
0.4 1.0
Fig. 7.4 Radial variation of ⟨T+⟩ and
√
σ2T for 2% TI case with algebraic model using 1.8M ( )
and 2.2M ( ) grids and with unstrained flamelets model using 1.8M ( ) and 2.2M ( ) grids are
compared with experimental data [176, 178] (symbols).
base. It is to be noted that the bluff-body base ends at r ≈ 0.5D as seen in Fig. 7.2. Although
the local values of Da∆ is low, the unstrained flamelet model is shown to work well for x = 2D
as it can handle finite rate chemistry effects. Also a reasonable estimate of the progress variable
variance plays a role and an accurate and robust modelling of SGS dissipation rate is important
for this. The comparison of radial variation of normalised temperature rms shown in Fig. 7.4 is
reasonable and satisfactory although there is some room to improve this. Possible avenues to
improve the rms values are yet to be explored.
7.2.2 24% approach turbulence
Increasing the level of turbulence entering the combustion chamber changes the flow dynamics
and thus the combustion behaviour significantly. The experimental data [176, 178] show that the
size of the recirculation region decreases from Lr ≃ 2D to 1.3D when the incoming turbulence
intensity is increased from 2% to 17%. Increasing this turbulence level further by 5%, gives
150 Bluff body stabilised flame
0 0.1 0.2
0.5
0.52
0.54
0.56
0.58
0.6
r
/
D
u′/Ub
v ′r/Ub
(a)
0 0.5 1 1.5
0.5
0.52
0.54
0.56
0.58
0.6
r
/
D
U/UbVr/Ub
(b)
Fig. 7.5 Radial profiles of (a) rms and (b) mean velocities, obtained using the algebraic model on 1.8M
grid ( ) and unstrained flamelet model on 2.2M grid ( ), are compared with experimental data
(symbols) for 24% TI at x/D = 0.
Lr ≃ 1.1D, i.e. the recirculation zone length decreases by about 15%. In a recent LES [9] of
this flame, the turbulence at the inlet was underestimated leading to larger recirculation length.
The sensitivity of Lr to incoming turbulence level and the lack of a full characterisation of this
turbulence in the experiment poses additional challenges for CFD simulations. An attempt is
made in this work to simulate the case with 24% approach turbulence intensity since a good set
of scalar measurements were reported for this flame in [168, 169, 176, 178]. The turbulence
levels reported in the experimental work by measuring the rms velocity in the axial direction at
the flame holder station are 18% and 22% of TI for r/D = 0.52 and r/D = 0.55 respectively.
This TI is defined in reference to the velocity, Uref. Mean and rms values of radial and axial
velocities from LES are compared with the experimental data in Fig. 7.5. The 24% nominal T.I.
has been varied by a small amount (≈ 3%) until the condition at the burner entrance for the
cold flow is matched with the experimental measurements, yielding the results in Fig. 4.12b.
This condition is then maintained for the reacting flow simulation, which produces a mean
level of turbulence intensity of 19% (for the region 0.5≤ r/D≤ 0.7) but turbulence is higher
closer to the bluff-body edge before vanishing very quickly on the wall, as seen in Fig. 7.5.
This mean value is about 2% smaller than that obtained for the cold flow, which is probably
because of the heat release effects. About 2% and 10% difference exists between the algebraic
and unstrained flamelet models, respectively for the axial velocity, u′, and the radial velocity,
7.2 Flame and recirculation region 151
v′r, rms values, as seen in Fig. 7.5a. The results of the algebraic model are obtained using 1.8M
grid for the reason noted earlier. The differences observed for the combustion models are thus
due to the differences in flow evolution along the bluff body wall for the two grids, with better
estimates on the 2.2M grid. Mean velocities from the two models are very similar, but both
models overestimate the experimental values, as seen in Fig. 7.5b. These results suggest that
the computed boundary layer is somewhat thinner than in experiments. These differences at
the burner entrance may have a non-negligible effect on the flow field. Improved comparisons
would be possible if one uses non-homogeneous inlet turbulence, which is not easy to control
in the digital filter framework [112] used for the simulations of this study. Instead, matching
u′ value at x/D = 0.52 was set as a target, since the turbulence level close to the bluff body is
expected to have stronger influence on the flow field than in farer regions. The overall estimates
of the flame and flow characteristics obtained using this approach is reasonable as one shall see
next.
A. Reaction zones
In Fig. 7.6 typical instantaneous contours of ω˙+ obtained with algebraic and unstrained
flamelets models, are shown on the mid-plane. The combustion is nearly complete in the
recirculation zone, as was observed in experiments [169]. Also, the flame front is much more
wrinkled and corrugated than for 2% TI case shown in Fig. 7.2a, because of the increased TI at
the inlet. This alters the flow dynamics in downstream regions due to the interplay of turbulence,
combustion, shear stresses and pressure gradients, resulting in an overall increase of turbulent
kinetic energy and reduction of the length of the recirculation zone [176], as will be observed
later. Another characteristic observed in Fig. 7.6 is that the flame predicted by the algebraic
model is thicker than that predicted by the unstrained flamelet model. Moreover, the algebraic
model suggests that high values of the reaction rate extend further downstream compared to the
unstrained flamelet model. In both cases, the flame is anchored to the bluff-body. However, very
large reaction rate, ω˙+ > 1.3, are in the flame anchoring region for the algebraic model, and
the unstrained flamelet model predicts these regions to be slightly in the downstream location
as observed in Fig. 7.6. The large scale wrinkling of the reaction zones is very different for
these two models as shown by ω˙+ = 0.1 contours in Fig. 7.6. This will reflect in the averages
of flame related quantities to be discussed in section 7.5.
152 Bluff body stabilised flame
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
1.5
2
2.5
3
y/D
x
/
D
 
 
ω˙
+
0
0.5
1
1.5Unstrained Algebraic
Fig. 7.6 Contours of filtered normalised reaction rate ω˙+ = ω˙δth/(ρusL) at an arbitrary chosen time.
The results are on the mid-plane for unstrained flamelet and algebraic models with 24% TI. Isolines
having ω˙+ = 0.1 are also shown.
B. Axial variations
The centreline variations of normalised mean axial velocity and temperature are shown in
Fig. 7.7 for the case with 24% turbulence intensity. The recirculation zone length is over-
estimated by ≈ 12% by both models because of its strong sensitivity to the inlet turbulence,
noted in the earlier part of this section. Other attributes of this comparison are similar to those
observed for the 2% case. The centreline variation of normalised mean temperature is almost
constant throughout the combustor, and it is underestimated by both models as seen in Fig. 7.7b,
This under-estimation was also observed in [9] and it was suggested that fluid dynamic strain
might be responsible for this. Here, it is observed that the SGS variance of temperature has
substantial influence on this underestimate. The normalised averaged temperature is computed
using Eq. (3.18), as noted for the 2% TI case, where the SGS variance is approximated using
the SGS variance of the progress variable in case of unstrained flamelets closure. Using this
approximation an analysis of the influence of SGS variance on temperature can be performed
7.2 Flame and recirculation region 153
0 1 2 3 4
−1
−0.5
0
0.5
1
1.5
2
2.5
x/D
〈U
〉/
U
r
e
f
(a)
0 2 4 60.6
0.8
1  
1.2
x/D
〈T
+
〉
(b)
Fig. 7.7 Centreline variation of (a) ⟨U⟩/Uref and (b) ⟨T+⟩ obtained with algebraic model using
1.8M ( ) grid, and with unstrained flamelet model using 2.2M ( ) grid are compared to ex-
perimental data [176, 178] (symbols) for 24% TI case. The influence of ⟨σ2T,SGS⟩ on ⟨T+⟩ is shown for
the unstrained flamelet ( ) and algebraic ( ) models.
and results of this analysis are discussed first before discussing Fig. 7.7b. Figure 7.8 is a scatter
plot of the Favre averaged progress variable, indicated as ⟨c˜⟩, and the Reynolds averaged values,
indicated as ⟨c⟩. The Reynolds averaged value is obtained using Eq. (3.18). If the SGS variance
is excluded while calculating ⟨c⟩ then the difference between ⟨c˜⟩ and ⟨c⟩ is not large as seen in
Fig. 7.8a. However, a significant difference is seen if one includes the SGS contribution in the
analysis, and thus this contribution cannot be neglected for this flame.
Figure 7.9 shows a scatter plot of the total variance of progress variable obtained as a sum
of resolved and SGS part, σ2c =
(⟨c˜2⟩−⟨c˜⟩2)+ ⟨σ2c,sgs⟩, and ⟨c˜⟩. The total variance increases
along the axial direction until x/D ≈ 3 and then decreases. Nevertheless, the total variance
is very high in the combusting region and close to the maximum possible value, ⟨c˜⟩(1−⟨c˜⟩).
This suggests that the maximum possible variance may be used as a good approximation if the
sub-grid variance is unavailable, as in the case of the algebraic model used here.
The mean temperature obtained including the SGS variance is shown as dash-dotted line in
Fig. 7.7b for both models. A good improvement is seen for x≤ 2D because the temperature
variance was observed to be high in these regions in experiments [168, 169, 178] and a similar
observation is also made, which will be discussed later while discussing Fig. 7.11. Thus, one
cannot ignore the influence of variance, specifically the sub-grid scale variance, ⟨σ2c,sgs⟩. This
information is not available for the algebraic model used in this study and so the maximum
variance, obtained as ⟨c˜⟩(1−⟨c˜⟩), is used to include the effect of SGS variance as suggested by
the results in Fig. 7.9. If this correction is made to ⟨T+⟩ obtained from the algebraic model
then a very good agreement with the experimental data is seen in Fig. 7.7b. Thus, the SGS
variance plays an important role to obtain the measured mean temperatures.
154 Bluff body stabilised flame
0 0.5 10
0.2
0.4
0.6
0.8
1
〈c˜〉
0 0.5 10
0.2
0.4
0.6
0.8
1
〈c˜〉
〈c
〉
(a) (b)
Fig. 7.8 Scatter plots of ⟨c˜⟩ and ⟨c⟩ for unstrained flamelet model with 24% TI. The Reynolds average
is obtained using Eq. (3.18) (a) without and (b) with SGS variance, ⟨σ2c,sgs⟩.
C. Statistics of velocity and temperature
The radial variations of ⟨U⟩/Uref and ⟨Vr⟩/Uref, where Vr is the radial component of velocity,
are shown in Fig. 7.10 for six axial locations inside the recirculation zone having a length of
Lr ≈ 1.1D [176]. The velocity variations are captured well and there is some underestimate at
the end of the recirculation zone, which is consistent with the centreline variation shown in
Fig. 7.7.
The radial profiles of ⟨T+⟩ and
√
⟨σ2T ⟩ are shown in Fig. 7.11. Experimental results for√
⟨σ2T ⟩ are unavailable for x > 0.6D and thus they are not shown in Fig. 7.11. Temperature
data reported for 24% approach turbulence level was obtained using Rayleigh technique and it
was observed not to change very much with turbulence level for x≤ 2D and r ≤ 0.3D [178]. A
similar behaviour is observed in this study only if the SGS temperature variance, ⟨σ2T,sgs⟩, is
included in the analysis. This is shown as dash-dotted line in Fig. 7.11 for both the algebraic
and the unstrained flamelet models. Similar values of mean temperature are observed for the
two closures considered for this study. When the effect of SGS variance is taken into account,
7.3 Post-flame region 155
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
〈c˜〉
〈σ
2 c
〉
 
 
x
D
0
1
2
3
4
5
Fig. 7.9 Scatter plot of mean progress variable and total variance, ⟨σ2c ⟩ (sum of resolved and SGS parts),
for unstrained flamelet model with 24% TI. Colours represent the axial locations, x/D, for the data. The
maximum possible variance, ⟨c˜⟩(1−⟨c˜⟩) is also shown ( ).
clear improvements are seen as in Fig. 7.11, confirming that the SGS contribution cannot
be neglected. Some overestimation is seen in r/D > 0.5 for the algebraic model, because
the maximum variance is used. This can be observed in the radial variation of normalised
temperature rms, shown in Fig. 7.11. The solid lines denote the resolved part and the dash-
dotted lines include the SGS part as noted in Eq. (7.1). The maximum possible variance used for
the algebraic model does not vanish at r/D = 0 because ⟨T˜+⟩ predicted by the algebraic model
at this position is slightly smaller than unity. The rms values computed from the unstrained
flamelet model are reasonable for x = 0.1D and some overestimates are seen for x = 0.3D and
x = 0.6D in the region 0.2D≤ r≤ 0.5D. In the shear region, 0.4≤ r/D≤ 0.6, the temperature
rms values are underestimated substantially when the contribution from the SGS part is ignored.
Overall, the various comparisons shown in this figure are acceptable.
7.3 Post-flame region
At the end of the recirculation zone the combustion is nearly complete [169] and for this reason
the region downstream of the recirculation zone is referred as ‘post-flame’ region. The flame
156 Bluff body stabilised flame
−1 0 1
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
−1 0 1
0.3
−1 0 1
0.6
−1 0 1
0.8
−1 0 1
1.0
−1 0 1
1.2
−0.5 0   0.5 
0
0.2
0.4
0.6
0.8
r
/
D
0  0.5 0  0.5 0  0.5 0.20.4 0  0.5
〈U 〉/Uref
〈Vr〉/Uref
Fig. 7.10 Radial variation of ⟨U⟩/Uref and ⟨Vr⟩/Uref, obtained with algebraic model using 1.8M ( ) grid
and with unstrained flamelets model using 2.2M ( ) grid are compared with measurements [176, 178]
(symbols) for 24% TI within the recirculation region.
front in this region the flame front has been thickened by turbulent eddies and the combustion
condition lies in the distributed reaction zones of the premixed regime diagram [176, 178]. LES
results of mean radial profiles of velocities are compared with experimental data in Fig. 7.12,
which are available up to x = 2D for these quantities [176]. Some underestimation is seen
for the axial velocity obtained from the unstrained flamelet model and it is related to a small
overestimation of the recirculation zone length (see Fig. 7.7a). The algebraic model is observed
to overestimate this velocity and this overestimate is expected to increase further in downstream
regions, as suggested by the centreline variation in Fig. 7.7a. However, the computed radial
velocity are observed to agree very well with measured values for both models.
The computed radial profiles of ⟨T+⟩ and
√
⟨σ2T ⟩ are compared with experimental re-
sults [168, 169] in Fig. 7.13. Experimental data for
√
⟨σ2T ⟩ are available only for x = 6D
location. A close agreement with the experimental data when the SGS variance is included in
the analysis suggests the importance of this variance, specifically for locations x= 1.5D and 2D.
This is because the level of fluctuation increases until x≈ 3D and then decreases, as discussed
7.4 Implication of local Damköhler and Karlovitz numbers 157
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
0.5 1  1.5
0.3
0.5 1  1.5
0.6
0.5 1  1.5
0.8
0.5 1  1.5
1.0
0.5 1  1.5
1.2
0.2 0.4 0.6
r
/
D
0  0.2 0.4 0.6
0
0.2
0.4
0.6
0.8
r
/
D
0.2 0.4 0.6
〈T˜+〉
〈T+〉
〈T+〉
√〈σ2T 〉
√
〈σ2T , res〉
√〈σ2T 〉
Fig. 7.11 Radial variation of normalised averaged temperature and its rms, obtained using the algebraic
model with 1.8M ( ) grid and unstrained flamelet model with 2.2M ( ) grid are compared with
measurements [168, 169, 176, 178] (symbols) for 24% TI case within the recirculation region. The
influence of ⟨σ2T,sgs⟩ on ⟨T+⟩ is shown for the unstrained flamelet ( ) and algebraic ( ) models.
in Fig. 7.9. The predictions from both models are very good despite the distributed reaction
zones regime combustion suggested in the experimental study [168, 169]. The total variance
calculated using the unstrained flamelet model is good for x = 6D location, despite some over-
estimation for 0.4≤ r/D≤ 0.6. The maximum variance approximation used for the algebraic
model yields an overestimate as seen in Fig. 7.13, especially close to the centreline. This is
because of the following reason. As ⟨T+⟩ approaches unity, the maximum variance is expected
to approach zero. An error of 1% in ⟨T+⟩ would produce
√
⟨σ2T ⟩ ≈
√⟨T+⟩(1−⟨T+⟩)≈ 0.1,
a 10% error in the rms value. This must be kept in mind while studying these results.
7.4 Implication of local Damköhler and Karlovitz numbers
Local SGS Damköhler, Da∆, and Karlovitz, Ka∆, numbers can be used to individuate the local
combustion regime at the SGS level. In Fig. 7.14, PDFs of Da∆ and Ka∆ are shown. Only points
158 Bluff body stabilised flame
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
0.5 1  1.5
1.7
0.5 1  1.5
2.0
−0.5 0   0.5 
0
0.2
0.4
0.6
0.8
r
/
D
0  0.5 0  0.5
〈U 〉/Uref
〈Vr〉/Uref
Fig. 7.12 Radial variation of ⟨U⟩/Uref and ⟨Vr⟩/Uref are compared with measurements [176, 178] for
24% TI in the post-flame region. The legend is as in Fig. 7.10.
where u′∆ > 0 and ω˙ > 0 have been used to compute the PDFs to be physically meaningful
because Da∆ and Ka∆ are undefined outside the flame. PDFs obtained using the unstrained and
algebraic model are similar to each other, and the effect of increasing TI is of narrowing the
PDF width towards lower Damköhler numbers and higher Karlovitz numbers. Representative
values suggested by this distributions are Da∆ ≈ 0.1 and Ka∆ ≈ 35 for 24% TI case. Since
LES involves the modelling of the SGS, these global values may affect model performance
rather than Da and Ka at the integral scale, which would partly explain why the flamelet models
used in this study remain accurate in the distributed reaction zones regime. In case of the
unstrained flamelet model, local values of c˜ and its SGS variance, σ2c,sgs, imply a certain shape
of the β -PDF used to compute ω˙ , which in turn affects the statistical behaviour at large scales.
Depending on local values of c˜ and σ2c,sgs, the β -PDF can assume a shape close to bimodal or
Gaussian distribution. However, the assumption of high Damköhler number in the algebraic
model implies that the statistical behaviour at SGS level implied by this model is bimodal.
The statistical behaviour in the distributed reaction zones regime is far from being bimodal,
and this seems to set limits to the algebraic model. On the other hand, predictions shown
in section 7.3 using the algebraic model are reasonable. In order to explain this, a statistical
7.4 Implication of local Damköhler and Karlovitz numbers 159
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
0.5 1  1.5
2.0
0.5 1  1.5
6.0
0  0.2 0.4 0.6
0  
0.2
0.4
0.6
0.8
〈T+〉
〈T+〉
〈T˜+〉
√〈σ2T 〉
√〈σ2T 〉
√
〈σ2T , res〉
Fig. 7.13 Radial variation of mean normalised temperature and its rms are compared with measure-
ments [168, 169] for 24% TI in the post-flame region. The legend is as in Fig. 7.11.
analysis of c˜ has been done and its results are shown in Fig. 7.15. The PDFs in this figure have
been computed using the algebraic model results at two axial locations with different variance,
one within and one downstream of the recirculation region. Three radial positions are shown
within the shear layer in order to span the flame brush from ⟨c˜⟩ ≈ 0.2 to ⟨c˜⟩ ≈ 0.8. β -PDFs
computed using ⟨c˜⟩ and its resolved variance are also shown. These PDFs match very well
those obtained from LES data and are representative of the large scales. The SGS PDFs are
also shown and they are computed as β -functions using the values of ⟨c˜⟩ and its mean SGS
variance, ⟨σ2c,sgs⟩. The SGS variance is estimated using results from the unstrained flamelet
model, since this quantity is not computed in the algebraic model framework. The SGS PDFs
are close to bimodal distribution because the SGS variance is high. Thus, this plots suggest
that non-bimodal behaviours at resolved scales can be captured using the algebraic model as
long as the SGS statistical behaviour of the flame under investigation is bimodal, i.e. the SGS
variance is high. However, this condition verified for the flame configuration studied in this
work may not work for configurations having different values of Da∆ and Ka∆. Moreover, the
velocity field estimated using the algebraic model becomes inaccurate for x> 2D, as observed
in Fig. 7.7a, suggesting some limitation of the algebraic model. The unstrained flamelet model
160 Bluff body stabilised flame
−2 −1 0 1 2
0
0.2
0.4
0.6
0.8
1
log10 Da∆
P
(l
o
g 1
0
D
a
∆
)
(a)
−2 −1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
log10 Ka∆
P
(l
o
g 1
0
K
a
∆
)
(b)
Fig. 7.14 PDF of (a) SGS Damköhler, Da∆, and (b) Karlovitz, Ka∆, numbers in the combusting region
are shown for the unstrained flamelet model on 2.2M grid with 2% ( ) and 24% ( ) TI, and algebraic
model on 1.8M grid with 24% TI ( ).
gives good predictions for all quantities investigated and it is more robust from this point of
view.
7.5 Species comparisons
Species mass fractions are provided within the experimental dataset [168, 169]. LES statistics
are compared with measurements for CH4, O2, CO2, H2O, H2, CO and OH. Spontaneous
Raman scattering was used to measure the first six, while Laser Induced Fluorescence (LIF)
was used to measure OH [169]. Species were found at adiabatic equilibrium values in the
exhaust at x/D = 6. Super-equilibrium values were observed for OH and CO in the shear layer
region.
Based on the analysis in section 7.2.2, SGS variance must be taken into account when
comparing mean values. Reynolds averaged mass fraction of species i can be computed by
accessing the pre-filtered table a posteriori with local values of ⟨c⟩ in case of the algebraic
model, and as
⟨Y i⟩=
1∫
0
Yi P(c;⟨c⟩,⟨c′2⟩) dc (7.2)
in case of the unstrained flamelet model, where P(c;c,c′2) is the β -PDF of c, parameterised on
the Reynolds averaged quantities ⟨c⟩ and ⟨c′2⟩. The first is obtained using Eq. (3.18), where
7.5 Species comparisons 161
2
4
6
P
(c˜
)
r/D = 0.4 r/D = 0.5 r/D = 0.6
0  0.2 0.4 0.6 0.8 1  
0
2
4
6
c˜
P
(c˜
)
0.2 0.4 0.6 0.8 1  
c˜
0.2 0.4 0.6 0.8 1  
c˜
0.64
0.048
0.100
0.22
0.038
0.044
0.31
0.033
0.125
0.53
0.045
0.145
0.72
0.031
0.131
〈c˜〉 = 0.83
〈σ2c , res〉 = 0.021
〈σ2c , sgs〉 = 0.097
Fig. 7.15 PDFs of c˜ from LES using algebraic model on 1.8M grid ( ) at x/D = 0.6 (top) and
x/D = 2.0 (bottom) and for three radial positions along the shear are compared with β -PDFs computed
using ⟨c˜⟩ and resolved variance ⟨σ2c,res⟩= ⟨c˜2⟩−⟨c˜⟩2 ( ), and using ⟨c˜⟩ and mean SGS variance, ⟨σ2c,sgs⟩
( ).
the total variance is substituted by the maximum possible variance in case of algebraic model
for the reason explained in section 7.2.2. The Reynolds variance, ⟨c′2⟩ = (⟨c2⟩−⟨c⟩2), is
computed as [122]
⟨c′2⟩= ⟨c˜⟩
2+σ2c
1+ τ⟨c˜⟩
[
1+ τ(⟨c˜⟩2(1−⟨c˜⟩)+σ2c (2−⟨c˜⟩)
σ2c + ⟨c˜⟩(1−⟨c˜⟩)
]
−⟨c⟩2 (7.3)
Mean quantities have to be used as the resolved variance must also be taken into account in
Eq. (7.3), and it is not defined for filtered quantities. An additional analysis has been done
using Eq. (3.18) with mean SGS variance instead of total variance, and comparing the results
with those obtained by applying Eq. (3.18) to the filtered quantities before averaging. The
differences in the results obtained using these two ways of computing the species statistics are
small (not shown), and thus computing species as a post-processing step is acceptable.
Using the property ρP(c) = ρP˜(c) [122], ⟨Y i⟩ can be also computed, in case of the un-
strained flamelet model, as
⟨Y i⟩
⟨ρ⟩ =
1∫
0
Yi
ρ
P˜(c;⟨c˜⟩,⟨σ2c,sgs⟩) dc (7.4)
162 Bluff body stabilised flame
A comparison between Eqs. (7.2), (7.4) and experimental data [168, 169] is shown for CH4
mass fraction in Fig. 7.16. The relative error increases in the radial direction, as the level of
0 2 4
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
2 4
0.3
2 4
0.6
2 4
0.8
2 4
1.0
2 4
1.2
〈YCH4〉%
0 1 2
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
1 2
〈YCH4〉%
2.0
1 2
6.0
Fig. 7.16 Radial profiles of CH4 mass fraction, obtained using the algebraic model with 1.8M grid ( )
and the unstrained flamelet model on 2.2M grid using Eq. (7.2) ( ), are compared with experimental
data [168, 169] (symbols) for (a) six axial locations within the recirculation region and (b) three locations
in the post-flame region. Results obtained using Eq. (7.4) ( ) are also shown.
fluctuation is also increasing in this direction. Since YCH4 is directly related to c by its definition,
⟨Y CH4⟩ can be also computed directly from ⟨c⟩ as
⟨Y CH4⟩= (1−⟨c⟩)Y uCH4 (7.5)
7.5 Species comparisons 163
where Y uCH4 is the value of YCH4 in unburnt mixture. Profiles obtained using Eq. (7.5) have
been found to be very similar to those obtained using Eq. (7.2) and thus they are not shown.
Moreover, using Eq. (7.2) is more consistent with the definition used for algebraic model, and
is therefore preferred for comparison purposes.
The algebraic and unstrained flamelet model predictions of YCH4 are similar and they match
well the experimental data in both recirculation and post-flame region. Other major species
predictions are shown in Figs. 7.17 to 7.19. Comparisons with experimental results are very
good using both models and for both recirculation and post-flame region, with the unstrained
flamelet model giving some better predictions. The equilibrium value at x/D = 6 is also very
well captured by both the models. Some underestimation is seen for CO2 for both models.
This is not unexpected, as similar underestimations were seen for this quantity on a different
configuration using either RANS [122] or LES in Chapter 5. Thus, this underestimate seems to
be related to the flamelet framework and an additional insight is given in section 7.5.1. Overall
the agreement is very good.
Radial profiles of minor species mass fractions, obtained using algebraic and unstrained
flamelets models, are compared with experimental data [168, 169] in Figs. 7.20-7.22. Error
bars are shown when the measure error is larger than the symbol size. Prediction of CO
from unstrained flamelet model is reasonable and improves moving in the axial and radial
direction, where the level of fluctuations increases. A significant overestimation is seen for
x ≤ 0.6D, where the regime is between flamelet regime and thin reaction zones regime. A
similar overestimation was seen in both RANS [122] and LES in Chapter 6 using the same
flamelet framework for a piloted flame lying in the thin reaction zones regime. This may
suggest that the unstrained flamelet model predictions of CO improve with high level of SGS
variance. The equilibrium value at x = 6D is also overestimated. H2 mass fraction prediction
from the unstrained flamelet model improves moving in the axial direction up to x = 2D, while
the equilibrium value is under-predicted at x = 6D, where the level of fluctuation has decreased
(see Fig. 7.9). The algebraic model results for CO and H2 are similar to those of the unstrained
flamelet model at x = 0.1D, but large overestimates are seen overall when moving downstream,
suggesting some limitation in the assumptions of high Damköhler number.
Predictions of OH mass fraction are reasonable for x/D≤ 0.8≤ 2, but large inaccuracies
are seen from both models outside this region and close to the centreline. The value of YOH at
x = 0.1D and r/D = 0 is severely overestimated despite the flow contains fully burnt mixture.
This inaccuracy is related to the choice of methane as progress variable, as will be discussed
below.
164 Bluff body stabilised flame
0 10 20
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
0 10 20
0.3
0 10 20
0.6
0 10 20
0.8
0 10 20
1.0
0 10 20
1.2
〈YO2〉%
0 5 10 15 20
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
5 10 15 20
〈YO2〉%
2.0
5 10 15
6.0
Fig. 7.17 Radial profiles of O2 mass fraction are compared with experimental data [168, 169] for six
axial locations within the recirculation region (top) and three locations in the post-flame region (bottom).
The legend is as in Fig. 7.16.
7.5 Species comparisons 165
0 3 6 9 12
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
3 6 9 12
0.3
3 6 9 12
0.6
3 6 9 12
0.8
3 6 9 12
1.0
3 6 9 12
1.2
〈YCO2〉%
0 5 10
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
5 10
〈YCO2〉%
2.0
5 10
6.0
Fig. 7.18 Radial profiles of CO2 mass fraction are compared with experimental data [168, 169] for six
axial locations within the recirculation region (top) and three locations in the post-flame region (bottom).
The legend is as in Fig. 7.16.
166 Bluff body stabilised flame
0 2 4 6 8
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
2 4 6 8
0.3
2 4 6 8
0.6
2 4 6 8
0.8
2 4 6 8
1.0
2 4 6 8
1.2
〈YH2O〉%
0 2 4 6 8
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
2 4 6 8
〈YH2O〉%
2.0
2 4 6 8
6.0
Fig. 7.19 Radial profiles of H2O mass fraction are compared with experimental data [168, 169] for six
axial locations within the recirculation region (top) and three locations in the post-flame region (bottom).
The legend is as in Fig. 7.16.
7.5 Species comparisons 167
0 5 10
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
5 10
0.3
5 10
0.6
5 10
0.8
5 10
1.0
5 10
1.2
10 · 〈YCO〉%
0 5 10
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
5 10
10 · 〈YCO〉%
2.0
5 10
6.0
Fig. 7.20 Radial profiles of CO mass fraction are compared with experimental data [168, 169] for six
axial locations within the recirculation region (top) and three locations in the post-flame region (bottom).
The legend is as in Fig. 7.16.
168 Bluff body stabilised flame
0 1 2
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
1 2
0.3
1 2
0.6
1 2
0.8
1 2
1.0
1 2
1.2
100 · 〈YH2〉%
1 2
100 · 〈YH2〉%
2.0
1 2
6.0
0 1 2
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
Fig. 7.21 Radial profiles of H2 mass fraction are compared with experimental data [168, 169] for six
axial locations within the recirculation region (top) and three locations in the post-flame region (bottom).
The legend is as in Fig. 7.16.
7.5 Species comparisons 169
0 5 10
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
5 10
0.3
5 10
0.6
5 10
0.8
5 10
1.0
5 10
1.2
100 · 〈YOH〉%
0 5 10
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 1.5
5 10
100 · 〈YOH〉%
2.0
5 10
6.0
Fig. 7.22 Radial profiles of OH mass fraction are compared with experimental data [168, 169] for six
axial locations within the recirculation region (top) and three locations in the post-flame region (bottom).
The legend is as in Fig. 7.16.
170 Bluff body stabilised flame
7.5.1 Sensitivity to choice of progress variable
Strong gradients of species mass fractions can be found in laminar flamelets depending on the
particular choice of the progress variable. To exemplify this situation, variation of major and
minor species from laminar flamelets for three different definitions of the progress variable
are shown in Fig. 7.23. These progress variable are c = 1−YCH4/Y uCH4 , used for this study,
c=YH2O/Y
b
H2O and c= (YCO2 +YCO)/(Y
b
CO2 +Y
b
CO), where superscripts u and b refer to unburnt
and burnt conditions respectively. Gradients of YCO, YCO2 and YOH are very strong close to c= 1
when the methane-based progress variable is used. An enlargement of the region 0.999≤ c≤ 1
H2O based (CO2+ CO) based
0.2 0.4 0.6 0.8 1  
0
0.5
1
1.5
2
c
Y
i%
0.2 0.4 0.6 0.8 1  
c
0.2 0.4 0.6 0.8 1  
c
0 
5 
10
15
20
25
CH4 based
Y
i%
CH4
H2O
100 · H2
O2
CO
CO2
10 · OH
Fig. 7.23 Mass fractions variation of major and minor species using different progress variable definitions
for a stoichiometric methane-air flamelet.
is shown for OH in Fig. 7.24, using the three definitions of progress variable described earlier.
The pre-integrated table used for species computation in the unstrained flamelet model uses
1000 points to discretise c and linear interpolation is performed between two values of c. When
σ2c,sgs → 0 in a LES, c˜→ c and the flamelet behaviour is laminar. An error of 0.01% on c will
produce an error of 90% on YOH. The value of ⟨c⟩ at x/D = 0.1 and r/D = 0 is ⟨c⟩ ≈ 0.9998,
therefore an error of 0.02% is present and this explains why OH mass fraction does not vanish
for r → 0 close to the bluff body in Fig. 7.22. Similar considerations can be done for CO and
CO2 for locations where ⟨c⟩ → 1.
7.5 Species comparisons 171
0.999 0.9995 1
0
0.5
1
1.5
c
1
0
·Y
O
H
%
 
 
Fig. 7.24 Enlargement of OH mass fraction in Fig. 7.23 on the burnt side using different definitions of c:
c = 1−YCH4/Y uCH4 ( ), c = YH2O/Y bH2O ( ) and c = (YCO2 +YCO)/(Y bCO2 +Y bCO) ( ).
Radial profiles of major and minor species mass fractions, obtained using the unstrained
flamelet model with c based on CH4, are compared with those obtained using definitions of
c based on H2O and (CO2+CO) and with experimental results in Figs. 7.25 to 7.31 for six
axial locations in the combusting region. The computed profiles converge on each other for all
species for r/D≥ 0.6, i.e. when ⟨c⟩ → 0 or ⟨σ2c ⟩ → 0.25. Major species predictions are almost
insensitive to the progress variable choice, apart for CO2 in the region where ⟨c⟩ → 0, because
of the presence of strong gradients in the laminar flamelet. Under-prediction of YCO2 , seen for
methane- and water-based flamelets for 0.6≤ x/D≤ 2, are much smaller when the flamelets is
based on (CO2+CO) instead. Significant improvements are seen in CO predictions when H2O
and (CO2+CO) based flamelets are used, and results are very good, apart some over-prediction
at x = 0.6D. Similar results are obtained using the three definitions of c for H2 mass fraction,
with some better predictions for the methane-based c at x/D = 0.1 and x/D = 6. Also, clear
improvements are seen for OH mass fractions at x/D= 0.1, x/D= 0.6 and x/D= 6 when H2O
and (CO2+CO) based progress variables are used. However, there is a large underestimate for
1≤ x/D≤ 2, although the qualitative behaviour is captured. Equilibrium values at x/D = 6
are well captured for major species using any of the definitions of c, while minor species
seem to be more sensitive to the choice of the progress variable. Overall the matches are very
good irrespective of the combustion regime, suggesting that the unstrained flamelets model
investigated here is robust and accurate.
172 Bluff body stabilised flame
0 2 4
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
2 4
0.6
2 4
1.0
1 2
1.5
1 2
2.0
1 2
6.0
〈YCH4〉%
Fig. 7.25 Radial profiles of CH4 mass fraction, obtained using the unstrained flamelets model with 2.2M
grid using Eq. (7.2) for three different definition of c, c = 1−YCH4/Y uCH4 ( ), c =YH2O/Y bH2O ( ) and
c = (YCO2 +YCO)/(Y
b
CO2 +Y
b
CO) ( ), are compared with experimental data [168, 169] (symbols) for
six axial locations.
7.6 Analysis of βc PDF
The PDF of the combustion parameter βc in Eq. (2.22), computed dynamically, is shown in
Fig. 7.32 for the 24% TI case. The result is shown for both algebraic and unstrained flamelet
closures, and the data is taken from the entire domain obeying the conditions noted in §6.5. The
realisability limit for algebraic model [39], 2/(2Cm−1), is also shown in the figure. Both PDFs
have their maximum for βc ≈ 4. The PDF has a longer tail for the unstrained flamelet closure,
which is related to the different flow dynamics in the region downstream of the recirculation
region, as was shown in Fig. 7.6. Because of this difference, the mean value of βc is ⟨βc⟩ ≈ 4
for the algebraic model, which is close to the maximum probable value, whereas ⟨βc⟩ ≈ 12 for
the unstrained flamelet model. Additional simulations with constant βc have shown that an
optimum value for this parameter is also around 4 for both algebraic and unstrained flamelets
closures. DNS analyses [77] show that βc increases with the heat release parameter, τ , which
is τ ≈ 4.2 for the lean premixed configuration studied here. This is consistent with the LES
analysis on stoichiometric piloted flames in Chapter 5, as an optimum value of βc = 7.5 was
found in that case for τ ≈ 6.4. This analysis is an useful indication of which value to set for βc
when a dynamic formulation is not possible, i.e. in cases of low turbulence, as will be discussed
in the next chapter.
7.6 Analysis of βc PDF 173
0 10 20
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
0 10 20
0.6
0 10 20
1.0
0 10 20
1.5
0 10 20
2.0
0 10 20
6.0
〈YO2〉%
Fig. 7.26 Radial profiles of O2 mass fraction from LES, obtained using three different definitions of the
progress variable, are compared with experimental data [168, 169] for six axial locations. The legend is
as for Fig. 7.25.
0 3 6 9 12
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
3 6 9 12
0.6
3 6 9 12
1.0
3 6 9 12
1.5
3 6 9 12
2.0
3 6 9 12
6.0
〈YCO2〉%
Fig. 7.27 Radial profiles of CO2 mass fraction from LES, obtained using three different definitions
of the progress variable, are compared with experimental data [168, 169] for six axial locations. The
legend is as for Fig. 7.25.
174 Bluff body stabilised flame
0 2 4 6 8
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
2 4 6 8
0.6
2 4 6 8
1.0
2 4 6 8
1.5
2 4 6 8
2.0
2 4 6 8
6.0
〈YH2O〉%
Fig. 7.28 Radial profiles of H2O mass fraction from LES, obtained using three different definitions
of the progress variable, are compared with experimental data [168, 169] for six axial locations. The
legend is as for Fig. 7.25.
0 5 10
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
5 10
0.6
5 10
1.0
5 10
1.5
5 10
2.0
5 10
6.0
10 · 〈YCO〉%
Fig. 7.29 Radial profiles of CO mass fraction from LES, obtained using three different definitions of the
progress variable, are compared with experimental data [168, 169] for six axial locations. The legend is
as for Fig. 7.25.
7.6 Analysis of βc PDF 175
0 1 2
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
1 2
0.6
1 2
1.0
1 2
1.5
1 2
2.0
1 2
6.0
100 · 〈YH2〉%
Fig. 7.30 Radial profiles of H2 mass fraction from LES, obtained using three different definitions of the
progress variable, are compared with experimental data [168, 169] for six axial locations. The legend is
as for Fig. 7.25.
0 5 10
0
0.2
0.4
0.6
0.8
r
/
D
x/D = 0.1
5 10
0.6
5 10
1.0
5 10
1.5
5 10
2.0
5 10
6.0
100 · 〈YOH〉%
Fig. 7.31 Radial profiles of OH mass fraction from LES, obtained using three different definitions of the
progress variable, are compared with experimental data [168, 169] for six axial locations. The legend is
as for Fig. 7.25.
176 Bluff body stabilised flame
0 5 10 15 200  
0.1
0.2
0.3
0.4
0.5
0.6
0.7
βc
P
(β
c
)
2
2Cm−1
Fig. 7.32 PDF of βc for algebraic ( ) and unstrained flamelet ( ) models and 24% TI. The realisability
limit, 2/(2Cm−1), is also marked.
7.7 Summary
The unstrained flamelet closure is found to be reasonably accurate in this configuration which
involves different regimes. The effect of finite rate chemistry is found to be important in the
region of distributed reactions downstream of the recirculation region, where the unstrained
flamelet closures is indeed more accurate than the algebraic closure, which assumes fast
chemistry instead. However, the flames analysed here and in the previous chapters involve
relatively high turbulence. It is interesting to assess the potentiality of the unstrained flamelet
closure in conditions of low turbulence, where effects of thermal dilatation become strong and
may substantially affect the combustion and thus its modelling. This analysis is done in the
next chapter.
Chapter 8
Non piloted jet flame
In this chapter a low-turbulence, lean propane-air flame is simulated. This flame lies in the
corrugated flamelets regime of the combustion diagram. Flames in this regime are likely to
experience turbulence resulting from flame induced effects such as heat release and flame
intermittency because of low levels of background turbulence. The series of experiments
in [72–75] is a good example for this case. Recently, Furukawa et al. [76] reported statistics of
gas velocities and reaction progress variable fields in a non-piloted lean propane-air premixed
Bunsen flame in quiescent ambient environment, and remarked that those statistics and cer-
tain aspects such as “bubble formation” and “bimodal radial velocity” are useful for testing
combustion modelling. Thus, these premixed flames offer additional challenges for numerical
modelling and the sub-grid scale models must cater for the subtleties of turbulence-combustion
interaction physics in the corrugated flamelets regime. It is worth noting that the air entrainment
does not affect the flame in this configuration as one shall see next, which thus operates in a
fully premixed mode.
The aim of this chapter is to shed light on this interaction by simulating the lean propane-air
flame in [76]. Despite the low velocities involved, this flame is prone to instabilities and flow
transition which are challenging for combustion modelling. This provides a good opportunity
to gather insights on the capabilities and limitations of the statistical flamelet model for SGS
combustion in LES of this flame. Since the shear-produced turbulence does not affect this
flame, as one shall see in the results section, the turbulence level experienced by the flame
results mainly from flame oscillation and thermal expansion effects. These are sub-grid scale
(SGS) phenomena in a practical LES and thus they would affect the SGS velocity which needs
to be modelled. It may be debatable whether one needs to include dilatation effect in this
velocity modelling or not as propositions have been made in the past to exclude this effect
in general [47]. The SGS velocity models of Eqs. (3.50), (3.51) and (3.56) are considered to
assess their influence on the premixed combustion, as these models treat the thermal-expansion
178 Non piloted jet flame
(a) (b)
Fig. 8.1 A direct photograph (a) and a schematic (b) of the lean propane-air flame of [76].
effects differently. Models of Eqs. (3.52) and (3.55) are respectively similar to models of
Eqs. (3.51) and (3.50) from this point of view and thus will not be considered in this analysis.
8.1 Experimental case
The experimental dataset of [76] includes a lean and a rich flames of propane-, and methane-air
mixtures and these non-piloted flames belong to the corrugated flamelets regime of the turbulent
premixed combustion regime diagram. These reactant mixtures have the same laminar flame
speed, sL = 0.34 m/s and these flames were observed to be similar except for the differential
diffusion effects in the rich propane flame. So, the lean propane-air flame having an equivalence
ratio of φ = 0.85, having a laminar flame thermal thickness of δth = 0.4 mm, is chosen for this
study to better understand the turbulence-flame interaction in corrugated flamelets and their
modelling implications without differential diffusion effects. The influences of differential
diffusion will be studied in future.
A direct photograph of the lean flame of [76] is shown in Fig. 8.1a. The reactant mixture
issuing from a pipe of diameter D = 26 mm spreads into an open environment as shown
schematically in Fig. 8.1b. The bulk mean velocity based on the mass flow rate at cold
8.2 Numerical details 179
condition is Ub = 4 m/s. The air entrainment caused by the reactant jet creates a shear layer
surrounding the flame as sketched in this figure and this is quite different from the case of
piloted turbulent premixed flames where the flame resides inside the shear layer between
reactant and pilot streams. Since the flame is not in the shear layer for this configuration the
turbulence experienced by the flame comes from the inlet, produced by thermal expansion
effects and flame-front intermittency. This insight is obtained indeed through LES to be
discussed in later sections. Furthermore, the turbulence produced by the shear through the
Kelvin-Helmoltz instability is not expected to affect the flame as one shall see later. A 3D
LDV system having a spherical probe of about 0.2 mm diameter was used to measure three
components of velocity, which were used to deduce various characteristics of the flame. These
measurements along the centreline were taken from h = 10 to 110 mm and along the radius
for r ≤ 15 mm at given height, h, ranging from 30 to 90 mm. These measurement regions did
not include the shear layer or region where the potential core of the jet starts to breakdown
(marked as "Transition" in Fig. 8.1b). In other words, the measurements were focused in regions
where shear generated turbulence effects are weaker. This experiment offers a good challenge
for combustion modelling because the flame induced turbulence and mutual interactions are
important in these flames and thus the combustion and other submodels have to cater for this
physics. This flame is simulated here using LES paradigm which is described next.
8.2 Numerical details
The algebraic closure in Eq. (2.19) will not be used for this flame because the assumption
in this closure involves relatively high Reynolds numbers, which is not met. The unstrained
flamelets model described in §3.3.2 is used along with the numerical procedure of §3.4. The
dilution effect due to air entrainment is accounted by using a transport equation for a passive
marker as described in that section. As noted before, the air entrainment does not affect
the flame structure in this configuration, as will be also shown next, thus additional terms
in Eq. (3.26) accounting for partial premixing effects are not required and the procedure of
§3.4 is adequate. The instantaneous progress variable is c = 1−Yf /Y f ,u, where Yf ,u is the
fuel mass fraction in the unburnt mixture. The sub-grid stresses are closed using a localised
dynamic Smagorinsky model [81, 141] and the SGS scalar fluxes are computed using gradient
hypothesis with dynamic Schmidt number approach [141]. The SGS velocity, u′∆, is modelled
using the models of Eqs. (3.50), (3.51) and (3.56) as these models represent differently the
dilatational effects which are of interest in this study. The dynamic approach [79, 130] for
the scale-dependent parameter βc in Eq. (2.22) cannot be used for the flame investigated here
because of the low turbulence. Thus, a static value is used as detailed in section 8.3.1.
180 Non piloted jet flame
The cylindrical computational domain with Dc× Lc = 1m× 1m is discretised using a
non-uniform grid having 278×506 cells in the radial and axial directions respectively, with 44
cells for the jet diameter, D. This grid results in 5.4M cells in total for the chosen computational
volume. The filter size, based on the numerical cell volume, has a minimum of ∆≈ 1.3δth close
to the jet exit. Another refined grid having about 23M cells, by increasing the number of cells
by about 60% in all three directions, is used to test grid dependency of the flame statistics. The
progress variable is assigned to be 0 and 1 for the reactant jet and entrainment inlet boundaries
respectively. A small velocity of 0.1 m/s and a temperature of 298 K are assigned for the
entrained air at the respective inlet boundary. The inlet jet velocity cannot be prescribed using
the bulk mean, Ub, and centreline, U0 = 4.5m/s, velocities reported by [76] from cold flow
measurements for the reasons discussed below.
The measured centreline variation of axial velocity is shown in Fig. 8.5, to be discussed
in detail later, using symbols. The increase in the axial velocity at around 100 mm is due
to acceleration through the flame tip. The axial velocity at the centreline is about 5.5 m/s at
about one diameter distance from the jet exit implying a strong acceleration over this distance
if it were 4.5 m/s at the jet exit. A careful considerations and further measurements of jet
tube wall temperature suggested that this results from heating of reactant mixture inside the
tube when the flame is present. To capture these influences correctly, a preliminary RANS
simulation using Fluent (a commercial CFD code) is performed for the computational domain
and conditions illustrated in Fig. 8.2. This cylindrical domain is discretised using 31×775 cells
in radial and axial directions respectively. The measured mass flow rate of propane-air mixture
at 298 K is specified at the inlet with flat profiles. A linear temperature profile as shown in the
figure is specified near the exit based on the experimental estimates. This linear profile gives a
temperature gradient of about 3 K/mm. The radial variations of temperature and axial velocity
at the jet exit obtained from this RANS calculation are shown in Fig. 8.3. The temperature at
the jet exit is nonuniform which accelerates the flow. The centreline value of axial velocity is
about 5.4 m/s, which is close 5.5 m/s observed in the experiment at 1D from the jet exit. These
computed profiles are specified as boundary conditions for the LES.
Although a rms value of 0.22 m/s reported by [76] is low at the jet exit, it is important to
include this because the flame experiences this turbulence and does not see the shear generated
turbulence as noted earlier. So, a synthetic turbulence having similar characteristics as in the
experiment is obtained using the digital filter technique [112] and is added to the velocity U
shown in Fig. 8.3a for the jet exit. The values of rms velocities needed for the digital filter
method are taken from the RANS simulation along with the measured length scale.
The grid size is also increased by 1.6 in all directions resulting in 23M cells for the same
computational volume as noted earlier to test grid sensitivity. The CFL number is kept below
8.3 Results 181
Fig. 8.2 Numerical domain for the preliminary RANS.
−0.5 0 0.50
3
6
r/D
〈U
〉[
m
/
s]
(a)
−0.5 0 0.5300
350
400
450
r/D
〈T
〉[
K
]
(b)
Fig. 8.3 Profiles of (a) mean axial velocity and (b) mean temperature at the jet exit, computed using
RANS.
0.3 everywhere in the domain using a constant time step of 30µs and 50µs respectively for
23M and 5.4M grids. The LES are run for 1.6s of real time using 208 cores for 23M and
32 cores for 5.4M grids, of which the later half (0.8s) is used for collecting statistics. These
simulations took about 72h on a wall clock.
8.3 Results
8.3.1 Role of SGS velocity modelling
The SGS velocity will be influenced by dilatation resulting from combustion and thus u′∆ values
obtained using Eqs. (3.50), (3.51) and (3.56) can be significantly different as these models
presume the role of the dilatation differently. The difference in u′∆ gives different ε˜c, see
182 Non piloted jet flame
0 1 2 3 4 5
3.5
4
4.5
5
5.5
6
6.5
7
u′+∆
ε˜
+ c
Fig. 8.4 Variation of ε˜+ = ε˜δth/sL, with u′+∆ = u
′
∆/sL, for ∆/δth = 1.4.
Eq. (2.22), influencing σ2c,sgs. This can influence the filtered reaction rate which can affect the
u′∆. Thus, the combustion model used here has two-way coupling between flame and turbulence.
Figure 8.4 shows the sensitivity of ε˜c, normalised using δth/sL, to u′∆ while keeping other
parameters in Eq. (2.22) to be constant and ∆+ = 1.4 to be a representative value for the flame
regions based on the grid used here. The quantity σ2c,sgs/βc is taken to be unity for the purpose
of this analysis. The variation is almost sublinear for the range shown and a 100% change
in u′+∆ produces only about 20% change in ε˜
+. This implies that the same variation could
be obtained by changing βc by about 20%. Since the incoming turbulence influences u′∆ any
difference in its value obtained synthetically can also influence ε˜+. The sub-grid velocity can
be written as u′∆ =Cu
′∗
∆ with a constant C ̸= 0 and u′∗∆ given by Eqs. (3.50) to (3.56) without
their respective model constants. The ε˜c in Eq. (2.22) can then be expressed as
ε˜c =F f (u′∆,∆)
σ2c,sgs
βc
=F f1(u′∗∆ ,∆) f2(C)
σ2c,sgs
βc
(8.1)
where f (u′∆,∆) = f2(C) f1(u
′∗
∆ ,∆) is found by comparing Eqs. (8.1) and (2.22). The variation of
ε˜c for f2(C) = 1 is shown in Fig. 8.4. From Eq. (8.1) one can define β ′c = βc/ f2(C) which can
be used instead of C and βc and this approximation is plausible because of the quasi-linearity of
ε˜c with u′∆ observed in Fig. 8.4. The value of β
′
c for the various u
′
∆ models is chosen by matching
the computed and measured flame lengths and this length is estimated to be the location of the
peak axial velocity from the jet exit along the centreline for a meaningful comparative analysis.
The values of β ′c obtained using the three u′∆ models given in Eqs. (3.50) to (3.56) are shown in
Table 8.1. The β ′c for Lilly’s model is larger than those for the other two models suggesting that
the SGS velocity estimated by it is also different as one shall see later. The computed axial
velocity along the centreline is compared to the measured values in Fig. 8.5 for the three u′∆
8.3 Results 183
models. Practically there is no difference among the computed values which also agree well
with measurements.
The variation of ε˜+c within the filtered flame is shown in Fig. 8.6 for the three u′∆ models.
The values obtained using Eqs. (3.51) and (3.56) for u′∆ look similar, with the latter exhibiting
more scatter. These large values are due to the effects of intermittency of the interface between
the reacted jet fluid and entrained air at locations closer to the jet exit. The sub-grid SDR for
Lilly’s model, Eq. (3.50), is smaller due to the higher value of β ′c. This can be also interpreted
that this model needs reduced dissipation to achieve the same flame length or this model
overestimates u′∆.
Further insights can be obtained by studying the spatial variation of ⟨u′∗∆ ⟩, which is time
averaged u′∗∆ , and a typical variation is shown in Fig. 8.7 for the mid-plane. Formally, this
quantity is u′∗∆ (see Eq. 8.1), which is normalised using the respective maximum value so that
it varies from 0 to 1 and to make the comparison sensible because of the difference in β ′c. A
significantly larger value and wider region is seen for the shear layer for Colin et al. model
compared to Pope model and it seems to be the case even for non-reacting regions. Although
Colin et al. model in Eq. (3.56) is derived from Eq. (3.51) there is no guarantee that the SGS
velocities predicted with these two models will be the same for non-reacting regions. The
maximum values of u′∗∆ given in the figure occur near the jet exit. The same values inside
the shear layer at x ≈ 150 mm are 0.15, 0.3 and 1.19 m/s for Eqs. (3.50), (3.51) and (3.56)
respectively. The peak value from the model in Eq. (3.56) is nearly 4 times larger compared to
that from Eq. (3.51) for non-reacting regions where similar values are expected.
A second region having non-zero u′∆ is seen in Fig. 8.7 for Eq. (3.51) and it results from
flame induced effects. This flame induced u′∆ inside the flame brush is of similar magnitude
of those produced in the shear layer region. The flame induced SGS velocity is not visible
for Colin et al. model as it is designed to exclude the dilatation effect. Nevertheless, the
dimensional ⟨u′∗∆ ⟩ inside the flame brush obtained using Eqs. (3.51) and (3.56) are found to be
about 0.2 m/s , which is masked by the very large value in non-reacting region.
For Lilly’s model in Eq. (3.50), the value of ⟨u′∗∆ ⟩ in the region of the flame brush is about
0.02, which is an order of magnitude smaller than those of the other two models. Since the
viscosity is computed dynamically, the dilatational part in Lilly’s model coming from the strain
rate is masked by the fact that the Smagorinsky constant goes to zero in quasi-laminar laminar
Table 8.1 Values of β ′c obtained using the SGS velocity models in Eqs. (3.50), (3.51) and (3.56).
Model Lilly (Eq. 3.50) Pope (Eq. 3.51) Colin et al. (Eq. 3.56)
β ′c 10.0 6.0 6.0
184 Non piloted jet flame
region. For the same reason u′∆ approaches zero even in the shear region close to the jet exit as
seen in Fig. 8.7b. It is worth noting that Lilly’s model was originally derived from the stresses
without involving any dynamic constant.
The above analysis shows the potentials of the various SGS velocity models considered
here. The flame length in this analysis is tuned by selecting an appropriate value for β ′c and this
parameter alters the amount of sub-grid SDR in Eq. (8.1), which in turn alters the value of σ2c,sgs
and thus ω˙ . Since β ′c = βc/ f (C), the role of β ′c can be interpreted as that of controlling the
magnitude of u′∆ to maintain the correct value of ε˜c. The above analysis of u
′
∆ models suggests
that either the dilatation effect should be included in or a different value for c2 is required for
Eq. (3.56). Lilly’s model in Eq. (3.50) is not suitable for the present flame conditions because
the dynamic viscosity used in this model goes to zero as the magnitude of u′∆ goes to zero in
laminar regions irrespective of flame induced effects.
8.3.2 Comparison of statistics
Figure 8.8 shows axial velocity contours, both instantaneous and time averaged, along with a
contour of mean progress variable of ⟨c˜⟩= 0.5. These contours indicate the positions of the
shear layer and flame, and it is seen that these two regions do not overlap. The jet becomes
highly turbulent at x≈ 200 mm but the flame height is only about 100 mm. The acceleration
through the flame tip is visible at x≈ 100mm, which is consistent with the experimental data
as seen in Fig. 8.5.
The computed rms of radial velocity fluctuation along the centreline is compared to the
experimental data in Fig. 8.9 for the three SGS velocity models. The values shown here are
only the resolved part, which is found to be insensitive to the u′∆ model and the computed
values agree well with the experimental data. A small underestimate is seen toward the tip of
the flame and the reason for this will become clear in the next section.
20 40 60 80 100 1204
5
6
7
x [mm]
〈U
〉[
m
/
s]
Fig. 8.5 Comparison of measured (symbols) and computed axial velocity along the centreline;
Eq. 3.50 (dashed line), Eq. 3.51 (continuos line) and Eq. 3.56 (dash-dotted line).
8.3 Results 185
Fig. 8.6 Scatter plot of normalised sub-grid SDR versus progress variable, obtained using models of
Eq. (3.50) (Lilly), Eq. (3.51) (Pope) and Eq. (3.56) (Colin et al.) for u′∆.
x [mm]
r
[m
m
]
 
 
0 50 100 150−50
0
50
Colin et al.
Pope
x [mm]
r
[m
m
]
 
 
0 50 100 150
−50
0
50
0
0.2
0.4
0.6
0.8
1
Lilly
Pope
Fig. 8.7 Contours of ⟨u′∗∆ ⟩/⟨u′∗∆ ⟩max in the mid-plane. The values of ⟨u′∗∆ ⟩max are 0.15, 0.80 and 2.03m/s
respectively for Lilly, Pope and Colin et al. models.
The radial variations of computed mean and rms velocities are compared to the experimental
results in Fig. 8.10. These statistics are also insensitive to the u′∆ model, and ⟨U⟩,
√
⟨u′2⟩, and√
⟨v′2t ⟩ compare quite well with the experimental data. Some under-prediction is seen in both
mean and rms radial velocities close to the jet exit and for 10≤ r ≤ 15 mm, i.e in the flame
region. This under-prediction is related to the dilatation effects and bimodal radial velocity,
which is investigated in detail in the next section.
Reynolds- and Favre-averaged progress variable statistics are also available from the
experiments by deducing the progress variable using bimodal velocity data [76]. This approach
could not be used for the LES because the flame is broadened at least to size of ∆ > δth.
The transported methane-based progress variable is used here. The radial variations of the
Favre-averaged progress variable are compared to those from the experiment in Fig. 8.11. Error
186 Non piloted jet flame
y [mm]
x
[m
m
]
 
 
−50 0 500
100
200
300
400
500
U˜
[m
/
s]
0
1
2
3
4
5
6
(a)
y [mm]
x
[m
m
]
 
 
−50 0 500
100
200
300
400
500
〈U
〉[
m
/
s]
1
2
3
4
5
(b)
Fig. 8.8 Contours of (a) filtered, U˜ , and (b) mean, ⟨U⟩, axial velocity, obtained using Eq. (3.51) for SGS
velocity. The mean progress variable isoline ⟨c⟩= 0.5, is also shown (black line).
bars are shown for the experimental data at x = 40 mm and similar errors can be expected
for other locations. All SGS velocity models predict similar variation with some difference
in the flame region (10 ≤ r ≤ 15 mm). However, the variations are within the error limits.
The Reynolds averaged values are also shown for x = 40 mm and these values are computed
using Eq. (3.18) [122]. Furukawa et al. [76] used an expression similar to Eq. (3.18) while
analysing their experimental data. The significant difference between Favre- and Reynolds-
averaged values observed in the figure implies that the variance of the progress variable in
both experiments and LES is large. Since the flame is quasi-laminar at these locations, this
variance is predominantly produced by combustion, which is at SGS level for the LES. The
radial variations of resolved and SGS variance are shown in Fig. 8.12. The results from 23M
grid will be discussed later in section 8.3.4. The SGS variance is nearly 4 to 5 times the
resolved part for the location close to the jet exit and the resolved part is smaller than SGS
part for all locations shown here. The relative contribution of the resolved part is increasing as
one moves downstream because of a relative increase in turbulence level and flame thickness.
8.3 Results 187
20 40 60 80 100 1200
0.2
0.4
0.6
0.8
x [mm]
√ 〈v
′2 r
〉[m
/
s]
 
 
Fig. 8.9 Comparison of measured and computed centreline variation of radial rms velocity. The legend
is as in Fig. 8.5.
These observations suggest that models for σ2c,sgs deduced from passive scalar scenario would
be inappropriate, as discussed in the order analysis of §3.3.2.
Figure 8.13a compares the flame position deduced from the experiments by [76] using
Favre-averaged progress variable of 0.5 with those obtained in the computations using ⟨c˜⟩= 0.5.
The flame brush width computed as δt = 1/|∂ ⟨c˜⟩/∂ r|max is compared in Fig. 8.13b. These
comparisons are good and the curves marked as 23M will be discussed in section 8.3.4. The
flame brush thickness is slightly thinner for the Lilly and Colin et al models for u′∆, which is
consistent with results in Fig. 8.11. The bubble in the measured flame position, suggested by
the local minimum for r locations at x = 60 in Fig. 8.13a is not captured in the LES even with
a grid of 23M. The reasons for this difference is unclear at this time and will be explored in
future.
In summary, the results suggest that the flame statistics are not influenced by u′∆ modelling if
the parameter βc is chosen carefully to match the measured flame length and the SGS variance
of c is obtained using its transport equation. Thus, the unstrained flamelet model can be used
with careful selection of SGS sub-modelling to capture the relevant physical aspects.
8.3.3 Bimodal of radial velocity
Figure 8.14, taken from [76] and shown here for the sake of convenience, shows that the
bimodal behaviour is seen only for the radial velocity. This bimodal behaviour results from
the flame intermittency, back and forth flame movement in the radial direction, as suggested
by [73]. Since the flame is very thin because of the weak turbulence, a probe in the region of
this flickering would register a fast turnover of cold and hot gases provided the probe size is
small enough for the flame to be completely on either side of it at a given time. It is worth to
note that this behaviour is less likely when the flame is situated inside the shear layer because
the shear-generated turbulence is likely to increase the front thickness smearing the flame
effects. The probe size, which is also the sampling size in the radial direction denoted as
188 Non piloted jet flame
5 10 154
5
6
7
r [mm]
〈U
〉[
m
/
s]
 
 
5 10 150
0.2
0.4
0.6
0.8
r [mm]
√ 〈v
′2 r
〉[m
/
s]
5 10 150
0.2
0.4
0.6
0.8
r [mm]
√ 〈u
′2
〉[m
/
s]
 
 
5 10 150
0.2
0.4
0.6
0.8
r [mm]
√ 〈v
′2 t
〉[m
/
s]
5 10 150
1
2
3
 
 
5 10 150
1
2
3
〈V
r
〉[
m
/
s]
5 10 150
1
2
3
r [mm]
x = 30 mm
50 mm
70 mm
Fig. 8.10 Radial variations of mean and rms velocities in axial and radial directions. The rms of
tangential velocity is also shown. The legend is as in Fig. 8.5.
8.3 Results 189
5 10 150
0.2
0.4
0.6
0.8
1
r [mm]
〈c
〉
 
 
5 10 150
0.2
0.4
0.6
0.8
1
r [mm]
〈c
〉
5 10 150
0.2
0.4
0.6
0.8
1
r [mm]
〈c
〉
5 10 150
0.2
0.4
0.6
0.8
1
r [mm]
〈c
〉
 
 
70 mm50 mm
Favre
average
40 mm
Reynolds
average
x = 30 mm
Fig. 8.11 Comparison of measured and computed Reynolds- and Favre-averaged progress variable
variation. The legend is as in Fig. 8.5. Error bars are also shown.
δsamp below, is a sphere with a diameter of about 0.2 mm in the experiments of Furukawa et
al. [76]. A strong bimodal behaviour is observed in the experiments for r = 11 mm, where
the probability of finding radial velocity equal to its mean is about one fifth of the maximum
probability (or N/Ns) seen in Fig. 8.14. Let us denote this ratio using a symbol Ψ̂ which is 0.2
for the experimental flame.
Furukawa et al. [76] observed that the tangential rms velocity had a local maximum for
r = 11 mm at x = 50 mm. LES results are shown in Fig. 8.15 only for the radial position of
r ≈ 10.2 mm, as at this position the strongest bimodal behaviour is observed. This position
is slightly different from the experimental location of 11 mm, but within the experimental
uncertainty of about 2 mm shown in Figure 8.13. These results show a weak sensitivity to
the u′∆ modelling. As for the experiments, the bimodal behaviour is seen only for the radial
component and this behaviour is seen to depend on the sampling width, δsamp, used for data
collection in the LES. The bimodality is not seen when the δsamp = 0.25 mm for the LES is
about that for the experiments, implying that the sampling point is always within the filtered
flame which is suggested by the most probable radial velocity seen in the top row of Fig. 8.15.
The results for the other two velocity components are similar to those from experiments. The
bimodal behaviour starts to appear when the sampling width is increased gradually so that the
radial movements of the filtered flame could be captured through the collected velocity data.
The minimum sampling width for which bimodality occurs is found for δsamp ≈ 3mm for the
190 Non piloted jet flame
5 10 15 200  
0.1
0.2
r [mm]
〈σ
2 c
〉
 
 
5 10 15 200  
0.1
0.2
r [mm]
〈σ
2 c
〉
5 10 15 200  
0.1
0.2
r [mm]
〈σ
2 c
〉
 
 
5 10 15 200  
0.1
0.2
r [mm]
〈σ
2 c
〉
x = 30 mm
SGS
Resolved
50 mm
70 mm 90 mm
Fig. 8.12 Resolved and SGS variance of the progress variable. The legend is as in Fig. 8.5 and the dotted
line is for 23M grid.
5.4M grid. When the sampling width is increased to 6 mm, which is 15δth whereas it is about
0.5δth for the experiments, then the bimodal behaviour is seen clearly with Ψ̂≈ 0.5, which is
larger than the experimental value of 0.2. This is because, only the resolved velocity is used to
construct the results shown in Figure 8.15. Strictly, one must use instantaneous velocity which
includes the SGS fluctuations and these fluctuations are unavailable in LES framework unless
the grid is refined to a level so that they become negligible. Indeed, one can refine the grid to
have ∆< δth so the filtered flame width become comparable to the flamelet thickness to capture
the bimodal behaviour. The minimum filter width based on the numerical cell volume is about
∆+ = 1.4 for the 5.2M grid and to have a filter width of ∆+ ≈ 0.5, which is comparable to
the experimental probe size, one needs a grid with about 120M cells which would be a very
expensive calculation and thus it is not attempted here. However, another grid with about 23M
cells are used to test the sensitivity of the statistics shown above and this sensitivity is discussed
next.
It is worth to note that the rms of radial velocity fluctuation is observed to have a local
maximum near the sampling location and not the tangential component as noted for the
experiment.
8.3 Results 191
0 5 10 150
20
40
60
80
100
r [mm]
x
[m
m
]
 
 
(a)
0 5 10 150
20
40
60
80
100
δt [mm]
x
[m
m
]
 
 
(b)
Fig. 8.13 The computed isolines of ⟨c˜⟩= 0.5 are compared to the experimental result in (a) along with
the uncertainty for the measurements. A comparison of flame brush thickness is shown in (b). The
legend is as in Fig. 8.5. The dotted line is for 23M grid.
8.3.4 Influence of numerical grid
The first order statistics obtained using 23M grid are very similar to those from the 5.4M grid,
for example see ⟨c˜⟩= 0.5 contour shown in Fig. 8.13, and thus they are not shown here. The
second order statistics show some sensitivity to grid resolution as seen in Fig. 8.16. Since the
filter size decreases with increase in grid resolution the SGS velocity fluctuations are expected
to decrease which is seen in this figure. However, u′∆ values are still high compared to the rms
values for the three velocity components and this is because the flame effects are responsible
for predominant part of the SGS kinetic energy in these regions. The rms of axial velocity
fluctuation for 23M grid compares well with the experimental data, tangential component does
not show any sensitivity, and there is some improvement for the radial component. However,
there is substantial difference between the computed and measured
√
⟨v′2r ⟩ because of the
missing subgrid part which is related to the flame intermittency effects. The probability
distribution of Vr (indeed only the resolved part) does not seem to change between 5.4M
192 Non piloted jet flame
Fig. 8.14 Normalised histograms of [76] for three velocity components at x = 50 mm for various radial
locations.
and 23M grids as shown in Fig. 8.15, suggesting that the statistics influenced by background
turbulence effects are captured quite well and depends weakly on the grid. It is worth to note
that the minimum sampling width at which bimodality in the radial velocity occurs has been
observed to decrease in case of the 23M grid from 3mm to 2.4mm because the filtered flame
width is also smaller in this case, as one would expect.
The SGS variance, σ2c,sgs, does not seem to be sensitive to the grid resolution and the
resolved part increases with grid resolution as one would expect. The difference in the resolved
part also becomes less sensitive to the grid as one moves downstream. All of these points can
be observed in results for 23M grid shown in Fig. 8.12. This suggests that the SGS combustion
process is captured well by the reaction rate closure used for this study. However, the influence
8.3 Results 193
4 6
0
0.02
0.04
−1 0 1
0
0.02
0.04
0 1 2
0
0.02
0.04
4 6
0
0.02
0.04
N
/
N
s
U˜
−1 0 1
0
0.02
0.04
V˜t
0 1 2
0
0.02
0.04
V˜r
δsamp= 0.2 mm
3 mm
Fig. 8.15 Normalised histograms from LES for the three components of velocity at x = 50mm and
r = 10.2mm for various values of δsamp. The legend is as in Fig. 8.5. The result for 23M grid (•••) is
also shown for vr for comparison.
of the flame intermittency and resulting acceleration effects on the SGS velocity fluctuation is
not represented directly by the commonly used SGS velocity models. Thus, the experimentally
observed bimodality of the radial velocity is not captured fully.
8.3.5 Influence of bimodal PDF on statistics
The probability density function (PDF) constructed using the normalised histograms in Figs. 8.14
and 8.15 for the radial velocity is shown in Fig. 8.17 for the experimental and simulated flames.
Although the shapes of measured, in Fig. 8.17a, and computed, the solid line in Fig. 8.17b,
compare quite well there is a substantial difference. The analysis discussed below outlines
reasons for this and tries to identify limitations of the LES and SGS models used in this study.
The bimodal PDF shown in the above figure can be seen as a double-Gaussian PDF for the
given conditional means and variances. The first Gaussian results when the flame is completely
on one side of the sampling region suggesting that the sampling point is in the unburnt mixture.
The second Gaussian arises when the flame is on the other side, i.e. the sampling point lies
completely in the burnt mixture. The required conditional means and variances of the radial
velocity are obtained by collecting samples satisfying the conditions c˜ ≤ 0.05 and c˜ ≥ 0.95
for the two branches respectively. The PDF constructed thus is shown in Fig. 8.17b and it
is worth noting that the velocities collected are only resolved components. The left branch
has a mean of µL = 0.25 and a standard deviation of σL = 0.143, and the right branch has
194 Non piloted jet flame
5 10 150
0.4
0.8
r [mm]
√ 〈u
′2
〉[m
/
s]
5 10 150
0.4
0.8
r [mm]
√ 〈v
′2 t
〉[m
/
s]
5 10 150
0.4
0.8
r [mm]
√ 〈v
′2 r
〉[m
/
s]
5 10 150
0.1
0.2
r [mm]
〈u
′ ∆
〉[
m
/
s]
Fig. 8.16 RMS and SGS velocity profiles for 5.4M (continuous line) and 23M (dotted line) grids are
compared to the experimental data (symbols) at x = 50 mm.
µR = 1.06 and σR = 0.176. These two standard deviations are due to the turbulence effect
(resolved part only) in unburnt and burnt conditions, and σR > σL suggesting that the turbulent
kinetic energy is increased across the flame as one would expect for the corrugated flamelets
regime of premixed combustion. The computed radial velocity rms at the sampling location of
r = 10.2 mm is about 0.33 as seen in Fig. 8.16 and this value is very close to the (σR+σL).
This observation can be further ascertained by collecting sample from a single numerical cell
located at r = 10.2 mm and x = 50 mm, and constructing the PDF which is shown in Fig. 8.18.
This PDF is compared to the a Gaussian model having the same mean as for the sample and a
standard deviation of (σL+σR). The mean value is almost equal to ⟨V˜r⟩ shown in Figure 8.10
for r = 10.2 mm and x = 50 mm.
The computed PDF in Fig. 8.17b suggests that the peak-to-peak velocity difference is about
0.98 m/s which compares well with that of about 1.1 m/s for the experimental data. However,
the whole PDF is shifted to the lower values of the radial velocity in the computations, which
comes from the missing flamelet physics in the SGS velocity models although the acceleration
across the flame brush is captured quite well as seen in Figure 8.10. Reason for the above shift
and the difference in peak-to-peak values is as follows. The measured Vr is the instantaneous
velocity and it would not be easy to reconstruct this from LES because of the missing sub-grid
8.3 Results 195
0 0.5 1 1.5 2 2.50
0.5
1
1.5
Vr [m/s]
P
(V
r
)
r = 11mm
(a)
−0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
V˜r [m/s]
P
(V˜
r
)
r = 10.2mm
(b)
Fig. 8.17 The PDF of (a) measured and (b) computed radial velocity component vr at x = 50 mm. The
modelled values using two Gaussians are shown in (b) for the mean and variance of unburnt (c˜≤ 0.05)
and burnt (c˜≥ 0.95) mixtures in LES data.
−0.5 0 0.5 1 1.5 20
0.5
1
1.5
V˜r [m/s]
P
(V
r
)
r = 10.2mm
Fig. 8.18 The PDF (continuous line) obtained with samples collected from a single numerical cell
(δsamp < ∆) at x = 50 mm and r = 10.2 mm is compared to a Gaussian model with the same mean and
standard deviation of σL+σR (dashed line).
fluctuation and the u′∆ models discussed in section 3.3.2 gives only a statistical measure of this
fluctuation. Thus, one must be cautious while comparing PDFs generated from practical LES to
those from measurements. If one uses very fine grid to make these fluctuations to be negligible
then it is possible to compare the computed and measured PDFs and such LES, indeed it must
be at least a coarse DNS (direct numerical simulation), is very expensive for the present flame
conditions as noted in section 8.3.3.
The difference of 0.12 m/s in the peak-to-peak value is almost equal to the difference
between the measured
√
⟨v′2r ⟩ ≈ 0.46 (see Fig. 8.16) and (σL +σR), where the later is the
resolved part. This can be seen clearly if one consider the joint PDF P(ϕ,ζ ) = P(ϕ|ζ )P(ζ ),
196 Non piloted jet flame
5 10 150
0.2
0.4
0.6
0.8
√ 〈v
′2 r
〉[m
/
s]
r [mm]
5 10 150
0.2
0.4
0.6
0.8
√ 〈v
′2 r
〉[m
/
s]
r [mm]
5.4M 23M
Fig. 8.19 Radial variation of measured (open circle) and computed (lines) rms of radial velocity at
x = 50 mm. Results are shown for 5.4M (left) and 23M (right) grids. The effect of δsamp is shown for
δsamp < ∆ (continuous line), δsamp = 3 mm for 5.4M or 2.4 mm for 23M grids (continuous line with
squares); δsamp = 6 mm for 5.4M or 5 mm for 23M grids (continuous line with triangles). The dashed
lines represent the corresponding curves when the v′r,∆ is included.
where ϕ is the sample space variable for the velocity. If one presumes the marginal PDF to be
BML as has been done by [136] and [29] then
⟨v′2r ⟩=
∫
(ϕr−⟨ϕr⟩)2 P(ϕr|ζ = 0) dϕr +
∫
(ϕr−⟨ϕr⟩)2 P(ϕr|ζ = 1) dϕr
+ γ
∫
(ϕr−⟨ϕr⟩)2 P(ϕr|ζ ) dϕr
=
(
σ2L +σ
2
R
)
+ γ σ2f (8.2)
where γ is the burning mode part of the marginal PDF, P(ζ ) which can be obtained by
integrating the β -PDF used for the reaction rate closure between the limits of ζ = 0.05 and
0.95, but it is not straightforward to get σ f as it is related to the effects of combustion on SGS
velocity fluctuation. Furthermore, this contribution will be highly anisotropic in corrugated
flamelets regime and a way to include this contribution to SGS modelling for a practical LES is
an open question.
The above difference in the rms value results from the missing flame intermittency effects
on the flow field in general for the SGS velocity models and grids used. The influence of
this effect on the resolved part can be seen by increasing the width for the data sampling as
one would expect. This is shown in Fig. 8.19 comparing the measured and computed rms of
radial velocity (compare the solid lines). The sub-grid contribution has to come through u′∆. A
good comparison of
√
⟨u′2⟩ shown in Fig. 8.16 suggests that axial SGS velocity fluctuation
is very small and thus the u′∆ computed using Eq. (3.51) is taken to be from the radial and
tangential fluctuations because the flame can be intermittent in these two directions. However,
8.3 Results 197
their relative contribution is unknown and is not easy to estimate. But, if one presumes, guided
by the experimental result in Fig. 8.14, that v′r,∆ = u
′
∆ then the comparison of computed rms of
radial velocity with experimental data improves further for flame regions. These observations
apply equally for the results shown in Figure 8.19b for the 23M grid also. It seems that the
flame induced anisotropy in the SGS velocity need to be captured as these effects are strong in
corrugated flamelets regime and thus improved modelling strategies are required for the SGS
velocity for practical LES.

Chapter 9
Concluding remarks
9.1 Summary and final considerations
The efficacy of sub-grid scale (SGS) reaction rate closure in the flamelet framework has been
investigated In this work. Different turbulent premixed flame configurations, spanning from the
corrugated flamelets regime to the distributed reaction zones regime of the premixed combustion
diagram [180], have been investigated in order to assess potentials and limitations of flamelets
modelling in the context of Large Eddy Simulation. As result of this investigation, two model
for LES of turbulent lean premixed combustion of industrial relevance have been identified and
analysed. These models involve algebraic closure and presumed PDF based closure, and they
need a closure for the scalar dissipation rate of a progress variable, which is modelled with a
newly developed expression [61]. The sensitivity to numerical and physical parameters has
also been investigated. These models have been tested in the context of LES for the first time.
All simulations have been performed using Precise-MB, which requires multi-block struc-
tured grids. The scalar dissipation rate (SDR) based flamelets approaches investigated are
shown to work well on relatively smaller grid sizes and with low computational cost.
The first investigation in this study has been on the efficacy of the SDR-based algebraic
model for the SGS reaction rate closure in the light of recent developments on the modelling of
filtered scalar dissipation rate of the reaction progress variable. The closure for the SGS scalar
dissipation rate needs a model for the SGS velocity scale, u′∆. Four different models for this
velocity scale have been studied and evaluated. These models are based on the shear stress
related closure of Lilly [140], scale-similarity for velocity [196] and kinetic energy [11] and a
hypothesis of constant kinetic energy transfer down to the filter scale, ∆. Results on piloted
Bunsen flames lying in the thin reaction zone regime using these four models for u′∆ suggests a
weak sensitivity of the combustion model to these closures, at least for what concerns first and
second order statistics. The model in Eq. (3.51) based on the scale-similarity of velocity works
200 Concluding remarks
well in general and thus it has been used for detailed analysis for further study. A modelled
transport equation for SGS kinetic energy, ksgs, is also used to obtain u′∆. The additional gain in
computational accuracy is found to be small and this could be because of various ad hoc models
used in ksgs transport equation. Further work to develop appropriate models for the unclosed
terms in ksgs transport equation is needed. On the other hand, results of a quasi-laminar jet
flame belonging to the corrugated flamelets regime show that these u′∆ models do not contain
physics relevant to represent the flame induced effects such as flame intermittency, thermal
expansion effects and the associated anisotropic fluctuating velocity fields.
The algebraic filtered reaction rate closure given in Eqs. (2.19) and (2.22) is assessed to
work well if the filter width normalised by the laminar flame thermal thickness, δth, is large,
which is consistent with past DNS studies [61, 77]. This assessment is made by using two
numerical grids having 1.5M and 4.2M cells on a piloted Bunsen flame configuration, and it
is further verified on a bluff-body configuration. This condition is barely met for the 4.2M
grid in the piloted Bunsen flames, thus the difference in the reacting flow results is because
of underestimate of the fuel consumption rate when ∆+ < 1 and this observation from LES is
consistent with a priori analysis using DNS data [130]. This underestimate leads to lower peak
temperature and product scalar mass fractions in a consistent manner. The results obtained
using 1.5M grid compare quite well with the measurements and earlier RANS and LES results,
and similar observation are done for the bluff body configuration when ∆ is kept larger than
the flame thickness. Such limit is not observed in case of unstrained flamelets model. These
two models behave similarly well when the Karlovitz number, Ka, is relatively small, while
effect of finite chemistry are evident when the Karlovitz number increases, which is the case
in the buff body configuration. In this case both algebraic and unstrained flamelet closures
have been observed to predict the recirculation zone and the statistics well when the local
Karlovitz number is relatively small. Some statistics obtained using algebraic closure in regions
downstream of the rear stagnation point of the recirculation zone do not compare well with
experimental data because the high Damköhler number assumption used for this model may
not hold, while the unstrained flamelet model do not show such sensitivity and is observed
to work reasonably well, suggesting that the effect of finite rate chemistry is to be taken into
account in these conditions. An additional analysis in this study has indicated that the local
SGS Damköhler and Karlovitz numbers, rather than their integral values, may be in control of
the model performance in a LES, and thus the correct statistical behaviour can be predicted
if the combustion model is consistent with the physics at SGS level, which may be different
to that at larger scales. This gives additional insights on flamelets capabilities in light of the
recent advancement in the literature.
9.1 Summary and final considerations 201
A weak sensitivity to the grid resolution has been observed for the unstrained flamelets
model for all the flames analysed, at least when the grid satisfies the 80% turbulent kinetic
energy criteria [196]. Other analysis show that the unstrained flamelets model is also insensitive
to the choice of the progress variable, but strong gradients in the laminar flamelets may affect
substantially minor species predictions when the variance is low on the burnt side. A substantial
difference is observed between the results obtained using the unstrained and strained flamelet
closures. These two closures are compared on the piloted flame configuration. Since the
numerical grids employed for the simulations resolve most of the turbulent kinetic energy
the SGS kinetic energy is expected to be small implying that the SGS eddies may be weak
to impart any stretching effects on the flame front. This is similar to the views expressed by
Poinsot et al. [189] while analysing their DNS data to investigate turbulent combustion regimes.
Thus, using the SGS dissipation rate or any parameter of strained flamelets established in
counterflow geometries to include the effects of SGS stretching is likely to overcompensate the
SGS stretch effects compared to the influences of resolved dynamic eddies on the unstrained
flamelet. Hence, the burning rate is underestimated by the strained flamelets leading to over
prediction of fuel mass fraction and under-estimates of various reactive scalar mass fractions,
suggesting that a dependence on the filter width should be included in the modelling of the
SGS strain. The influences of resolved dynamic scales are inherently included in the unstrained
flamelet closure through the various transport equations and their interactions. In particular, an
order of magnitude analysis indicates that simple algebraic models for the progress variable
variance would result in an inaccurate reaction rate. This is also verified by a posterori analysis
in LES of different flames investigated. This central role of the SGS variance implies that
it must be predicted accurately in the unstrained flamelets model and therefore a transport
equation is necessary in this case. When the SGS variance is well predicted, as long as the
most of the strain is at the resolved level, a simple unstrained flamelets model is sufficient to
predict the flow field over different regimes. The importance of the SDR is evident in this
case as it directly controls the SGS variance. Finally, the contribution of SGS variance of
temperature and species has also been found to play a critical role in some configurations and
under certain condition of turbulence, as the SGS level of fluctuation affects the macroscopic
scales. A simple scaling used in this thesis has been found to be effective in including this
influence into the main statistics.
The sensitivity to the model parameter βc is also investigated in both algebraic and un-
strained flamelets model. This parameter is related to the physics of flame front curvature
effects induced by turbulence, chemical reactions, molecular dissipation, and their interactions
at sub-grid level. This parameter was also shown to depend on heat release parameter and
202 Concluding remarks
turbulence Reynolds number in earlier studies [77, 79] and possibly on filter width also because
of the influence of the above physical processes. Thus, a dynamic evaluation of this parameter
is ideal and has been evaluated. A power-law based model has also been considered for the
filtered scalar dissipation rate. The various statistics, specifically, averaged fuel mass fraction
variation in radial direction at a number of axial positions has shown that the power-law based
model is unable to see a flame when the filter size is of order of δth because the scale-similarity
of the progress variable gradient does not hold. For this scale-similarity to hold, one needs
a very fine grid, which is impractical. However, the scale-similarity of SGS velocity and
filtered progress variable, c˜, required for the dynamic evaluation of βc is easy to satisfy on
relatively small grids, and the only limitation is found to be the level of turbulence, which must
be high enough in order to have a range o scales for the dynamic procedure to be effectively
used. The various statistics obtained from LES compare well with the measured values of
these quantities for the cases where this procedure is employed. The difference in various
statistics in respect to results obtained with the empirical (static) evaluation of βc is found
to be small because the empirically determined βc values for the configurations analysed are
similar to the averaged value of βc obtained in the dynamic evaluations, which is verified by
analysing the PDF of βc. The evolution of flame brush thickness with axial distance com-
puted from LES results also compares quite well with measured values where they are available.
Effects of thermal dilatation on the SGS modelling have also been investigated by simu-
lating a non-piloted flame lying in the corrugated flamelets regime. The turbulence-chemistry
interaction in this case is driven by flame induced effects rather than shear-produced turbulence.
The modelling of u′∆ is expected to play a role and three different models treating the dilation
effects differently have been considered for this analysis. The computed statistics compare well
with the experimental results and the grid sensitivity of the statistics is found to be negligible in
this configuration. There is some under-prediction for the statistics of radial velocity which was
observed to be bimodal in experiments. Results suggest that the bimodal behaviour can only be
seen if the sampling width is larger than ∆ for situations with ∆> δth, which is typical for LES,
and thus one would need very large grids to implicitly capture this behaviour in the current
LE framework. If this condition is not met then the PDF is observed to be Gaussian rather
than bimodal, because the filtered flame is thicker than the flamelet and will not move back
and forth adequately about the sampling point to give bimodal behaviour. Nevertheless, this
flame broadening due to filtering does not unduly influence the first order statistics, indicating
that these statistics are not a good indicator of the appropriateness of a LES. This analysis
suggests that the commonly used u′∆ models are to be improved to handle SGS physics and this
is specifically so for the corrugated flamelets regime because of the dominant effect of thermal
9.2 Future work 203
dilatation in this case.
In conclusion, the algebraic and unstrained flamelets models are found to be simple, accurate,
computationally efficient and robust over a range a different conditions and configurations.
The key role of the SDR has been demonstrated as it controls directly the reaction rate in
the algebraic model, and it controls the variance of the SGS progress variable variance in
the unstrained flamelets model, and this quantity has been shown to be fundamental for the
modelling. Since finite rate chemistry can be taken into account in the unstrained flamelets,
this model represents a valid choice in industrial and academic environments.
9.2 Future work
• The models developed in this thesis can be applied on a real combustor geometry in
order to assess their relative performance on practical configurations; the success of this
implementation will give additional useful information on the advantages and limitations
of the flamelets modelling;
• The SGS kinetic energy transport equation can be employed to further investigate on the
effects of dilation on the flow field. This investigation is conditional to the finding of
appropriate models for the thermal dilation, which possibly account for anisotropy and
are regime-adaptive. This will need investigations using both DNS and LES on different
configurations.
• Despite the β -PDF has been shown to work reasonably well, alternative choices of
the SGS PDFs can be used and tested for the pre-integration of the reaction rate, like
the modified laminar flamelet PDF, in order to assess the relative model sensitivity.
According to the literature, this type of PDFs may better represent bimodal behaviours at
SGS level;
• An analysis on the interplay between turbulence, strain, heat release and pressure dilation
and their influence on the size of the recirculation zone on bluff body configurationss
would be beneficial for the understanding of this mode of combustion. Ideally, a unique
scaling for both cold and reacting flow exists and can be found with this analysis;
• A significant contribution of rms temperature is lost at sub-grid scales in LES. Approxi-
mating this contribution using rms of progress variable is not always possible because
the Lewis number is not always unity or may change in the flame, and may lead to
inaccuracies as shown for example in §7.2.1 for Fig. 7.4. Improved models for rms
204 Concluding remarks
temperature which, for example, account for the covariance of temperature and progress
variable are worth an investigation;
• Identify avenues to extend these models for partially premixed combustion of practical
relevance would be worthwhile.
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Appendix A
Database
The most relevant simulations are listed here for each of the cases investigated in this thesis.
This data is freely accessible on-line and available for interested researchers. For informa-
tion about how to get access, please contact the author of this thesis at il246@cam.ac.uk or
ivanlang87@gmail.com.
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rβ
c
In
ac
cu
ra
ci
es
C
A
34
"
√ 2k
sg
s/
3
"
"
"
"
M
od
el
K
2
of
Ta
bl
e
5.
1
W
ea
kl
y
se
ns
iti
ve
223
Ta
bl
e
A
.2
Si
m
ul
at
io
ns
lis
tf
or
th
e
pi
lo
te
d
fla
m
es
ca
se
fr
om
C
he
n
et
al
.[
46
]-
un
st
ra
in
ed
fla
m
el
et
s
m
od
el
(U
FM
).
N
am
e
β c
Sc
T
G
ri
d
si
ze
T p
ilo
t
D
es
cr
ip
tio
n
R
em
ar
k
C
U
1
7.
5
dy
n
1.
5M
19
50
K
Se
e
§6
.4
Se
e
§6
.4
C
U
2
7.
5
"
4.
2M
"
Se
ns
iti
vi
ty
to
gr
id
si
ze
Se
e
§6
.4
C
U
3
6.
0
"
1.
5M
"
Se
ns
iti
vi
ty
to
β c
A
lm
os
ti
ns
en
si
tiv
e
up
to
20
%
va
ria
tio
n;
le
ss
se
ns
iti
ve
th
an
al
-
ge
br
ai
c
m
od
el
C
U
4
6.
0
"
4.
2M
"
"
"
C
U
5
6.
0
"
4.
2M
"
U
si
ng
g
=
σ
2 c,
sg
s/
[c˜
(1
−
c˜)
]
in
-
st
ea
d
of
σ
2 c,
sg
s
to
pa
ra
m
et
ri
se
th
e
lo
ok
-u
p
ta
bl
e
N
um
er
ic
al
is
su
es
an
d
in
ac
cu
ra
-
ci
es
C
U
6
7.
5
"
1.
5M
17
50
Se
ns
iti
vi
ty
to
pi
lo
tt
em
pe
ra
tu
re
Se
ns
iti
ve
cl
os
e
to
je
te
xi
t
C
U
7
7.
5
0.
7
1.
5M
19
50
Se
ns
iti
vi
ty
to
Sc
hm
id
tn
um
be
rs
In
se
ns
iti
ve
C
U
8
"
dy
n
"
19
50
Pr
e-
fil
te
ri
ng
th
e
lo
ok
-u
p
ta
bl
e
be
-
fo
re
in
te
gr
at
io
n
L
ar
ge
in
ac
cu
ra
ic
es
C
U
9
"
dy
n
"
"
Ve
rti
ca
lv
el
oc
ity
(1
m
/
s)
as
si
gn
ed
on
la
te
ra
lb
ou
nd
ar
ie
s
In
se
ns
iti
ve
C
U
10
dy
n
"
"
"
Se
e
§6
.4
Se
e
§6
.4
C
U
11
dy
n
"
4.
2M
"
Se
e
§6
.4
Se
e
§6
.4
224 Database
Ta
bl
e
A
.3
Si
m
ul
at
io
ns
lis
tf
or
th
e
pi
lo
te
d
fla
m
es
ca
se
fr
om
C
he
n
et
al
.[
46
]-
st
ra
in
ed
fla
m
el
et
s
m
od
el
(S
FM
).
N
am
e
β c
G
ri
d
si
ze
σ N
ψ
m
ax
D
es
cr
ip
tio
n
R
em
ar
k
C
S1
dy
n
1.
5M
0.
3
50
0
Se
e
§6
.4
Se
e
§6
.4
C
S2
7.
5
"
0.
3
50
0
Se
ns
iti
vi
ty
to
β c
W
ea
kl
y
se
ns
iti
ve
C
S3
6.
0
"
"
"
Se
ns
iti
vi
ty
to
β c
W
ea
kl
y
se
ns
iti
ve
C
S4
dy
n
"
0.
5
"
Se
ns
iti
vi
ty
to
σ N
D
ec
re
as
in
g
ac
cu
ra
cy
C
S5
dy
n
"
0.
3
70
0
Se
ns
iti
vi
ty
to
ψ
m
ax
In
se
ns
iti
ve
C
S6
"
4.
2M
0.
3
50
0
Se
ns
iti
vi
ty
to
gr
id
si
ze
In
se
ns
iti
ve
ψ
m
ax
is
th
e
up
pe
rb
ou
nd
of
th
e
in
te
gr
at
io
n
do
m
ai
n
in
E
q.
(3
.4
0)
.
σ N
is
th
e
st
an
da
rd
de
vi
at
io
n
of
ln
(ψ
|c)
in
E
q.
(3
.4
1)
.
225
Ta
bl
e
A
.4
Si
m
ul
at
io
ns
lis
tf
or
th
e
bl
uf
fb
od
y
ca
se
[1
68
,1
69
].
N
am
e
M
od
el
β c
G
ri
d
si
ze
S.
T.
W
.R
.
W
.F
.
D
es
cr
ip
tio
n
R
em
ar
k
N
1-
2
A
/U
dy
n
1.
8M
N
o
N
o
Y
es
Se
e
§7
.2
.1
Se
e
§7
.2
.1
N
3-
4
A
/U
"
2.
2M
"
Y
es
N
o
Se
e
§7
.2
.1
Se
e
§7
.2
.1
N
5-
6
A
/U
"
2.
2M
"
Y
es
Y
es
Se
ns
iti
vi
ty
to
w
al
lf
un
ct
io
ns
D
ec
re
as
in
g
pe
rf
or
m
an
ce
be
-
ca
us
e
gr
id
is
w
ith
in
su
bv
is
co
us
la
ye
r
N
7-
8
A
/U
"
7.
6M
"
N
o
Y
es
G
ri
d
se
ns
iti
vi
ty
Si
m
ila
rt
o
1.
8M
gr
id
N
9
U
"
11
.2
M
"
Y
es
N
o
G
ri
d
an
d
W
.R
.s
en
si
tiv
ity
Si
m
ila
rt
o
2.
2
M
gr
id
N
10
A
"
1.
8M
"
N
o
N
o
Se
ns
iti
vi
ty
to
w
al
lf
un
ct
io
ns
D
ec
re
as
ed
ac
cu
ra
cy
N
11
-1
2
A
"
1.
8M
"
N
o
Y
es
U
ai
r
va
ri
ed
fr
om
0.
02
to
2.
2
m
/s
In
se
ns
iti
ve
N
13
A
"
"
"
N
o
Y
es
Se
ns
iti
vi
ty
to
in
le
tv
el
oc
ity
(U
0
=
10
.5
m
/s
)
W
ea
kl
y
se
ns
iti
ve
N
14
-1
5
A
"
"
"
N
o
Y
es
C
ofl
ow
te
m
pe
ra
tu
re
10
00
/1
60
0K
In
se
ns
iti
ve
N
16
-1
7
A
/U
"
"
"
N
o
Y
es
U
si
ng
H
2O
as
pr
og
re
ss
va
ri
ab
le
W
ea
kl
y
se
ns
iti
ve
N
18
-1
9
A
/U
"
2.
2M
"
Y
es
N
o
"
W
ea
kl
y
se
ns
iti
ve
N
20
A
2.
4
1.
8M
"
N
o
Y
es
Se
ns
iti
vi
ty
to
β c
O
ve
r-
pr
ed
ic
tio
n
of
re
ac
tio
n
ra
te
N
21
U
2.
4
2.
2M
"
Y
es
N
o
"
"
N
22
A
dy
n
1.
8M
"
N
o
Y
es
In
cr
ea
se
d
co
m
bu
st
or
le
ng
th
W
ea
kl
y
se
ns
iti
ve
at
x/
D
=
6
N
23
U
dy
n
2.
2M
Y
es
Y
es
N
o
Se
e
§7
.2
.2
Se
e
§7
.2
.2
N
24
A
dy
n
1.
8M
Y
es
N
o
Y
es
Se
e
§7
.2
.2
Se
e
§7
.2
.2
N
25
A
4.
0
1.
8M
"
N
o
Y
es
St
at
ic
β c
fr
om
dy
na
m
ic
an
al
ys
is
Si
m
ila
rt
o
dy
na
m
ic
ev
al
ua
tio
n
N
26
U
4.
0
2.
2M
"
Y
es
N
o
"
"
N
27
A
12
.0
1.
8M
"
N
o
Y
es
Se
ns
iti
vi
ty
to
β c
U
nd
er
-p
re
di
ct
io
n
of
re
ac
tio
n
ra
te
N
28
U
12
.0
2.
2M
"
Y
es
N
o
"
"
N
29
U
dy
n
2.
2M
"
Y
es
N
o
Sc
=
0.
7
in
st
ea
d
of
dy
na
m
ic
fo
r-
m
ul
at
io
n
In
se
ns
iti
ve
N
30
A
dy
n
7.
6M
"
N
o
Y
es
Se
ns
iti
vi
ty
to
gr
id
si
ze
Sm
al
ld
iff
er
en
ce
s
N
31
U
dy
n
11
.2
M
"
Y
es
N
o
"
"
N
32
U
dy
n
2.
2M
"
Y
es
N
o
U
si
ng
H
2O
as
pr
og
re
ss
va
ri
ab
le
Se
e
§7
.5
.1
N
33
U
dy
n
2.
2M
"
Y
es
N
o
U
si
ng
C
O
2
+
C
O
as
pr
og
re
ss
va
ri-
ab
le
Se
e
§7
.5
.1
A
=
al
ge
br
ai
c;
U
=
U
ns
tr
ai
ne
d;
S.
T.
=
Sy
nt
he
tic
tu
rb
ul
en
ce
;
W
.R
.=
W
al
lr
efi
ne
m
en
t;
W
.F
.=
W
al
lf
un
ct
io
ns
.
226 Database
Ta
bl
e
A
.5
Si
m
ul
at
io
ns
lis
tf
or
th
e
lo
w
-t
ur
bu
le
nc
e
fla
m
e
ca
se
fr
om
Fu
ru
ka
w
a
et
al
.[
76
].
N
am
e
M
od
el
β c
G
ri
d
si
ze
u′ ∆
u′
[m
/
s]
D
es
cr
ip
tio
n
R
em
ar
k
W
1-
2
A
/U
dy
n
5.
4M
E
q.
(3
.5
1)
0
Te
st
in
g
dy
na
m
ic
β c
β c
la
rg
el
y
ov
er
-p
re
di
ct
ed
;
in
ad
eq
ua
te
be
ca
us
e
of
lo
w
-
tu
bu
le
nc
e
W
3-
4
A
/U
"
"
"
0.
22
"
"
W
5-
6
A
/U
"
"
"
0.
37
Se
ns
iti
vi
ty
to
hi
gh
er
tu
r-
bu
le
nc
e
"
W
7
U
6.
0
"
"
0.
22
Se
e
C
ha
pt
er
8
Se
e
C
ha
pt
er
8
W
8
U
6.
0
"
E
q.
(3
.5
6)
"
Se
e
C
ha
pt
er
8
Se
e
C
ha
pt
er
8
W
9
U
10
.0
"
E
q.
(3
.5
0)
"
Se
e
C
ha
pt
er
8
Se
e
C
ha
pt
er
8
W
10
-1
2
A
7.
5/
15
"
E
q.
(3
.5
1)
0
Te
st
in
g
al
ge
br
ai
c
m
od
el
Fl
am
e
to
o
sh
or
t
W
13
-1
4
A
12
.5
/1
5
"
"
0.
37
"
"
W
15
-1
7
U
7.
5
"
E
qs
.(
3.
50
,3
.5
1,
3.
56
)
0.
22
Se
ns
iti
vi
ty
to
β c
V
er
y
se
ns
iti
ve
W
18
-1
9
U
7.
5
"
E
q.
(3
.5
0)
0/
0.
15
Se
ns
iti
vi
ty
to
in
le
tt
ur
bu
-
le
nc
e
V
er
y
se
ns
iti
ve
W
20
U
4.
5
23
M
E
q.
(3
.5
1)
0.
22
G
ri
d
se
ns
iti
vi
ty
Se
e
§8
.3
.4
A
=
al
ge
br
ai
c;
U
=
U
ns
tr
ai
ne
d.