1. HIGH-FIDELITY SIMULATIONS AND MODELING OF
COMPRESSIBLE REACTING FLOWS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF MECHANICAL
ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Amirreza Saghafian
March 2014
3. I certify that I have read this dissertation and that, in my opinion, it is fully adequate
in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Heinz Pitsch, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequate
in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Matthias Ihme
I certify that I have read this dissertation and that, in my opinion, it is fully adequate
in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Parviz Moin
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in
electronic format. An original signed hard copy of the signature page is on file in
University Archives.
iii
9. Abstract
Scramjets are air-breathing propulsion devices and have long been recognized as suit-
able for hypersonic propulsion. Because of the high speed in scramjet combustors,
the flow has a very short residence time before leaving the engine, during which air
and fuel must mix on a molecular level and chemical reactions have to be completed.
Although some ground and flight experiments have successfully demonstrated the
feasibility of supersonic combustion, experimental testing requires a large investment
and presents numerous difficulties. Computational tools are thus a key element to-
ward the development of an efficient, high-performance scramjet engine, and because
mixing and heat release are at the heart of a scramjet operation, the development
and use of accurate combustion models for supersonic combustion are critical. The
open questions in supersonic combustion span the spectrum from scientific pursuit,
e.g., shock/flame interactions, to engineering applications like prediction of unstart
phenomena in scramjets. In this study, direct numerical simulations (DNS) of a com-
pressible reacting mixing layer with finite rate chemistry are performed. The DNS
databases are used to explore the physics of supersonic combustion. An efficient com-
bustion model based on the flamelet/progress variable is then introduced. In this
approach, only two or three additional scalar transport equations need to be solved,
independently of the complexity of the reaction mechanism. The proposed combustion
model is validated using DNS databases. Finally, the compressible flamelet/progress
variable model is applied to the case of an under-expanded hydrogen jet in a super-
sonic cross-flow and HIFiRE scramjet.
ix
11. Acknowledgments
I would like to express my special appreciation and gratitude to my advisor Professor
Heinz Pitsch. I would like to thank him for encouraging my research and for allowing
me to grow as a researcher. This research would not have been possible without
the help and endless support from Professor Parviz Moin. His attention to details,
commitment to research, and unyielding engagement in teaching have been my in-
spiration. I would also like to thank Professors Matthias Ihme, Sanjiva Lele, Robert
MacCormack, and Margot Gerritsen for their insightful comments and suggestions.
I am deeply indebted to Dr. Frank Ham and Professor Vincent Terrapon for sharing
with me their knowledge in numerical analysis and combustion. I am thankful to my
friends and group members: Christoph, Ed, Eric, Krithika, Mehdi, Michael, Saman,
Shashank, Varun, and Vincent.
I would like to extend my gratitude to my family for their love and encouragement.
Most of all, I am grateful to my best friend and wife Hoora for her unwavering
love and understanding. I could not have accomplished this without her constant
encouragement and support.
Financial support from the Department of Energy under the Predictive Science
Academic Alliance Program (PSAAP) at Stanford University and NASA’s combus-
tion noise research program is gratefully acknowledged.
xi
19. List of Figures
4.1 Temporal variation of temperature (top) and pressure (bottom) in a
homogeneous reactor with Tp0q “ 1200 K and pp0q “ 1 bar. . . . . . 33
4.2 Temporal evolution of the major and minor species mass fractions in
a homogeneous reactor with Tp0q “ 1200 K and pp0q “ 1 bar. . . . . 34
4.3 Temporal evolution of the major and minor species mass fractions in
a homogeneous reactor with Tp0q “ 1200 K and pp0q “ 1 bar. . . . . 35
4.4 Schematic of a deflagration wave using a reference frame fixed on the
flame. Subscript u corresponds to the unburned and b to the burned
condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5 Temperature (top) and pressure (bottom) variations in a premixed
flame with Tu “ 300 K and pu “ 1 bar. . . . . . . . . . . . . . . . . . 37
4.6 Spatial variations of the major and minor species mass fractions in a
premixed flame with Tu “ 300 K and pu “ 1 bar. . . . . . . . . . . . 38
4.7 Spatial variations of the major and minor species mass fractions in a
premixed flame with Tu “ 300 K and pu “ 1 bar. . . . . . . . . . . . 39
4.8 Schematic of a laminar reacting mixing layer. . . . . . . . . . . . . . 40
4.9 Contours of temperature (top), reaction progress variable (middle),
and OH mass fraction (bottom) in a laminar reacting mixing layer
with Mox “ 1.84. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.10 Effect of the oxidizer Mach number on the ignition point. Comparison
of the current study (blue symbols) and Ju and Niioka (1994) (red
symbols). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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20. 4.11 Iso-surface of vorticity magnitude colored by u1{∆u. Contours are
from ´0.6 (blue) to 0.6 (red). . . . . . . . . . . . . . . . . . . . . . . 44
4.12 Spatial variation of streamwise root-mean-square velocity in a turbu-
lent inert mixing layer. Lines are this work (black line) and past DNS
studies of Rogers and Moser (1994) and Pantano and Sarkar (2002);
symbols are experimental measurements of Jones and Spencer (1971)
and Bell and Mehta (1990). . . . . . . . . . . . . . . . . . . . . . . . 45
5.1 Schematic of a temporal mixing layer. The upper stream is oxidizer,
and the lower stream is fuel. Red color outline represents a diffusion
flame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Non-dimensional maximum wavenumber, κmaxη, plotted against x2{δθ:
——, κmaxη; ´ ´ ´, κmaxη “ 1.5 line. . . . . . . . . . . . . . . . . . . 53
5.3 Mass fractions of important radicals plotted against x2{δθ: ——, fine
grid; ´ ´ ´, coarse grid. . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Dissipation rate plotted against x2{δθ: ——, fine grid; ´ ´ ´, coarse
grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 Mass fractions of important radicals plotted against x2{δθ:——, big
domain; ´ ´ ´, small domain. . . . . . . . . . . . . . . . . . . . . . . 55
5.6 Normalized averaged velocity in the streamwise direction plotted against
x2{δθ:——, big domain; ´ ´ ´, small domain. . . . . . . . . . . . . . 56
5.7 Time evolution of the momentum thickness (left), and the vorticity
thickness (right) for the Mc “ 1.1 case. . . . . . . . . . . . . . . . . . 58
5.8 Time evolution of the integrated dissipation rate (left), and turbulent
production rate (right) for the Mc “ 1.1 case. . . . . . . . . . . . . . 58
5.9 Collapse of the Favre-averaged streamwise velocity component (left),
and streamwise Reynolds stress (right) for the Mc “ 1.1 case; —,
t∆u{δ0
θ “ 1900; ´ ´ ´, t∆u{δ0
θ “ 2000; ´.´, t∆u{δ0
θ “ 2100; ...,
t∆u{δ0
θ “ 2200; ´..´, t∆u{δ0
θ “ 2300. . . . . . . . . . . . . . . . . . . 59
5.10 For the caption see the next page. . . . . . . . . . . . . . . . . . . . . 60
5.10 For the caption see the next page. . . . . . . . . . . . . . . . . . . . . 61
xx
21. 5.10 Instantaneous species mass fractions in the x3 “ 0 plane for the Mc “
1.1 case. Blue line shows the Z “ Zst line. . . . . . . . . . . . . . . . 62
5.11 For the caption see the next page. . . . . . . . . . . . . . . . . . . . . 64
5.11 For the caption see the next page. . . . . . . . . . . . . . . . . . . . . 65
5.11 Instantaneous species mass fractions in x3 “ 0 plane for the Mc “ 2.5
case. Blue line shows the Z “ Zst. . . . . . . . . . . . . . . . . . . . . 66
5.12 Instantaneous temperature contours in the x3 “ 0 plane for the Mc “
2.5 case. Blue line shows the Z “ Zst. . . . . . . . . . . . . . . . . . . 67
5.13 Scatter plots of O2 (black), H2 (blue), and H2O (red) against mixture
fraction for the Mc “ 2.5 case. Lines are ensemble averages. . . . . . 67
5.14 Scatter plots of minor radicals O (black), H (blue), and OH (red)
against mixture fraction for the Mc “ 2.5 case. Lines are ensemble
averages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.15 Iso-surface of YOH “ 0.02 colored by temperature from 1500 K (blue)
to 2500 K (red) for the Mc “ 2.5 case. . . . . . . . . . . . . . . . . . 68
5.16 Vorticity magnitude and mixture fraction contours in x3 “ 0 plane at
t∆u{δ0
θ “ 2200 for the Mc “ 1.1 case. Blue line shows the Z “ Zst. . . 69
5.17 Iso-surface of the second invariant of the velocity gradient tensor Q col-
ored by u1{∆u from -0.6 (blue) to 0.6 (red). Iso-surface of the stoichio-
metric mixture fraction (red surface) indicates approximate location of
the flame for the Mc “ 2.5 case. . . . . . . . . . . . . . . . . . . . . . 70
5.18 Vorticity magnitude contours in x3 “ 0 plane (left), and streamwise
Reynolds stress, R11 (right) at t∆u{δ0
θ “ 2300 in the self-similar region.
Blue line shows the Z “ Zst, and green line represents ru1 “ 0. . . . . 71
5.19 Time evolution of the integrated production term (—), the integrated
viscous dissipation (´ ´ ´), and the integrated pressure-strain term
(´.´) for the reacting Mc “ 1.1 case. All the integrated terms are
rescaled with 1{∆u3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
xxi
22. 5.20 Time evolution of the integrated production term (—), the integrated
viscous dissipation (´ ´ ´), and the integrated pressure-strain term
(´.´) for the reacting Mc “ 2.5 case. All the integrated terms are
rescaled with 1{∆u3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.21 Contours of the dilatation (top) and heat release rate (bottom) in x3 “
0 plane. Two strong shocklets interacting with the flame are marked. 78
5.22 Heat release rate budget plotted against mixture fraction across a
shocklet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1 Species mass fractions of the major species (H2O, H2, O2) and im-
portant minor species (H, OH, O) obtained from the solution of the
flamelet equations for three values of the fuel temperature, Tf , fixed
oxidizer temperature, Tox “ 1550 K, and fixed background pressure,
p “ 1 bar. All flamelet solutions have the same value of the reaction
progress parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Species mass fraction of the major species (H2O, H2, O2) and important
minor species (H, OH, O) obtained from the solution of the flamelet
equations for three values of the oxidizer temperature, Tox, a fixed fuel
temperature, Tf “ 300 K, and fixed background pressure p “ 1 bar.
All flamelet solutions have the same value of the reaction progress
parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 Gas constant, R, obtained from the solution of the flamelet equations
for three values of the oxidizer temperature, Tox, fixed fuel temperature,
Tf “ 300 K, and fixed background pressure p “ 1 bar. All flamelet
solutions have the same value of the reaction progress parameter. . . 85
6.4 Species mass fractions of the major species (H2O, H2, O2) and im-
portant minor species (H, OH, O) obtained from the solution of the
flamelet equations for three values of the background pressure, p, fixed
fuel temperature, Tf “ 300 K, and fixed oxidizer temperature, Tox “
1550 K. All flamelet solutions have the same value of the reaction
progress parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
xxii
23. 6.5 Temperature versus mixture fraction (left), and the difference between
the temperature obtained from the analytical expression, Eq. (3.10),
and from solving iteratively Eq. (3.5) (right). Black line corresponds
to flamelet with Tox “ 2000 K, Tf “ 300 K, and p “ 1 bar. Red
circles show CFPV results computed from chemical library at reference
conditions, Tox “ 1550 K, Tf “ 300 K, p “ 1.0 bar. Blue triangles
represent temperature corresponding to reference condition. . . . . . 87
6.6 Specific heat ratio as function of mixture fraction. Black line corre-
sponds to flamelet with Tox “ 2000 K, Tf “ 300 K, and p “ 1 bar.
Red circles show CFPV results computed from chemical library at ref-
erence conditions, Tox “ 1550 K, Tf “ 300 K, p “ 1.0 bar. Blue
triangles represent values at reference condition. . . . . . . . . . . . . 88
6.7 Source term of the progress variable versus mixture fraction. Black
line is flamelet with Tox “ 2000 K, Tf “ 300 K, and p “ 1 bar.
Blue triangles show source term of the progress variable at reference
condition Tox “ 1550 K, Tf “ 300 K, and p “ 1.0 bar. Red circles show
CFPV results computed from chemical library at reference conditions
and compressibility correction, Eq. (3.14). . . . . . . . . . . . . . . . 89
6.8 Each blue circle represents a sample point of the reference condition
used to create the chemistry table based on the Gauss-Hermite quadra-
ture points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.9 Expectation (top) and standard deviation (bottom) of the progress
variable computed using RANS and Gauss-Hermite quadrature. . . . 92
6.10 Same as Fig. 6.9, but without the compressibility correction of the
source term of the progress variable (Eq. (3.14)). . . . . . . . . . . . . 94
6.11 Contours of the progress variable in the symmetry plane z “ 0 for
the case Tr
ox “ 1180 K and pr
tb “ 1.2 bar computed with RANS;
the compressibility correction of the source term of progress variable
(Eq. (3.14)) is activated (top) and deactivated (bottom). . . . . . . . 95
6.12 Contours of the temperature in the x3 “ 0 plane, computed using
Eq. (3.10) (top), and DNS field (bottom), for the Mc “ 2.5 reacting case. 97
xxiii
24. 6.13 Contours of the mixture fraction, Z, conditioned PDFs of rT (left), rγ
(middle), and rR (right) for the CFPV approach (top) and the FPV
approach (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.14 Comparison of the Favre averaged CFPV results (solid lines) and DNS
data (dashed lines) for the Mc “ 2.5 case. . . . . . . . . . . . . . . . . 99
7.1 Comparison of instantaneous OH mass fraction from LES (top) with
experimental OH PLIF signal (bottom) in the symmetry plane z “ 0.
Contours for LES results are from rYOH “ 0 (blue) to rYOH “ 0.024
(white). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.2 Instantaneous OH mass fraction without applying temperature correc-
tion for the source term of the progress variable (Eq. (3.14)). Contours
are from rYOH “ 0 (blue) to rYOH “ 0.024 (white). . . . . . . . . . . . . 108
7.3 Instantaneous OH mass fraction computed from LES in different planes
parallel to the plate. Contours are from rYOH “ 0 (blue) to rYOH “ 0.024
(white). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.4 Experimental OH PLIF signal in the plane y “ 0.5d. Contours are
from blue to white. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5 Side to side comparison between LES (top) and RANS (bottom) of
time-averaged OH mass fraction and mixture fraction in the plane y “
0.25d. Contours are from rYOH “ 0 (blue) to rYOH “ 0.024 (white) and
rZ “ 0 (blue) to rZ “ 1 (white). . . . . . . . . . . . . . . . . . . . . . . 112
7.6 Eleven-frame average of the experimental OH PLIF signal in the y “
0.5d plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.7 Time averaged OH mass fraction in the symmetry plane z “ 0; LES
(top), RANS (middle), and eleven-frame average experimental OH
PLIF measurement (bottom). Contours are from rYOH “ 0 (blue) to
rYOH “ 0.024 (white) in LES and RANS. . . . . . . . . . . . . . . . . 113
7.8 Time averaged mixture fraction in the symmetry plane z “ 0; LES
(top) and RANS (bottom). Contours are from rZ “ 0 (blue) to rZ “ 1
(white). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
xxiv
25. 7.9 Side to side comparison between time averaged LES (right half) and
RANS (left half) in the planes x “ 4d (top), x “ 8d (middle), and
x “ 12d (bottom). Contours are from rYOH “ 0 (blue) to rYOH “ 0.024
(white), rZ “ 0 (blue), to rZ “ 1 (white), and rC “ 0 (blue), to rC “ 0.25
(white), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.10 Schematic of HIFiRE 2 scramjet. . . . . . . . . . . . . . . . . . . . . 115
7.11 Wall pressure along the centerline between the injectors. LES results
(solid lines) are compared with the experimental measurements (sym-
bols). The black line shows the LES results without compressibility
correction of the source term of the progress variable, i.e., Eq. 3.14 is
deactivated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.12 Contours of the mixture fraction on six slices normal to the streamwise
direction at x1 “ 0.25, 0.3, 0.35, 0.4, 0.45, and 0.5 m; contours are
from Z “ 0 (blue) to Z “ 1 (red). Combustor wall is colored by
pressure; contours are from p “ 10 kPa (blue) to p “ 250 kPa (red). 121
7.13 Iso-surface of the stoichiometric mixture fraction colored by the progress
variable from C “ 0 (blue) to C “ 0.3 (red). . . . . . . . . . . . . . 121
7.14 Contours of the streamwise velocity component (top) from u1 “ ´500 m/s
(blue) to u1 “ 2500 m/s (red), where the thick white line shows the
u1 “ 0 iso-surface. Contours of the pressure (bottom) from p “ 5 kPa
(blue) to p “ 250 kPa (red). . . . . . . . . . . . . . . . . . . . . . . . 122
7.15 Contours of the progress variable from C “ 0 (black) to C “ 0.3
(white). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
xxv
27. Chapter 1
Introduction
1.1 Motivation
As part of the renewed interest in high-speed flight, a need was identified for the
development of hypersonic air-breathing propulsion systems using the ambient air
as oxidizer. These systems have long been recognized as well-suited for hypersonic
propulsion. Although a traditional ramjet is most appropriate in the supersonic
regime (Mach 3 to 5), hypersonic speeds (Mach 6 to 15) can be reached only with the
use of a scramjet, where combustion takes place in the supersonic regime. Because of
the high speed in scramjet combustors, the flow has a very short residence time inside
the engine, during which air and fuel must mix on a molecular level, and chemical
reactions have to be completed. Although some ground and flight experiments have
successfully demonstrated the feasibility of supersonic combustion (Bolender et al.,
2012; Hank et al., 2008; Schramm et al., 2008; Smart et al., 2006), experimental
testing requires a large investment and presents numerous difficulties. Computational
tools are thus a key element toward the development of an efficient, high-performance
scramjet engine, and because mixing and heat release are at the heart of a scramjet
operation, the development and use of accurate combustion models for supersonic
combustion are critical.
Supersonic combustion involves a complex interaction between supersonic flow and
chemistry. Heat release from combustion can dramatically change the behavior of
1
28. 2 CHAPTER 1. INTRODUCTION
flow inside the scramjet, even leading to catastrophic unstart of the engine due to
thermal choking. On the other hand, chemistry is influenced by supersonic flow, e.g.,
chemical reactions respond to the jump in temperature and pressure across a shock
wave.
The open questions in supersonic combustion span the spectrum from scientific
pursuit, e.g., shock/flame interactions, to engineering applications like prediction of
unstart phenomena in scramjets.
1.2 Current state-of-the-art
1.2.1 Compressible mixing layer
Although there is a large body of literature on the inert compressible mixing layers
(e.g. Day et al., 1998; Jackson and Grosch, 1989; Pantano and Sarkar, 2002; Sandham
and Reynolds, 1990, 1991; Sarkar, 1995; Vreman et al., 1996), fundamental studies
of supersonic combustion are scarce (see Day et al., 2001; Luo, 1999; Pantano et al.,
2003). Furthermore, even the few existing computations are typically based on sim-
ple combustion models like infinitely fast chemistry assumption or one-step global
reaction.
Low Mach number compressible mixing layers exhibit features similar to incom-
pressible mixing layers, where the primary instability is two-dimensional (Moser and
Rogers, 1993). However, at higher Mach numbers (Mc ą 0.6), the dominant instabil-
ity mode is three-dimensional (Vreman, 1995). At high Mach numbers, the mixing
layer growth rate is reduced. Vreman et al. (1996) show that the decrease in pressure
fluctuations leads to a reduction in the pressure-strain term, and in turn is responsible
for the changes in the growth rate of the compressible mixing layer.
Pressure fluctuations have a crucial role in distributing the turbulent energy in com-
pressible flows. Vreman et al. (1996) show that attenuation of pressure fluctuations
reduces the turbulent production and pressure-strain term. Sarkar (1995) also ob-
served a reduction in pressure fluctuations by increasing the gradient Mach number,
which leads to smaller pressure-strain term in the Reynolds transport equation and
29. 1.2. CURRENT STATE-OF-THE-ART 3
smaller turbulence intensity. Freund et al. (2000) have also observed the reduction in
the pressure-strain term in an annular mixing layer.
Pantano et al. (2003) studied mixing of a conserved scalar by using direct numerical
simulations of reactive and inert mixing layers. An infinitely fast irreversible chemistry
model based on the Burke-Schumann solution was employed. It was found that the
heat release does not significantly change the profile of unconditional scalar dissipation
rate. The probability density function of scalar and the conditioned scalar dissipation
rate are, nevertheless, affected by the heat release, which leads to a reduction in
overall reaction rate. Pantano et al. (2003) also analyzed the interactions between
compressibility, heat release, and turbulence.
Mahle et al. (2007) performed direct numerical simulations of reactive mixing layers
using infinitely fast chemistry. The analysis of Mahle et al. (2007) suggests that the
heat release and compressibility influence the mixing layer in a similar fashion, i.e.
momentum thickness growth rate, Reynolds stress, and pressure-strain term decrease.
This behavior in compressible reacting mixing layers is primarily due to the mean
density change. The turbulence intensity at the flame location is, however, very low
in their study (see Mahle, 2007, p. 87), especially for supersonic Mach numbers.
Luo (1999) has used a one-step global reaction to simulate a partially premixed
supersonic flame. His observations are different from Mahle et al. (2007). He reports
that the pressure-strain term in the Reynolds transport equation is the dominant
term, and becomes more dominant by increasing heat releasee. Based on Luo (1999),
heat release can increase or decrease the mixing layer growth rate. However, the
reactive flow-fields in this study had not reached self-similarity (see Luo, 1999, p.
426).
1.2.2 Combustion modeling
The vast majority of computational modeling work in supersonic turbulent combus-
tion so far has relied on simplified/reduced chemical mechanisms and the explicit
transport of the involved species (Bray, 1996). Such approaches require the closure
for the filtered (or ensemble averaged) chemical source term in the species transport
30. 4 CHAPTER 1. INTRODUCTION
equations. This can be achieved, for example, with simpler but less accurate models
such as the direct use of the Arrhenius law with the mean quantities (Davidenko
et al., 2003; Kumaran and Babu, 2009), which neglects turbulence closure, the Eddy
Dissipation Concept model (Chakraborty et al., 2000), or with closure based on as-
sumed (Baurle and Girimaji, 2003; Karl et al., 2008) or transported (Baurle et al.,
1995; Möbus et al., 2003) probability density functions (PDF). Also, the Linear Eddy
model (LEM) (Genin et al., 2003; Ghodke et al., 2011) has been applied for this
case. However, due to the strong non-linearity of the source term and the wide range
of time scales associated with the chemistry, reactive transport equations are very
stiff and difficult to solve in all of these approaches. Moreover, due to very short
residence times in high speed flows, flame stabilization mechanisms are often gov-
erned by auto-ignition. It is critical to model accurately such ignition and extinction
phenomena in order to predict the stability of scramjet combustion. Therefore, an
approach based on detailed chemical kinetics, which can predict flame stabilization,
is required. While a model transporting all involved species can easily be extended to
more detailed chemical mechanisms, it quickly becomes computationally intractable,
especially when complex fuels must be considered.
An alternative approach is based on the flamelet concept (Peters, 2000; Pitsch,
2006), which assumes that the chemical time scales are shorter than the turbulent
time scales so that the flame can be approximated as an ensemble of laminar flamelets.
The so-called steady flamelet approach allows the computation of the chemistry to
be performed independently of the flow simulation and stored in tabulated form as a
function of a limited number of scalars. During the actual simulation, the quantities
of interest are read and interpolated, thus, dramatically decreasing the computational
cost and allowing the use of complex chemical mechanisms. In the low Mach number
flamelet implementation, the temperature and the species mass fraction are assumed
to depend only on two scalars, traditionally the mixture fraction and its dissipation
rate. Chemical tables are then constructed assuming constant background pressure.
This formulation can also be extended to better reproduce the unsteady character of
combustion by replacing the scalar dissipation rate with a progress variable (Pierce
and Moin, 2004).
31. 1.2. CURRENT STATE-OF-THE-ART 5
Typical implementation of the flamelet model is based on a low Mach number
assumption, and there are limited number of studies of high-speed flows using this
approach (Berglund and Fureby, 2007; Kumar and Tamaru, 1997; Oevermann, 2000).
Kumar and Tamaru (1997) used a laminar flamelet model to simulate a compressible
ram combustor, where the temperature was interpolated from a flamelet library based
on the mixture fraction and scalar dissipation rate. Oevermann (2000) extended this
model by computing the temperature from the internal energy and the species mass
fractions, where the energy is determined from the solution of a transport equation,
and the species mass fractions were interpolated from a flamelet table, based on the
same parameters and without introducing any corrections to the low Mach number
flamelet model. Berglund and Fureby (2007) used a one-equation (mixture fraction)
and a two-equation (mixture fraction and progress variable) flamelet model in con-
junction with a two-step reaction mechanism. This combustion model is also based
on the low Mach number flamelet approach without any compressibility correction.
Vicquelin et al. (2011) also developed a formalism to couple chemistry tabulation to
solvers for mildly compressible flows. They computed temperature from the inter-
nal energy using a simple expansion, whereas no extension has been used for other
quantities. This approach is reasonable for low Mach number compressible flows, but
cannot accurately describe the high Mach number regime.
As shown in this study, the low Mach number assumption does not hold at su-
personic speed and strong compressibility effects and viscous heating start to play
important roles. For instance, the source term of the progress variable is extremely
sensitive to the variations in temperature and pressure, and appropriate modifications
should be implemented. Therefore, without appropriate compressibility corrections,
a combustion model based on the low Mach number assumptions is not plausible for
compressible flows, especially at high Mach numbers, where flows can admit shock
waves and expansion fans.
32. 6 CHAPTER 1. INTRODUCTION
1.3 Objectives
High quality data is required to fully understand the physics of supersonic com-
bustion. Obtaining this necessary data experimentally requires simultaneous mea-
surements of the species concentrations in addition to temperature, pressure, and
velocity fields which is very challenging especially in harsh supersonic environments.
Existing experimental data at best provides qualitative OH concentration obtained
through planar laser-induced fluorescence (PLIF), and wall pressure (Gamba et al.,
2012, 2011a; Heltsley et al., 2007). Three-dimensional direct numerical simulations
(DNS) with detailed chemistry can fill this gap by providing 3D time-dependent data
of quantities of interest.
As mentioned in the previous section, all the fundamental studies of the compress-
ible reacting flows employ low-fidelity combustion models based on the infinitely fast
chemistry, or one-step global reaction. These combustion models are not accurate
enough to explore the interplay between compressibility and combustion. Therefore,
one of the main objectives of this study is to perform high-fidelity direct numeri-
cal simulations of supersonic combustion using realistic finite-rate chemistry. The
DNS databases are used to study physics of supersonic combustion, e.g., eddy shock-
let/flame interaction and heat release/turbulence interaction.
Another objective is to devise an efficient flamelet-based combustion model for su-
personic regime. This model will be based on the flamelet/progress variable approach
(Ihme, 2007; Pierce and Moin, 2004), which allows for the use of a detailed chemistry
mechanism in a pre-processing step. In this approach, only two or three additional
scalar transport equations need to be solved, independently of the complexity of the
reaction mechanism; the required quantities of interest can then be interpolated from
a flamelet table. Here, the new combustion model will be formulated, validated, and
improved by using high-fidelity direct numerical simulation databases. The combus-
tion model is then applied to the case of a hydrogen jet in a supersonic cross-flow and
HIFiRE scramjet.
33. 1.4. ACCOMPLISHMENTS 7
1.4 Accomplishments
The major contributions of this work are summarized in the following list:
" Fundamental investigation of supersonic combustion by performing DNS studies
of reactive temporal mixing layers using finite-rate chemistry.
" Development of an efficient combustion model based on the flamelet/progress
variable approach for high-speed flows (CFPV). In this formulation, the temper-
ature is computed from the transported total energy and tabulated species mass
fractions. Combustion is thus modeled by three additional scalar equations and
a flamelet library that is computed in a pre-processing step.
" A priori analysis of the compressibility effects on the proposed combustion
model. This analysis reveals that mixture properties which are not function of
temperature (e.g., gas constant) have very small departure from the nominal
solution, whereas mixture properties that depend on temperature (e.g., specific
heat ratio and source term of the progress variable) show higher sensitivity to
the perturbations.
" A priori and a posteriori analyses of the CFPV model using DNS databases.
These studies show that the CFPV approach can model the interplay between
compressibility and combustion with good accuracy.
" A sensitivity study of the CFPV approach. This study confirms the importance
of the compressibility corrections, especially for the source term of the progress
variable.
" Development of a compressible DNS solver using finite rate chemistry (“Fer-
dowsi” solver)
" Development of an LES solver using the compressible flamelet/progress variable
approach, based on the unstructured platform “Charles”.
" Application of the CFPV model to the case of a hydrogen jet in a supersonic
cross-flow.
34. 8 CHAPTER 1. INTRODUCTION
" Application of the CFPV model to the HIFiRE scramjet.
35. Chapter 2
Governing Equations
The compressible Navier-Stokes equations are presented in the first section of this
chapter. Large eddy simulation equations are derived by applying a filtering procedure
to the compressible Navier-Stokes equations. The numerical schemes used in the
solution of these equations are briefly discussed is Chapter 4.
2.1 Direct numerical simulation
The conservation equations for mass, momentum, energy, and species are written in
terms of tρ, ρui, ρet, ρYαu, where ρ is the density, ui the components of the velocity
vector, et the total energy per unit mass, and Yα are species mass fractions. Using
the Cartesian tensor notation, the transport equations for these variables are
Bρ
Bt
`
Bρuj
Bxj
“ 0, (2.1)
Bρui
Bt
`
Bρuiuj
Bxj
“ ´
Bp
Bxi
`
Bτji
Bxj
, (2.2)
Bρet
Bt
`
Bρetuj
Bxj
“ ´
Bpuj
Bxj
`
Bτjiui
Bxj
´
Bqj
Bxj
, (2.3)
9
36. 10 CHAPTER 2. GOVERNING EQUATIONS
BρYα
Bt
`
BρYαuj
Bxj
“ ´
BρYαVαj
Bxj
` 9mα for α “ 1, 2, ..., Ns, (2.4)
where p is the mixture pressure, τij the viscous stress tensor, qi the heat flux vector, Vαi
the diffusion velocity, 9mα the reaction rate of species Mα (chemical symbol of species
α), and Ns is the total number of species. It should be noted that the summation
convention is only applied for repeated lower-case Latin indices not for the repeated
Greek symbols.
2.1.1 Caloric equation of state
The total energy in Eq. (2.3) also includes the chemical energy and is defined as
et “ e `
1
2
ukuk, (2.5)
where e is internal energy per unit mass defined by
e “
Nsÿ
α“1
Yαeα, (2.6)
where eα is the internal energy of species α. Using the ideal gas assumption, the
internal energy of each species is only function of temperature, and can be computed
from
eα “ eαpTref q `
ż T
Tref
cv,αpT1
qdT1
, (2.7)
where cv,α is the specific heat at constant volume for species Mα. The first term
on the right hand side of Eq. (2.7) is the energy of formation of species Mα, and
the second term is the sensible energy. The energy of formation of species Mα is
the change of energy that occurs for forming one mole of Mα from a set of reference
species at reference conditions. The specific heats of each species may be calculated
from NASA polynomials (McBride et al., 2005). The specific heat at constant volume
of the mixture can be determined as
37. 2.1. DIRECT NUMERICAL SIMULATION 11
cv “
Nsÿ
α“1
Yαcv,α. (2.8)
Introducing Eqs. (2.7) and (2.8) in Eq. (2.6) yields
e “
Nsÿ
α“1
YαeαpTref q `
ż T
Tref
cvpT1
qdT1
. (2.9)
Because of the non-linear dependence of specific heats on temperature, this caloric
equation of state cannot be inverted easily, and hence the computation of temper-
ature from internal energy requires an iterative approach. Since the derivative of
the internal energy with respect to temperature is known (i.e. specific heat ratio at
constant volume), the Newton-Raphson method is a plausible approach to compute
temperature.
2.1.2 Thermodynamic equation of state
Pressure is required to close Eqs. (2.2) and (2.3), and is computed from the equation
of state
p “ ρRT, (2.10)
where R is the gas constant of the mixture defined as the ratio of the universal gas
constant ˆRu and molecular weight of the mixture W, i.e. R “ ˆRu{W. The molecular
weight of the mixture is computed from
1
W
“
Nsÿ
α“1
Yα
Wα
, (2.11)
where Wα is the molecular weight of species Mα.
38. 12 CHAPTER 2. GOVERNING EQUATIONS
2.1.3 Diffusion velocity
The diffusion velocities are modeled by the Curtiss-Hirschfelder approximation (Cur-
tiss and Hirschfelder, 1949), and mass conservation is satisfied with a correction ve-
locity (see Coffee and Heimerl, 1981)
Vαi “ V D
αi ` V C
i . (2.12)
Here V D
αi is the diffusion velocity due to mole fraction gradient and is given by
V D
αi “ ´
Dα
Xα
BXα
Bxi
, (2.13)
where Dα is the diffusivity, and Xα is the mole fraction of species Mα. The diffusion
coefficient is computed from
Dα “
λ{ρcp
Leα
, (2.14)
by assuming that the Lewis number of all species are constant, where λ is the thermal
conductivity, and cp is the specific heat at constant pressure of the mixture.
The correction velocity, determined from the mass conservation condition
řNs
α“1 YαVαi “
0, is
V C
i “
Nsÿ
α“1
Yα
Xα
Dα
BXα
Bxi
. (2.15)
In this study, diffusion fluxes due to temperature gradient (Soret effect), and due
to pressure gradient are neglected.
2.1.4 Viscous stress tensor
In a Newtonian fluid, the viscous stress tensor is modeled by assuming a linear de-
pendence on the strain-rate tensor. In general, coefficients are a fourth-order tensor
(43
coefficients are required in the most general linear dependence); nevertheless, the
arguments of symmetry and isotropy reduce the number of independent coefficients
to two yielding
39. 2.1. DIRECT NUMERICAL SIMULATION 13
τij “ 2µSij `
ˆ
µB ´
2
3
µ
˙
Skkδij, (2.16)
where Sij “ 1
2
´
Bui
Bxj
`
Buj
Bxi
¯
is the strain-rate tensor, µ is the dynamic (shear) viscosity,
and µB is the bulk viscosity. Both coefficients depend on temperature, although the
bulk viscosity is often neglected (Stokes assumption). In this study, the bulk viscosity
is neglected.
2.1.5 Heat flux
The heat flux vector includes a heat diffusion term due to temperature gradient
(Fourier’s Law), and a second term due to different diffusion velocities of species
leading to
qi “ ´λ
BT
Bxi
` ρ
Nsÿ
α“1
YαVαihα, (2.17)
where λ is the heat conductivity, and hα is the enthalpy of species Mα. The radiative
heat flux is neglected in this study, albeit potentially important in simulation of large
scramjets.
2.1.6 Chemical source terms
Consider a chemistry mechanism with Nr reactions among Ns species
Nsÿ
α“1
ν1
αζMα é
Nsÿ
α“1
ν2
αζMα ζ “ 1, 2, ..., Nr, (2.18)
in which ν1
αζ and ν2
αζ are the molar stoichiometric coefficients of species α in reaction
ζ. These stoichiometric coefficients should satisfy the following relations to enforce
mass conservation
Nsÿ
α“1
ναζWα “ 0 ζ “ 1, 2, ..., Nr, (2.19)
40. 14 CHAPTER 2. GOVERNING EQUATIONS
where
ναζ “ ν2
αζ ´ ν1
αζ. (2.20)
The rate of change of concentration (mole per unit volume) of species α and β in
reaction ζ, denoted by ωαζ and ωβζ respectively, are related by
9ωαζ
ναζ
“
9ωβζ
νβζ
“ 9ωζ, (2.21)
where 9ωζ (mole per unit volume per second) is a species independent coefficient and
is called reaction rate.
The phenomenological law of mass action states that the reaction rate is propor-
tional to the product of concentration of reactants. For reversible reactions (e.g.
Eq. (2.18)), the net reaction rate can be computed from
9ωζ “ kfζ
Nsź
α“1
ˆ
ρYα
Wα
˙ν1
αζ
´ kbζ
Nsź
α“1
ˆ
ρYα
Wα
˙ν2
αζ
, (2.22)
where kfζ and kbζ are the specific reaction rate constants of forward and reverse
reactions, respectively. The specific reaction rate constants depends on temperature
and are usually computed from the so-called Arrhenius law
kf “ Af Taf
exp
ˆ
´
Ef
RuT
˙
, (2.23)
kb “ AbTab
exp
ˆ
´
Eb
RuT
˙
, (2.24)
where Ru is the universal gas constant, and Ef and Eb are forward and backward
activation energies (subscript ζ is omitted for clarity).
The net rate of production of species α (source term of Eq. (2.4)) is computed from
9mα “
Nrÿ
ζ“1
ναζ 9ωζ. (2.25)
Notice that the subscript on the left hand side of Eq. (2.25) is for species α, whereas
41. 2.2. LARGE EDDY SIMULATION 15
the subscript in Eq. (2.22) is for reaction ζ.
2.2 Large eddy simulation
In the large eddy simulation (LES) approach, the large energy-containing turbulent
structures are directly represented, whereas the effects of smaller eddies are modeled
(Rogallo and Moin, 1984). In LES, a low-pass filtering operation is applied to the
governing equations to derive the equations for large structures.
2.2.1 LES filtering
Filtered variables are obtained from
¯f pxi, tq “
ż
Ω
Gpxi, riqf pxi ´ ri, tq dri, (2.26)
where G is a filter kernel. In variable density flows, the filtered equations can be
greatly simplified by using the Favre filtering
rf pxi, tq “
1
¯ρ pxi, tq
ż
Ω
Gpxi, riqρ pxi ´ ri, tqf pxi ´ ri, tq dri. (2.27)
Equation (2.27) can be written more compactly as rf “ ρf
¯ρ
.
The instantaneous variable f can be decomposed as
f “ rf ` f2
, (2.28)
where f2
is used to denote the fluctuation with respect to the Favre filtered value.
2.2.2 Transport equations
Applying the filtering operation to Eqs. (2.1), (2.2), and (2.4) (assuming that filtering
can commute with differentiation), results in the following set of transport equations
for ¯ρ, ¯ρrui, and ¯ρĂYα
42. 16 CHAPTER 2. GOVERNING EQUATIONS
B¯ρ
Bt
`
B¯ρruj
Bxj
“ 0, (2.29)
B¯ρrui
Bt
`
B¯ρruiruj
Bxj
“ ´
B¯p
Bxi
`
B¯τji
Bxj
`
B¯τR
ji
Bxj
, (2.30)
B¯ρrYα
Bt
`
B¯ρrYαruj
Bxj
“ ´
B¯ρrYα
rVαj
Bxj
` 9mα `
B ¯JR
j
Bxj
for α “ 1, 2, ..., Ns, (2.31)
where
¯τR
ij “ ¯ρruiruj ´ ¯ρ Ąuiuj, (2.32)
¯JR
i “ ¯ρrui
rYα ´ ¯ρĆYαui. (2.33)
The energy equation can, however, be written in different forms based on the def-
inition of Favre filtered total energy. A filtered energy can be defined with either of
the following relations
˜et “ ˜e `
1
2
Ćukuk, (2.34)
˜E “ ˜e `
1
2
rukruk. (2.35)
A transport equation for ˜et is obtained by simply filtering Eq. (2.3). On the other
hand, deriving the transport equation for the second definition is more involved,
albeit easier to solve numerically. To derive the transport equation of ˜E, we add
the transport equations for the filtered internal energy and the mean kinetic energy
1
2
rukruk.
The transport equation for internal energy then reads
Bρe
Bt
`
Bρeuj
Bxj
“ ´p
Buj
Bxj
` τji
Bui
Bxj
´
Bqj
Bxj
. (2.36)
Notice that the RHS of Eqs. (2.36) and (2.3) are very similar except that pressure
43. 2.2. LARGE EDDY SIMULATION 17
and viscous stress tensor are outside of derivatives in Eq. (2.36). Applying the filter
operator on Eq. (2.36) results in
B¯ρ˜e
Bt
`
B¯ρ˜eruj
Bxj
“ ´p
Buj
Bxj
` τji
Bui
Bxj
´
B¯qj
Bxj
`
B¯πR
j
Bxj
, (2.37)
where
¯πR
i “ ¯ρ˜eruj ´ ¯ρ Ăeuj. (2.38)
The transport equation for the mean kinetic energy is derived by multiplying
Eq. (2.30) by rui
B¯ρ1
2
rukruk
Bt
`
B¯ρ1
2
rukrukruj
Bxj
“ ´ruj
B¯p
Bxj
` rui
B¯τji
Bxj
` rui
B¯τR
ji
Bxj
. (2.39)
Therefore, the transport equation for ˜E is obtained by adding Eqs. (2.37) and (2.39)
resulting in
B¯ρ ˜E
Bt
`
B¯ρ ˜Eruj
Bxj
“ ´
B¯pruj
Bxj
`
B¯τjirui
Bxj
´
B¯qj
Bxj
`
B¯πR
j
Bxj
` rui
B¯τR
ji
Bxj
, (2.40)
where the following assumptions have been made:
p
Buj
Bxj
` ruj
B¯p
Bxj
«
B¯pruj
Bxj
, (2.41)
τji
Bui
Bxj
` rui
B¯τji
Bxj
«
B¯τjirui
Bxj
. (2.42)
The residual stresses and scalar fluxes are modeled using the dynamic procedure
(Germano et al., 1991; Moin et al., 1991).
2.2.3 Heat flux
In this section, a simplified relation for the heat flux vector is introduced. Enthalpy of
the mixture is computed from h “
řNs
α“1 Yαhα. Taking the gradient of this equation
results
44. 18 CHAPTER 2. GOVERNING EQUATIONS
Bh
Bxi
“
Nsÿ
α“1
hα
BYα
Bxi
`
Nsÿ
α“1
Yα
Bhα
Bxi
. (2.43)
Note that for ideal gases Bhα
Bxi
“ cp,α
BTα
Bxi
, and similar to Eq. (2.8) the specific heat
at constant pressure of the mixture is defined as cp “
řNs
α“1 Yαcp,α. Using these two
relations and Eq. (2.43), the gradient of the temperature can be computed from
BT
Bxi
“
1
cp
Bh
Bxi
´
1
cp
Nsÿ
α“1
hα
BYα
Bxi
. (2.44)
Substituting Eq. (2.44) into Eq. (2.17) yields
qi “ ´
λ
cp
Bh
Bxi
`
Nsÿ
α“1
„
ρYαVαi ´
λ
cp
BYα
Bxi
hα. (2.45)
If the diffusion velocity is computed from Fick’s law,
V D
αi “ ´
Dα
Yα
BYα
Bxi
, (2.46)
instead of Eq. (2.13), the term in the brackets in Eq. (2.45) vanishes assuming unity
Lewis number.
It should be emphasized that Eq. (2.17) is used for DNS (without introducing any
simplifications), whereas the simplified expression, qi “ ´ λ
cp
Bh
Bxi
, is used for the LES
studies.
2.2.4 Thermodynamic equation of state
The filtered pressure is obtained by applying the filtering operator to Eq. (2.10)
¯p “ ¯ρĄRT. (2.47)
However, ĄRT cannot be computed directly. Instead, rR is defined using Eq. (2.11)
rR “ ˆRu
Nsÿ
α“1
rYα
Wα
, (2.48)
45. 2.3. FLAMELET-BASED COMBUSTION MODELING 19
and a gas-constant weighted temperature is introduced as
ˆT “
ĄRT
rR
. (2.49)
Equation (2.47) is then reads
¯p “ ¯ρ rR ˆT. (2.50)
The difference between rT and ˆT is, however, not significant for the cases considered
in this study (see Chapter 5), and in the LES, it is assumed that ˆT « rT.
2.3 Flamelet-based combustion modeling
Details of the flamelet-based models are discussed in Chapter 3. In this section,
additional transport equations (i.e. mixture fraction and progress variable) required
in flamelet-based approach are summarized.
2.3.1 Mixture fraction
In non-premixed combustion, fuel and oxidizer are injected separately into a combus-
tor. It is imperative that fuel and oxidizer mix on the molecular level before reactions
occur. Chemistry reactions are often faster than the mixing process. Therefore, the
diffusion of fuel and oxidizer is the rate-limiting process in such systems. Accurate
prediction of the mixing process is thus crucial.
Mixture fraction is a conserved scalar which indicates the level of mixing between
fuel and oxidizer streams. Mixture fraction can be defined in numerous ways. Its
value is usually set to zero in the oxidizer stream and one in the fuel stream.
For a one-step global reaction such as
νFF ` νOO Ñ νPP, (2.51)
where F, O, and P refer to fuel, oxidizer, and product, respectively, Burke and Schu-
mann (1928) defined a mixture fraction given by
46. 20 CHAPTER 2. GOVERNING EQUATIONS
Z “
νYF ´ YO ` YO,2
νYF,1 ` YO,2
, (2.52)
where YO,2 is the oxidizer mass fraction in the oxidizer stream, YF,1 is the fuel mass
fraction in the fuel stream, and ν “ νOWO
νFWF
.
Alternatively, a mixture fraction can be defined using elementary mass fractions
(Bilger, 1976)
Zβ “
Nsÿ
α“1
ηαβ
Yα
Wα
Wβ, (2.53)
where ηαβ denotes the number of β elements in Mα.
Pitsch and Peters (1998) defined the mixture fraction in a two-feed system as the
solution of the transport equation
BρZ
Bt
`
BρZuj
Bxj
“
B
Bxj
ˆ
ρDZ
BZ
Bxj
˙
, (2.54)
where DZ is the diffusion coefficient of the mixture fraction. This definition does
not have some of the limitations of the aforementioned ways to define the mixture
fraction, e.g. the equal lewis number assumption, and allowed Pitsch and Peters
(1998) to develop a consistent flamelet formulation for diffusion flames, which is used
in this study.
2.3.2 Reaction progress variable
A reaction progress parameter Λ has been defined by Pierce (2001) and Pierce and
Moin (2004). This parameter enabled Pierce and Moin (2004) to uniquely identify
each flamelet along the S-shaped curve including the unstable branch (refer to Chap-
ter 3 for more information). The definition of the reaction progress parameter is
arbitrary as long as it allows a unique identification of all flamelets (Ihme et al.,
2005). Pierce and Moin (2004) defined progress variables as the sum of the major
combustion products. Ihme (2007) solved an optimization problem to find the best
definition for the reaction progress variable, and concluded that summation of major
47. 2.3. FLAMELET-BASED COMBUSTION MODELING 21
products with unity weights is sufficiently accurate.
Therefore, the progress variable is defined as C “ YH2O ` YH2 ` YCO2 ` YCO for hy-
drocarbons and as C “ YH2O for hydrogen in this study. Consequently, the transport
equation for the progress variable is given as
BρC
Bt
`
BρCuj
Bxj
“
B
Bxj
ˆ
ρDC
BC
Bxj
˙
` 9ωC, (2.55)
where 9mC is the source term of the progress variable.
49. Chapter 3
Combustion Modeling for
Compressible Flows
We introduce here two different formulations of a compressible flamelet/ progress-
variable approach, where temperature is not given by a chemistry table, but computed
from the total energy and the species mass fractions. An analytical relationship is
derived to eliminate costly iterative steps during the temperature calculation. In
addition, the source term for the progress variable is rescaled by the mixture temper-
ature and density, providing better accounting for compressibility effects on chem-
istry. Compressibility corrections have also been devised for the mixture properties,
which are sensitive to the compressible variations of temperature and pressure. The
model is tested in both LES and RANS computations for a reacting hydrogen jet in
a supersonic transverse flow and LES of HIFiRE scramjet in chapter 7.
The low Mach number flamelet/progress variable approach is summarized in Sec-
tion 3.1. Two extensions of this model for compressible flows are then introduced
in sections 3.2 and 3.3. In all cases, the models are based on the flamelet/progress
variable approach, in which the chemistry is pre-computed and tabulated as a series
of laminar flamelet solutions for a given set of boundary conditions and background
pressure.
23
50. 24 CHAPTER 3. COMBUSTION MODELING FOR COMPRESSIBLE FLOWS
3.1 Low Mach number flamelet/progress variable ap-
proach
The low Mach number flamelet/progress variable (FPV) approach (Ihme, 2007; Ihme
et al., 2005; Pierce, 2001; Pierce and Moin, 2004) is based on the steady flamelet
equations Peters (1984, 2000), which are derived by transforming the species and
energy transport equations from the physical space to the mixture fraction space.
The solutions of the steady flamelet equations are
´ ρ
χZ
2
B2
Yα
BZ2
“ 9mα, (3.1)
here only shown for species mass fractions, can be represented by the so-called S-
shaped curve. The scalar dissipation rate, χZ “ 2D | Z|2
, that appears as a parame-
ter in Eq. (3.1), describes the local effect of diffusion on the chemistry, thus displaying
the interplay between turbulence and combustion.
In order to solve Eq. (3.1), the scalar dissipation rate, a set of boundary conditions
(including temperature and mixture composition) at the fuel and oxidizer sides, and
a background pressure should be specified. Different models for the scalar dissipation
rate can be devised resembling the local structure of a counterflow diffusion flame
Peters (2000) or a semi-infinite mixing layer Pitsch (1998b).
A reaction progress parameter, Λ, is introduced based on the progress variable
Ihme (2007); Pierce and Moin (2004) to uniquely identify an associated flamelet. For
each flamelet, Λ is defined as the value of the progress variable C at stoichiometric
condition. The reaction progress parameter then replaces the scalar dissipation rate
in the chemistry parametrization used in the flamelet model. In a CFD simulation,
the equations for mixture fraction Z and the progress variable C are solved. Given
a flamelet table in the form C “ FC pZ, Λq, for a given state, defined by Z and C,
the reaction progress parameter, Λ, and hence the corresponding flamelet solution,
can be determined by inverting the flamelet table. This inversion and associated
assumptions and difficulties are discussed in Ihme and Pitsch (2008). Assumed PDFs
51. 3.2. COMPRESSIBLE EXTENDED FLAMELET TABLE APPROACH 25
are introduced to account for the turbulence/chemistry interaction. Typically, a β-
PDF is assumed for the mixture fraction Z and a δ-PDF for the progress variable
C. Therefore, the flamelet library is parametrized by the mean mixture fraction rZ,
the variance of the mixture fraction ĄZ22, and the mean progress variable rC. The
temperature is usually looked up from the flamelet library.
3.2 Compressible extended flamelet table approach
A fluid particle in a compressible flow can experience strong variations in temperature
and pressure due to viscous heating, shock waves, or strong expansions. To account
for these compressibility effects, two additional degrees of freedom could be added to
the manifold representing the thermochemical state of the flamelets. Typically, pres-
sure and a temperature-related quantity (e.g. internal energy or enthalpy) could be
considered as additional dimensions to the flamelet library. Although the dimension-
ality of the problem increases in such an approach, it still remains a low-dimensional
representation of combustion.
In the compressible extended flamelet table (CEFT) approach, the S-shaped curve
solutions to the flamelet equations typically used in the FPV model is expanded to
an S-shaped hypersurface. This hypersurface is defined by all the flamelet solutions
of Eq. (3.1) considering all possible variations of the boundary temperatures and the
background pressure. However, the memory requirement of a five-dimensional table
is often prohibitive, especially for hydrocarbon fuels. This motivates the development
of a simplified version of this model that limits the dimensionality of the chemistry
table, as explained next.
3.3 Compressible flamelet/progress variable approach
The main idea of the compressible flamelet/progress variable (CFPV) approach is
to reduce the dimensionality of the chemistry table by representing the S-shaped
hypersurface as a perturbation around a nominal low Mach number flamelet solution.
In other words, the additional dimensions introduced in the CEFT approach above are
52. 26 CHAPTER 3. COMBUSTION MODELING FOR COMPRESSIBLE FLOWS
approximated by an analytical surface. The exact analytical form of this expression
can vary depending on the quantity considered, but as an example, a quantity of
interest φ that needs to be tabulated, can be written as
φ
´
rZ, ĄZ22, rC, ¯p, re
¯
“ φ0
´
rZ, ĄZ22, rC; ¯p0, re0
¯
` Φ
´
¯p, re; rZ, ĄZ22, rC
¯
, (3.2)
where e represents the internal energy including the chemical energy, and φ0 is the
value of φ obtained at the nominal conditions corresponding to p0 and e0. On the
other hand, Φ is an analytical expression that represents the deviation of φ at the local
conditions p and e with respect to the nominal conditions. Note that the analytical
expression Φ depends on parameters that are typically functions of rZ, ĄZ22, and rC,
and are thus tabulated in the chemistry table, as illustrated below. This leads to
a chemistry table with three dimensions, as in the low Mach number FPV method,
which is computationally tractable. Such expansions have been proposed by Wang
et al. (2011), and have been used by Mittal and Pitsch (2013) for low Mach number
flows.
Typical quantities of interest that are not known from the transport equations in-
clude species mass fractions, gas constant, temperature, pressure, molecular viscosity,
and thermal diffusivity. The remaining part of this section describes the analytical
perturbations used for these quantities.
The first assumption is that the species mass fractions do not strongly vary if the
deviation in pressure and temperature from nominal conditions is not too large. In
other words, it is assumed that the species mass fractions are frozen at their nominal
level:
Yα
´
rZ, ĄZ22, rC, ¯p, re
¯
“ Yα,0
´
rZ, ĄZ22, rC; ¯p0, re0
¯
, (3.3)
This is a rather strong assumption, which is discussed in the next section. A direct
consequence is that the gas constant becomes
R
´
rZ, ĄZ22, rC, ¯p, re
¯
“ R0
´
rZ, ĄZ22, rC; ¯p0, re0
¯
, (3.4)
and can be directly tabulated.
53. 3.3. COMPRESSIBLE FLAMELET/PROGRESS VARIABLE APPROACH 27
Given a mixture composition, the temperature can be computed from the total
energy. The total energy rE of the N species mixture is defined as the sum of the
internal energy, re, the kinetic energy 1
2
ruj ruj, and the turbulent kinetic energy k. The
internal energy is
re “ rh ´ ĄRT “
Nÿ
α“1
ĆYαhα ´ ĄRT, (3.5)
where R “ Ru
ř
α Yα{Wα is the gas constant, Ru the universal gas constant, Wα
the molecular weight of species α, and T the temperature. The specific enthalpy of
species α is computed as
hαpTq “ h0
αpTref q `
ż T
Tref
cp,αpT1
qdT1
, (3.6)
using its specific heat capacity cp,α and heat of formation h0
αpTref q. Due to the wide
range of temperature variations caused by compressibility in a high-speed flow, the
dependence of the heat capacity on temperature must be considered.
Since the total energy is a non-linear function of the temperature, an iterative
method like Newton-Raphson is required to compute the temperature given the energy
rE and the mixture composition. In order to eliminate this expensive iterative step,
an alternate approach is proposed. For a given mixture, i.e. for fixed rZ, ĄZ22, and rC,
the internal energy can be approximated as
re “ re0 `
ż rT
T0
rcvpTqdT “ re0 `
ż rT
T0
rR
rγpTq ´ 1
dT, (3.7)
where γ is the ratio of the specific heats. rγ can then be expanded about the flamelet
solution using a linear expansion in temperature,
rγpTq “ rγ0 ` aγp rT ´ T0q. (3.8)
This equation is validated in Chapter 6. The values of re0, rR0, rγ0, aγ and T0 can be
computed during a preprocessing step and tabulated in the flamelet library. This ap-
proximation then leads to an analytical relationship between temperature and specific
54. 28 CHAPTER 3. COMBUSTION MODELING FOR COMPRESSIBLE FLOWS
energy
re “ re0 `
rR
aγ
ln
˜
1 `
aγp rT ´ T0q
rγ0 ´ 1
¸
, (3.9)
which can be inverted to yield the temperature as
rT “ T0 `
rγ0 ´ 1
aγ
´
eaγpre´re0q{ rR
´ 1
¯
. (3.10)
The temperature dependencies of the molecular viscosity and thermal diffusivity
are described by the following power-law corrections,
rµ
rµ0
“
˜
rT
T0
¸aµ
, (3.11)
rλ
rλ0
“
˜
rT
T0
¸aλ
, (3.12)
where aµ and aλ are functions of rZ, ĄZ22, and rC, and are stored in the flamelet library
along with T0, µ0, and λ0.
In order to completely close the system of equations (7.1)-(7.6), the equation of
state for an ideal gas is used to compute the pressure
¯p “ ¯ρĄRT. (3.13)
Although mixture properties not depending on temperature (e.g., gas constant) are
not very sensitive to the compressibility effects (see Section 6.1), the source term of
the progress variable is very sensitive to these perturbations. Therefore, a rescaling
for this term is introduced to account for these perturbations due to compressibility.
The filtered source term in Eq. (2.55) is thus rescaled as
9ωC
9ωC0
“
ˆ
¯ρ
¯ρ0
˙aρ
exp
„
´Ta
ˆ
1
rT
´
1
T0
˙
, (3.14)
where ¯9ωC0 is the tabulated source term computed at a background pressure p0. As
55. 3.3. COMPRESSIBLE FLAMELET/PROGRESS VARIABLE APPROACH 29
before, the values T0, ρ0, aρ, and Ta are computed in a preprocessing step to describe
the dependency of the source term on the mixture temperature and pressure, and are
then tabulated as a function of rZ, ĄZ22, and rC.
56. 30 CHAPTER 3. COMBUSTION MODELING FOR COMPRESSIBLE FLOWS
57. Chapter 4
Verification and Validation of DNS
Solver
The DNS solver with finite-rate chemistry “Ferdowsi”, developed in this study, is based
on the CTR solver “Charles”, which has been extensively validated for turbulent flows.
Ferdowsi uses an unstructured finite volume method with explicit third-order Runge-
Kutta time integration. The spatial discretization relies on a hybrid central and ENO
method, in which a shock sensor is used to identify the cells where the ENO scheme
should be applied. In all cases considered here, a hydrogen chemistry mechanism
(Hong et al., 2011) based on an improved GRI3.0 chemical kinetics mechanism (Bates
et al., 2001; Herbon et al., 2002) has been used. This mechanism is based on nine
species, i.e., O2, H2, O, H, OH, H2O, HO2, H2O2 and N2, and 20 reactions, but does
not include the nitrogen chemistry.
Four different verification and validation studies relevant to this work are summa-
rized in this Chapter testing different capabilities of the code. First, a homogeneous
combustion case will be considered followed by a premixed one-dimensional flame.
Subsequently, results for laminar and turbulent mixing layers are presented.
31
58. 32 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
4.1 Homogeneous reactor
Consider a homogeneous mixture of fuel and oxidizer in an enclosed adiabatic vessel.
Initial temperature and pressure of the mixture are assumed to be Tp0q and pp0q,
respectively. During the auto-ignition process, mixture pressure and temperature
increase due to the conversion of chemical energy to sensible energy (i.e. heat release),
while the mixture density remains constant. The mixture reaches equilibrium after
a sufficiently long time, where forward and reverse reactions balance each other, and
consequently the heat release rate is zero.
Auto-ignition of the stoichiometric mixture of hydrogen and air at pp0q “ 105
Pa
and Tp0q “ 1200 K are considered here. The results of the Ferdowsi solver are com-
pared with the results of the FlameMaster program (Pitsch, 1998a). It should be
noted that Ferdowsi solves a full, three-dimensional set of equations (see Section 2.1),
while FlameMaster solves the zero-dimensional equations in time. Although the con-
vection and diffusion terms are not neglected in Ferdowsi, they are zero because the
flow remains homogeneous.
Figure 4.1 compares the temperature and pressure from Ferdowsi to the FlameMas-
ter results. Figures 4.2 and 4.3 show the temporal evolution of the minor and major
species. Ferdowsi and FlameMaster results are almost identical. Even minor species
like H2O2 (Fig. 4.3c) and HO2 (Fig. 4.3d) are predicted accurately with the Ferdowsi
solver. It should be noted that Ferdowsi is an explicit solver which uses a constant
time step, whereas FlameMaster is implicit with adaptive time stepping.
59. 4.1. HOMOGENEOUS REACTOR 33
Figure 4.1: Temporal variation of temperature (top) and pressure (bottom) in a
homogeneous reactor with Tp0q “ 1200 K and pp0q “ 1 bar.
60. 34 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
(a) YH2 (b) YO2
(c) YH2O (d) YOH
Figure 4.2: Temporal evolution of the major and minor species mass fractions in a
homogeneous reactor with Tp0q “ 1200 K and pp0q “ 1 bar.
61. 4.1. HOMOGENEOUS REACTOR 35
(a) YH (b) YO
(c) YH2O2 (d) YHO2
Figure 4.3: Temporal evolution of the major and minor species mass fractions in a
homogeneous reactor with Tp0q “ 1200 K and pp0q “ 1 bar.
62. 36 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
4.2 Premixed flame
Figure 4.4 illustrates a one-dimensional planar wave (flame) moving in a mixture of
fuel and oxidizer. Diffusion and reaction only occur in the vicinity of the flame. Using
a reference frame fixed at the flame location, the problem is simplified to a steady
one-dimensional system. Depending on the mixture conditions, the flame could be
either a compression wave or an expansion wave. Expansion waves propagate at
subsonic speed into the unburned mixture and are called deflagration waves, whereas
detonation waves (compression waves) propagate at supersonic speed.
Figure 4.4: Schematic of a deflagration wave using a reference frame fixed on the
flame. Subscript u corresponds to the unburned and b to the burned condition.
Figure 4.5 shows temperature and pressure as a function of the spatial coordinate
x for a deflagration premixed flame in a stoichiometric mixture of hydrogen and air
with Tu “ 300 K and pu “ 1 bar. The FlameMaster code uses the low Mach number
approximation, which assumes that the thermodynamic pressure remains constant,
while Ferdowsi solves a compressible system. Although pressure increases in the
preheat zone due to heat release and decreases as the flow expands, the maximum
change in the pressure is less than 0.1%.
Figures 4.6 and 4.7 show the major and minor species mass fraction profiles. Al-
though Ferdowsi uses a much coarser grid (100 cells uniformly spaced in the stream-
wise direction), results show good agreement even for minor species like H2O2 and
HO2
63. 4.2. PREMIXED FLAME 37
Figure 4.5: Temperature (top) and pressure (bottom) variations in a premixed flame
with Tu “ 300 K and pu “ 1 bar.
64. 38 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
(a) YH2 (b) YO2
(c) YH2O (d) YOH
Figure 4.6: Spatial variations of the major and minor species mass fractions in a
premixed flame with Tu “ 300 K and pu “ 1 bar.
65. 4.2. PREMIXED FLAME 39
(a) YH (b) YO
(c) YH2O2 (d) YHO2
Figure 4.7: Spatial variations of the major and minor species mass fractions in a
premixed flame with Tu “ 300 K and pu “ 1 bar.
66. 40 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
4.3 Laminar reacting mixing layer
Ju and Niioka (1994) studied ignition processes in a two-dimensional spatially evolving
mixing layer (see Fig. 4.8) using finite-rate chemistry. The fuel stream is a mixture
of hydrogen and nitrogen with XH2 “ 0.62 and XN2 “ 0.38, and the oxidizer stream
is air; both streams are supersonic.
Figure 4.8: Schematic of a laminar reacting mixing layer.
The computational mesh has 300 grid points in each direction, where about 50
grid points are located in the mixing layer (based on the vorticity thickness at the
end of the domain). The mixing layer reaches a steady state after 10 flow-through
times. Contours of temperature, H2O mass fraction (i.e. reaction progress variable),
OH mass fraction are displayed in Fig. 4.9.
Figure 4.10 shows the effect of the oxidizer Mach number on the ignition point.
Results of the current study demonstrate the same behavior as observed by Ju and
Niioka (1994). It should be noted that chemistry mechanisms and numerical schemes
are not identical, which can explain the observed differences. Figure 4.10 suggests
that the ignition point (xign) is moved downstream by increasing the oxidizer Mach
number for Mox ă 3.0. This behavior is due to an increase in the convective velocity,
which moves the ignition point downstream. However, for Mox ą 3.0, the ignition
point moves upstream. The viscous heating is stronger at higher Mach numbers, and
thus the temperature is increased, which in turn promotes ignition.
67. 4.3. LAMINAR REACTING MIXING LAYER 41
(a) T
(b) YH2O
(c) YOH
Figure 4.9: Contours of temperature (top), reaction progress variable (middle), and
OH mass fraction (bottom) in a laminar reacting mixing layer with Mox “ 1.84.
68. 42 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
Figure 4.10: Effect of the oxidizer Mach number on the ignition point. Comparison
of the current study (blue symbols) and Ju and Niioka (1994) (red symbols).
4.4 Turbulent inert mixing layer
DNS of turbulent reacting mixing layers are presented in Chapter 5. Here, the Fer-
dowsi solver is validated against experimental measurements of Jones and Spencer
(1971) and Bell and Mehta (1990) for a low Mach number (Mc “ 0.3) inert tem-
poral mixing layer. DNS results are also compared with those of Rogers and Moser
(1994), who performed DNS of an incompressible mixing layer, and with those of a
compressible mixing layer with Mc “ 0.3 by Pantano and Sarkar (2002). It should be
69. 4.4. TURBULENT INERT MIXING LAYER 43
noted that both streams are assumed to be air (mixture of nitrogen and oxygen) at
300 K. The full set of equations (Section 2.1) are solved. Although the mixture is not
assumed to be calorically perfect (i.e. specific heat are function of temperature), the
variations in the value of specific heats are not significant. For more details about the
initial conditions, boundary conditions, and computational mesh refer to Chapter 5.
Figure 4.11 shows an instantaneous snapshot of vortical structures colored by
u1{∆u. Since the densities of the upper and lower streams are the same, the mixing
layer remains at the center of the domain.
The cross-stream profile of self-similar velocity fluctuations
b
Ću2
1u2
1 is compared
against experiments (symbols) and previous DNS studies (lines) in Fig. 4.12. The
current DNS study shows better agreement with the experimental measurements,
especially for the peak value, which is significantly under-predicted by both Rogers
and Moser (1994) and Pantano and Sarkar (2002).
70. 44 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
Figure 4.11: Iso-surface of vorticity magnitude colored by u1{∆u. Contours are from
´0.6 (blue) to 0.6 (red).
71. 4.4. TURBULENT INERT MIXING LAYER 45
Figure 4.12: Spatial variation of streamwise root-mean-square velocity in a turbu-
lent inert mixing layer. Lines are this work (black line) and past DNS studies of
Rogers and Moser (1994) and Pantano and Sarkar (2002); symbols are experimental
measurements of Jones and Spencer (1971) and Bell and Mehta (1990).
72. 46 CHAPTER 4. VERIFICATION AND VALIDATION OF DNS SOLVER
73. Chapter 5
Direct Numerical Simulation of
Reacting Mixing Layer
Supersonic combustion is a complex process, which involves turbulent flows, chem-
istry, shock/expansion waves, and their interactions. Understanding these compli-
cated phenomena requires high quality data. Obtaining these data experimentally
is extremely challenging in the harsh supersonic environment. A detailed numerical
simulation with accurate chemistry, and flow characterization is therefore necessary
to investigate the physics and interactions in supersonic combustion.
In this chapter, governing equations discussed in section 2.1 are numerically solved
using the Ferdowsi solver for a temporal mixing layer configuration. Chapter 4 shows
validation studies for the the Ferdowsi solver, for both laminar reacting mixing layers
(section 4.3) and a turbulent inert mixing layer (section 4.4). All the simulations were
checked to make sure that they are grid-independent, domain-size independent, and
self-similar. These databases are used in Chapter 6 to validate the CFPV approach.
47
74. 48 CHAPTER 5. DNS OF REACTING MIXING LAYER
5.1 Problem formulation
5.1.1 Flow configuration
A schematic of the considered case is shown in Fig. 5.1. The flow configuration is a
temporal mixing layer, where the upper stream is air at 1500 K, and the lower stream
is fuel, which is an equimolar mixture of hydrogen and nitrogen at 500 K. The pressure
is initially 2 bar in the fuel and oxidizer streams. The fuel and oxidizer streams have
an equal velocity magnitude, but flow in opposite directions, Uox “ ´Uf “ U. The
computational domain is a rectangular box with length L1, width L2, and depth L3.
Figure 5.1: Schematic of a temporal mixing layer. The upper stream is oxidizer, and
the lower stream is fuel. Red color outline represents a diffusion flame.
5.1.2 Simulation parameters
The convective Mach number, introduced by Bogdanoff (1983), characterizes the
compressibility effects, and is defined as
Mc “
∆u
paox ` af q
, (5.1)
where ∆u “ Uox ´ Uf is the velocity difference of the upper and lower streams, and
a is the speed of sound.
75. 5.1. PROBLEM FORMULATION 49
The turbulent Mach number is computed from
Mt “
b
2
3
k
˜a
, (5.2)
where k and ˜a are the turbulent kinetic energy and Favre-averaged speed of sound at
the center of mixing layer.
The momentum thickness, δθ, is defined as
δθ “
1
4
ż 8
´8
¯ρ
ρO
«
1 ´
ˆ
ru1
U
˙2
ff
dx2, (5.3)
and the vorticity thickness is computed using the maximum slope of the streamwise
velocity profile
δω “
∆u
pB¯u1{Bx2q |max
. (5.4)
The Reynolds number is typically computed based on a length scale of the mixing
layer (e.g. vorticity thickness) and the velocity difference such as
Reδω “
δω∆u
˜ν
, (5.5)
where ˜ν is the kinematic viscosity at the center of mixing layer.
The simulations cover convective Mach numbers from subsonic to supersonic. Ta-
ble 5.1 summarizes some of the final parameters of the supersonic simulations.
Mc Mt Reδω
1.1 0.21 18036
1.8 0.28 15635
2.5 0.41 23057
Table 5.1: Parameters of the simulations.
76. 50 CHAPTER 5. DNS OF REACTING MIXING LAYER
5.1.3 Mathematical description
Transport equations for mass, momentum, total energy, and species mass fractions
presented in Chapter 2 are solved numerically using the Ferdowsi solver (Chapter 4).
A detailed chemistry mechanism with 9 species and 29 reactions for hydrogen is
used (Hong et al., 2011). Diffusion velocities (Section 2.1.3) are computed assuming
constant Lewis numbers, which are presented in Table 5.2. The parallel DNS solver
is based on an unstructured finite volume method with explicit third-order Runge-
Kutta time integration. The spatial discretization relies on a hybrid central and ENO
method, in which a shock sensor is used to identify the faces where the ENO scheme
should be applied. An efficient sensor based on the dilatation and gradient of species
has been developed to minimize the numerical dissipation and dispersion errors.
Species H2 O2 N2 H O OH HO2 H2O2 H2O
Lewis number 0.32 1.15 1.32 0.19 0.75 0.76 1.16 1.17 0.86
Table 5.2: Species Lewis numbers.
5.1.4 Initial condition
The mean streamwise velocity component is initialized with a hyperbolic-tangent
profile ru1 “ ∆u{2 tanhpx2{2δθ0 q, where δθ0 is the initial momentum thickness. Mean
velocities in the lateral and spanwise directions are assumed to be zero.
Two different approaches have been used to generate artificial velocity fluctuations
superimposed on the mean velocity to accelerate transition to turbulence. In the
first approach, perturbations are obtained from a linear stability analysis (Sandham
and Reynolds, 1990, 1991). This approach has been used by Vreman et al. (1996)
for various convective Mach numbers. In the second approach, synthetic turbulence
is generated using a digital filtering approach (di Mare et al., 2006; Klein et al.,
2003). Profiles of the Reynolds stress tensor components from the self-similar inert
compressible mixing layers (section 4.4) are rescaled and used as inputs. The peak
77. 5.1. PROBLEM FORMULATION 51
values of the initial Reynolds stress components are assumed to be 1% of their self-
similar counterpart. It is found that mixing layers reach self-similarity faster using
the digital filtering approach. Therefore, the second approach is used in the results
presented in this chapter.
Mixture composition, density, and internal energy are initialized using a simplified
version of the problem: either by solving the steady flamelet equations in mixture frac-
tion space assuming a constant background pressure, or by solving a one-dimensional
mixing layer with the same parameters and boundary conditions in the x2 direction.
The corresponding one-dimensional mixing layer grows in time until ignition occurs.
Both approaches have been found equally effective to ignite the mixing layer, and the
resulting self-similar mass fraction profiles are almost identical. However, the latter
approach provides more consistent flow fields, and is used in this study.
5.1.5 Boundary conditions
The flow is periodic in the streamwise and spanwise directions, and characteristic
boundary conditions (see Baum et al., 1995; Poinsot and Lele, 1992), are used in the
lateral direction to let acoustic waves exit the domain.
5.1.6 Ensemble averaging
The mean of any variable φ is defined by taking the ensemble average over the ho-
mogeneous directions x1 and x3
¯φ px2q “
1
L1L3
ż ż
φ px1, x2, x3q dx1dx3. (5.6)
The Favre average of φ is computed from rφ “ ρφ{¯ρ. Perturbations with respect to
the Favre average are then computed similar to Eq. (2.28).
5.1.7 Computational mesh
The base numerical domain is a rectangular box with 7.5 mm length, 5.0 mm width,
and 1.875 mm depth. About 47 million cells (768 ˆ 320 ˆ 192) are used to discretize
78. 52 CHAPTER 5. DNS OF REACTING MIXING LAYER
the domain. The grid is uniform in the streamwise and spanwise directions and
slightly stretched in the lateral direction. DNS is supposed to resolve all length and
time scales, and therefore the grid spacing should be fine enough to capture small
dissipative eddies and flame structures. The ratio of grid spacing ∆ to the Kolmogorov
length scale η should be sufficiently small, or, equivalently, a sufficiently large κmaxη is
necessary, where κmax, is the maximum wavenumber that the grid can handle. From
the analysis of the dissipation spectrum it was suggested that κmaxη ě 1.5 is a good
criterion for the grid resolution in DNS of inert isotropic turbulence (see Pope, 2000;
Yeung and Pope, 1989). κmaxη is plotted against the self-similar lateral coordinate
x2{δθ in Fig. 5.2. This criterion was developed for low Mach number non-reacting
turbulent simulations, and it does not consider combustion-relevant scales. Therefore,
an additional simulation with the same parameters, but with 50% more grid points
in each direction (a total of 159 million cells) was performed and compared with the
original results. Figures 5.3 and 5.4 show important radical mass fractions and the
non-dimensional dissipation rate, respectively. It is clear that the original grid spacing
is fine enough to capture both turbulence and combustion scales.
79. 5.1. PROBLEM FORMULATION 53
Figure 5.2: Non-dimensional maximum wavenumber, κmaxη, plotted against x2{δθ:
——, κmaxη; ´ ´ ´, κmaxη “ 1.5 line.
-10 -5 0 5 10
0
0.005
0.01
0.015
0.02
Figure 5.3: Mass fractions of important radicals plotted against x2{δθ: ——, fine grid;
´ ´ ´, coarse grid.
80. 54 CHAPTER 5. DNS OF REACTING MIXING LAYER
Figure 5.4: Dissipation rate plotted against x2{δθ: ——, fine grid; ´ ´ ´, coarse grid.
5.1.8 Domain size effect
The domain size should be large enough to allow the large-scale structures to evolve
in time without interference. Another simulation with a larger domain, 50% increase
in size in each direction, was performed to study the effect of the domain size on
the results. Mass fractions of important minor species and average velocity in the
streamwise direction are plotted in Figs. 5.5 and 5.6, respectively. The increase in
the domain size does not have a strong effect on the results.
81. 5.1. PROBLEM FORMULATION 55
Figure 5.5: Mass fractions of important radicals plotted against x2{δθ:——, big do-
main; ´ ´ ´, small domain.
82. 56 CHAPTER 5. DNS OF REACTING MIXING LAYER
Figure 5.6: Normalized averaged velocity in the streamwise direction plotted against
x2{δθ:——, big domain; ´ ´ ´, small domain.
83. 5.2. STRUCTURE OF THE MIXING LAYER 57
5.2 Structure of the mixing layer
5.2.1 Self-similarity
It has been observed that after an initial transient, the mixing layer thickness (e.g.,
momentum thickness δθ, or vorticity thickness δw) grows linearly (e.g. Ho and Huerre,
1984; Pantano and Sarkar, 2002; Rogers and Moser, 1994). This region is often
considered the self-similar region, where the ensemble averaged profiles (by scaling
with the velocity difference and mixing layer length scale) collapse onto a single profile.
The time history of the non-dimensional momentum thickness, δθ{δ0
θ , and the non-
dimensional vorticity thickness are shown in Fig. 5.7. After t∆u{δ0
θ „ 1000 both
curves show a linear growth in time. The momentum thickness, however, varies
smoothly in time, and is chosen as the mixing layer length scale in this study.
A more accurate criterion to check the self-similarity is the collapse of rescaled
profiles. For instance, the integral of the dissipation rate and production rate, defined
in section 5.3, should remain constant after the flow evolves self-similarity. The
integral of the dissipation rate and production rate in the lateral direction x2
E “
ż 8
´8
dx2, (5.7)
P “
ż 8
´8
P dx2, (5.8)
scale with ∆u3
, and, thus, should remain constant after reaching self-similarity. A
much longer time is required for the flow to evolve self-similarity based on Fig. 5.8,
where the profiles reach a plateau after t∆u{δ0
θ „ 1900. Self-similarity is also ob-
served by the collapse of the Favre-averaged streamwise velocity component, ru1, and
streamwise Reynolds stress, Ću2
1u2
1, in Fig. 5.9.
Note that although self-similarity analysis has only been illustrated for the Mc “ 1.1
case, the same conclusions are obtained for all the cases. All the profiles and flow
fields presented in this study are sampled after the flow evolves self-similarity.
84. 58 CHAPTER 5. DNS OF REACTING MIXING LAYER
0 500 1000 1500 2000 2500
0
5
10
15
20
25
0 500 1000 1500 2000 2500
0
5
10
15
20
25
Figure 5.7: Time evolution of the momentum thickness (left), and the vorticity thick-
ness (right) for the Mc “ 1.1 case.
0 500 1000 1500 2000 2500
0
0.0025
0.005
0.0075
0.01
0 500 1000 1500 2000 2500
0
0.0025
0.005
0.0075
0.01
Figure 5.8: Time evolution of the integrated dissipation rate (left), and turbulent
production rate (right) for the Mc “ 1.1 case.
85. 5.2. STRUCTURE OF THE MIXING LAYER 59
-15 -10 -5 0 5 10 15
-0.5
-0.25
0
0.25
0.5
(a)
-15 -10 -5 0 5 10 15
0
0.005
0.01
0.015
0.02
(b)
Figure 5.9: Collapse of the Favre-averaged streamwise velocity component (left), and
streamwise Reynolds stress (right) for the Mc “ 1.1 case; —, t∆u{δ0
θ “ 1900; ´ ´ ´,
t∆u{δ0
θ “ 2000; ´.´, t∆u{δ0
θ “ 2100; ..., t∆u{δ0
θ “ 2200; ´..´, t∆u{δ0
θ “ 2300.
5.2.2 Mixture composition
Figure 5.10 shows the species mixture fractions for the Mc “ 1.1 reacting case.
These snapshots correspond to a non-dimensional time in the region of self similarity
(t∆u{δ0
θ “ 2200). The blue line represents the Z “ Zst surface, which approximately
marks the location of the flame. The stoichiometric line works as a sink for H2 and O2
(Figs. 5.10a and 5.10b). Notice that all minor species shown in Figs. 5.10e–5.10i are
only present in the vicinity of the flame, and more towards the oxidizer side, except
for H radicals which are observed almost everywhere in the mixing layer. In fact,
the mass fraction of water (Fig. 5.10d) is correlated with the H radical mass fraction
(Fig. 5.10f).
The mixture composition for the Mc “ 2.5 reacting case is shown in Fig. 5.11. These
snapshots correspond to a time when both cases have almost an equal momentum
thickness. Notice that there are more small scales in the Mc “ 1.1 case, although its
Reynolds number is lower. At higher convective Mach number, the compressibility
effects decrease the shear layer growth rate (see Freund et al., 2000; Pantano and
Sarkar, 2002; Sarkar, 1995; Vreman et al., 1996), and also the viscosity increases due
86. 60 CHAPTER 5. DNS OF REACTING MIXING LAYER
(a) YH2
(b) YO2
(c) YN2
Figure 5.10: For the caption see the next page.
87. 5.2. STRUCTURE OF THE MIXING LAYER 61
(d) YH2O
(e) YOH
(f) YH
Figure 5.10: For the caption see the next page.
88. 62 CHAPTER 5. DNS OF REACTING MIXING LAYER
(g) YO
(h) YHO2
(i) YH2O2
Figure 5.10: Instantaneous species mass fractions in the x3 “ 0 plane for the Mc “ 1.1
case. Blue line shows the Z “ Zst line.
89. 5.2. STRUCTURE OF THE MIXING LAYER 63
to higher levels of temperature, which in turn leads to an increase in the Kolmogorov
length scale. Relatively higher levels of OH mass fraction are observed in Fig. 5.11e
for the Mc “ 2.5 case. The reason of this observation is again the higher temperature
levels in the Mc “ 2.5 case. Temperature contours are shown in Fig. 5.12. Note that
the maximum temperature is higher than 3000 K.
Scatter plots of the major and important minor species versus mixture fraction are
plotted in Figs. 5.13 and 5.14. Figure 5.15 shows the iso-surface of the OH mass
fraction suggesting local regions of extinction.
90. 64 CHAPTER 5. DNS OF REACTING MIXING LAYER
(a) YH2
(b) YO2
(c) YN2
Figure 5.11: For the caption see the next page.
91. 5.2. STRUCTURE OF THE MIXING LAYER 65
(d) YH2O
(e) YOH
(f) YH
Figure 5.11: For the caption see the next page.
92. 66 CHAPTER 5. DNS OF REACTING MIXING LAYER
(g) YO
(h) YHO2
(i) YH2O2
Figure 5.11: Instantaneous species mass fractions in x3 “ 0 plane for the Mc “ 2.5
case. Blue line shows the Z “ Zst.
93. 5.2. STRUCTURE OF THE MIXING LAYER 67
Figure 5.12: Instantaneous temperature contours in the x3 “ 0 plane for the Mc “ 2.5
case. Blue line shows the Z “ Zst.
Figure 5.13: Scatter plots of O2 (black), H2 (blue), and H2O (red) against mixture
fraction for the Mc “ 2.5 case. Lines are ensemble averages.
94. 68 CHAPTER 5. DNS OF REACTING MIXING LAYER
Figure 5.14: Scatter plots of minor radicals O (black), H (blue), and OH (red) against
mixture fraction for the Mc “ 2.5 case. Lines are ensemble averages.
Figure 5.15: Iso-surface of YOH “ 0.02 colored by temperature from 1500 K (blue) to
2500 K (red) for the Mc “ 2.5 case.
95. 5.2. STRUCTURE OF THE MIXING LAYER 69
5.2.3 Flame/turbulence interactions
Interactions between flame and turbulence are relatively strong in the Mc “ 1.1
and Mc “ 2.5 cases, as shown in Figs. 5.16 and 5.17. Diluting the fuel with ni-
trogen increases the value of the stoichiometric mixture fraction moving the flame
closer to the center of mixing layer, where turbulence is stronger, thereby intensifying
flame/turbulence interactions. This is the main reason that hydrogen is diluted with
nitrogen in this study.
(a) Vorticity magnitude
(b) Z
Figure 5.16: Vorticity magnitude and mixture fraction contours in x3 “ 0 plane at
t∆u{δ0
θ “ 2200 for the Mc “ 1.1 case. Blue line shows the Z “ Zst.
96. 70 CHAPTER 5. DNS OF REACTING MIXING LAYER
Figure 5.17: Iso-surface of the second invariant of the velocity gradient tensor Q
colored by u1{∆u from -0.6 (blue) to 0.6 (red). Iso-surface of the stoichiometric
mixture fraction (red surface) indicates approximate location of the flame for the
Mc “ 2.5 case.
97. 5.3. REYNOLDS STRESS BUDGET 71
0 0.005 0.01 0.015 0.02
Figure 5.18: Vorticity magnitude contours in x3 “ 0 plane (left), and streamwise
Reynolds stress, R11 (right) at t∆u{δ0
θ “ 2300 in the self-similar region. Blue line
shows the Z “ Zst, and green line represents ru1 “ 0.
After the flow reaches self-similarity, the streamwise Reynolds stress has two max-
ima as shown in Figs. 5.18 and 5.9b, which are approximately located at x2{δθ „ 1,
and x2{δθ „ 4. The first maximum, x2{δθ „ 1, is located at the center of mixing layer,
where the mean velocity gradient is maximum. At this location, the production of
the R11 is the dominant term in the Reynolds stress transport equation. The second
maximum is approximately located at the stoichiometric mixture fraction iso-surface,
Z “ Zst, which corresponds to the location of the maximum heat release.
5.3 Reynolds stress budget
Transport equation for the Reynolds stress tensor reads
B
Bt
´
ρu2
i u2
j
¯
`
B
Bxk
´
rukρu2
i u2
j
¯
“ ρ pPij ´ ijq ` Tij ` Πij ` Σij, (5.9)
where the terms on the right hand side are the turbulent production, dissipation rate,
turbulent transport, pressure-strain, and mass flux coupling, respectively, and are
defined as
Pij “ ´Ću2
j u2
k
Brui
Bxk
´ Ću2
i u2
k
Bruj
Bxk
, (5.10)
98. 72 CHAPTER 5. DNS OF REACTING MIXING LAYER
ij “
1
ρ
˜
τ1
ki
Bu2
j
Bxk
` τ1
kj
Bu2
i
Bxk
¸
, (5.11)
Tij “
B
Bxk
”
´ρu2
i u2
j u2
k ` u2
j σ1
ki ` u2
i σ1
kj
ı
, (5.12)
Πij “ p1
ˆ
Bu2
i
Bxj
`
Bu2
j
Bxi
˙
, (5.13)
Σij “ u2
j
Bσki
Bxk
` u2
i
Bσkj
Bxk
. (5.14)
The mass flux coupling term is relatively smaller than the other terms in Eq. (5.9).
The transport term is divergence of the energy flux, which merely redistributes
Reynolds stress components in space. The remaining terms could act as sources
or sinks of the Reynolds stress tensor.
The transport equation of the turbulent kinetic energy is derived by multiplying
Eq. (5.9) by contracting the free indices and dividing by two
B
Bt
pρkq `
B
Bxj
prujkq “ ρ pP ´ q ` T ` Π ` Σ, (5.15)
where k “ 1{2Ću2
i u2
i , and each term on the right hand side is derived by contracting
the indices of its counterpart in Eqs. (5.10)–(5.14) and dividing by two.
Integrating the Reynolds transport equation in the x2 direction results in
d
dt
ż 8
´8
Rijdx2 “
ż 8
´8
Pijdx2 ´
ż 8
´8
ijdx2 `
ż 8
´8
Πij
ρ
dx2, (5.16)
which indicates that the rate of change of the integrated Reynolds stress tensor only
depends on the production, dissipation, and pressure-strain terms.
Figures 5.19 and 5.20 show the integrated Reynolds stress budgets for the reacting
Mc “ 1.1 and Mc “ 2.5 cases, respectively. The same overall behavior is observed
for both cases. All the profiles reach a constant value after a sufficiently long time,
which confirms the self-similarity of the flow. The entire production of the turbulent
energy occurs in the R11 component of the Reynolds stress, i.e. this term is zero
in the R22, R33 transport equations, and is negative in the R12 transport equation.
99. 5.3. REYNOLDS STRESS BUDGET 73
The nonlinear pressure-strain interactions exchange energy between Reynolds stress
components. The pressure-strain term is a drain of turbulent energy in the streamwise
direction, while it works as the only source of the turbulent energy for other directions.
100. 74 CHAPTER 5. DNS OF REACTING MIXING LAYER
0 500 1000 1500 2000 2500
-0.01
0
0.01
0.02
(a) R11
0 500 1000 1500 2000 2500
-0.01
0
0.01
0.02
(b) R22
0 500 1000 1500 2000 2500
-0.01
0
0.01
0.02
(c) R33
0 500 1000 1500 2000 2500
-0.01
0
0.01
0.02
(d) R12
Figure 5.19: Time evolution of the integrated production term (—), the integrated
viscous dissipation (´ ´ ´), and the integrated pressure-strain term (´.´) for the
reacting Mc “ 1.1 case. All the integrated terms are rescaled with 1{∆u3
.
101. 5.3. REYNOLDS STRESS BUDGET 75
0 1000 2000 3000 4000
-0.01
0
0.01
0.02
(a) R11
0 1000 2000 3000 4000
-0.01
0
0.01
0.02
(b) R22
0 1000 2000 3000 4000
-0.01
0
0.01
0.02
(c) R33
0 1000 2000 3000 4000
-0.01
0
0.01
0.02
(d) R12
Figure 5.20: Time evolution of the integrated production term (—), the integrated
viscous dissipation (´ ´ ´), and the integrated pressure-strain term (´.´) for the
reacting Mc “ 2.5 case. All the integrated terms are rescaled with 1{∆u3
.
102. 76 CHAPTER 5. DNS OF REACTING MIXING LAYER
5.4 Shocklet/flame interaction
This configuration is not perfectly suited for the shock-flame interaction study due
to the absence of strong shocks. There are some shocklets, nevertheless, impinging
the flame as shown in Fig. 5.21. These shocklets should have all the characteristics
of a typical shock wave, e.g. satisfying the jump conditions (Lee et al., 1991); the
shocklets in this work satisfy these conditions.
Figure 5.21 reveals that the heat release rate is strongly negative in the shock-
let/flame interaction regions. The heat release rate budget, i.e. the contribution of
different reactions to the overall heat release rate, will be studied to further investi-
gate this observation. Some of the important reactions for the hydrogen chemistry
are shown in Table 5.3. These bimolecular reactions are written such that the for-
ward reaction is exothermic; the heats of reaction are shown in the third column of
Table 5.3. Consider a typical exothermic bimolecular reaction
A ` B é C ` D. (5.17)
The reaction rate can be computed from Eq. (2.22)
9ω “ 9ωf ´ 9ωb, (5.18)
where 9ωf and 9ωb are the forward and backward reaction rates
9ωf “ kf
ρYA
WA
ρYB
WB
, (5.19)
9ωb “ kb
ρYC
WC
ρYD
WD
. (5.20)
The specific reaction rate constants are computed from the Arrhenius law (Eqs. (2.23)
and (2.24))
kf “ Af Taf
exp
ˆ
´
Ef
RuT
˙
, (5.21)
