Thesis augmented

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

http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/sm021qx0479
© 2014 by Amirreza Saghafian. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-
Noncommercial 3.0 United States License.
ii

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
Matthias Ihme
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

To the cornerstones of my heart
Hoora and Nika
v

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

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

Contents
Abstract ix
Acknowledgments xi
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Current state-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Compressible mixing layer . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Combustion modeling . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Governing Equations 9
2.1 Direct numerical simulation . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Caloric equation of state . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Thermodynamic equation of state . . . . . . . . . . . . . . . . 11
2.1.3 Diffusion velocity . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.4 Viscous stress tensor . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.5 Heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.6 Chemical source terms . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Large eddy simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 LES filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Transport equations . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
xiii

2.2.4 Thermodynamic equation of state . . . . . . . . . . . . . . . . 18
2.3 Flamelet-based combustion modeling . . . . . . . . . . . . . . . . . . 19
2.3.1 Mixture fraction . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Reaction progress variable . . . . . . . . . . . . . . . . . . . . 20
3 Combustion Modeling for Compressible Flows 23
3.1 Low Mach number flamelet/progress variable approach . . . . . . . . 24
3.2 Compressible extended flamelet table approach . . . . . . . . . . . . . 25
3.3 Compressible flamelet/progress variable approach . . . . . . . . . . . 25
4 Verification and Validation of DNS Solver 31
4.1 Homogeneous reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Premixed flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Laminar reacting mixing layer . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Turbulent inert mixing layer . . . . . . . . . . . . . . . . . . . . . . . 42
5 DNS of Reacting Mixing Layer 47
5.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.1 Flow configuration . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . 48
5.1.3 Mathematical description . . . . . . . . . . . . . . . . . . . . 50
5.1.4 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.6 Ensemble averaging . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.7 Computational mesh . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.8 Domain size effect . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Structure of the mixing layer . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.2 Mixture composition . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.3 Flame/turbulence interactions . . . . . . . . . . . . . . . . . . 69
5.3 Reynolds stress budget . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Shocklet/flame interaction . . . . . . . . . . . . . . . . . . . . . . . . 76
xiv

6 Validation Studies of CFPV Approach 81
6.1 A priori analysis of compressibility effects . . . . . . . . . . . . . . . 81
6.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Validation of the CFPV model using DNS data . . . . . . . . . . . . 96
6.3.1 A priori analysis . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.2 A posteriori analysis . . . . . . . . . . . . . . . . . . . . . . . 98
7 Application of CFPV Approach 101
7.1 Mathematical framework . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Hydrogen jet in a supersonic cross-flow . . . . . . . . . . . . . . . . . 103
7.2.1 Flow configuration . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . 105
7.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 HIFiRE scramjet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3.1 Wall modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 117
7.3.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . 118
7.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8 Conclusions 125
xv

List of Tables
5.1 Parameters of the simulations. . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Species Lewis numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Some of the reactions in the chemistry mechanism. Third column
shows the heat released in the forward direction. . . . . . . . . . . . . 77
7.1 Inﬂow boundary condition parameters. . . . . . . . . . . . . . . . . . 118
xvii

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
xix

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
xx

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 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

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

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

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

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

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

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

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

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).

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.

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.

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.

" Application of the CFPV model to the HIFiRE scramjet.

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

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

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.

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

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)

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

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

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

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

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)

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

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

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.

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

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

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

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
´
¯
“ Yα,0
´
¯
, (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
´
¯
“ R0
´
¯
, (3.4)
and can be directly tabulated.

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

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

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.

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

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.

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.

(a) YH2 (b) YO2
(c) YH2O (d) YOH
Figure 4.2: Temporal evolution of the major and minor species mass fractions in a

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

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

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.

(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.

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.

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.

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.

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

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).

Figure 4.11: Iso-surface of vorticity magnitude colored by u1{∆u. Contours are from
´0.6 (blue) to 0.6 (red).

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).

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

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 “ Úf “ 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.

5.1. PROBLEM FORMULATION 49
The turbulent Mach number is computed from
Mt “
b
2
3
k
ã
, (5.2)
where k and ã 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.

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

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

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.

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.

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.

Figure 5.5: Mass fractions of important radicals plotted against x2{δθ:——, big do-
main; ´ ´ ´, small domain.

Figure 5.6: Normalized averaged velocity in the streamwise direction plotted against
x2{δθ:——, big domain; ´ ´ ´, small domain.

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.

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.

-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

(a) YH2
(b) YO2
(c) YN2
Figure 5.10: For the caption see the next page.

(d) YH2O
(e) YOH
(f) YH

(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.

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.

(a) YH2
(b) YO2
(c) YN2

(d) YH2O
(e) YOH
(f) YH

(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.

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.

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.

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.

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.

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)

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.

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.

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
.

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
.

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)

5.4. SHOCKLET/FLAME INTERACTION 77
kb “ AbTab
exp
ˆ
´
Eb
RuT
˙
. (5.22)
Note that the backward activation energy Eb is larger than the forward activation
energy Ef , since species A and B have a higher level of energy than species C and D.
Consequently, a sudden increase in the temperature (e.g. due to shocklets) ampliﬁes
kb more than ka. This change could potentially make the reaction rate negative if the
jump in the temperature is large enough. Thus, reaction (5.17) is forced to proceed
in the endothermic (backward) direction. This analysis has some similarities to the
Le Chatelier’s principle, which states that a system at equilibrium counteracts the
imposed change.
The heat release budget across one of the shocklets shown in Fig. 5.21 is plotted in
Fig. 5.22. Reactions (1), (2), and (4) contribute most to the overall heat release rate.
It should be noted that the heat release rates for the other reactions in the chemical
mechanism (Hong et al., 2011) are at least one order of magnitude smaller than the
heat release rates of the reactions in Table 5.3, and, therefore, are not plotted for
clarity. The source term of H2O is negative because of the dissociation of water in
reactions (2) and (4). Note that in the shocklet/ﬂame interaction regions, reactions
(1)–(4) proceed in the direction where OH radicals are produced.
Also notice that the abundance of H radicals and N2 as a third body reactant in
the rich side of the mixing layer, where temperature is lower, promotes the radical
recombination reaction H + H + M Ñ H2 ` M, which in turn produces heat in this
region.
No. Reaction Q (KJ/mol)
1 OH+O é H+O2 68.4
2 OH+OH é O+H2O 67.4
3 OH+H é O+H2 6.2
4 OH+H2 é H+H2O 61.1
Table 5.3: Some of the reactions in the chemistry mechanism. Third column shows
the heat released in the forward direction.

Figure 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.

5.4. SHOCKLET/FLAME INTERACTION 79
0.05 0.06 0.07 0.08 0.09
-50
-40
-30
-20
-10
0
10
Reaction (1)
Reaction (2)
Reaction (3)
Reaction (4)
Net heat release rate
Figure 5.22: Heat release rate budget plotted against mixture fraction across a shock-
let.

Chapter 6
Validation Studies of CFPV
Approach
The CFPV model described in Chapter 3 relies on different assumptions that are
quantified and discussed here.
6.1 A priori analysis of compressibility effects
As explained in Chapter 3, the reference condition is controlled by the boundary
temperatures and background pressure of the flamelet solutions. These quantities
should be chosen such that they represent the conditions of the flow as good as
possible. For example, the oxidizer temperature Tox can be set to the free stream
temperature and the fuel temperature to the fuel total temperature.
The assumption that the composition is not changed by disturbances of energy and
pressure around the reference state is assessed first. Figure 6.1 shows the species
mass fractions of the major and important minor species for three flamelet solutions
computed with different values of the fuel temperature. To be consistent with the
flamelet/progress variable approach, the flamelets used in this and the following com-
parisons are chosen such that they have an equal reaction progress parameter defined
as the value of the progress variable at stoichiometric condition. It can be seen that
the mixture composition changes little with varying fuel temperature. This can be
81

82 CHAPTER 6. VALIDATION STUDIES OF CFPV APPROACH
explained by the fact that the flame is located very close to the oxidizer side, be-
cause the stoichiometric mixture fraction is 0.03 for hydrogen. Consequently, larger
changes in the mixture composition are observed when the oxidizer temperature is
varied, as shown in Fig. 6.2, but these variations remain small for the major species.
Although minor species show a stronger departure from their nominal values, their
contributions to the mixture properties are much smaller than those of the major
species. For instance, Fig. 6.3 shows that the gas constant of the mixture does not
significantly vary with the oxidizer temperature. Finally, Fig. 6.4 compares the mix-
ture composition of three flamelet solutions with different values of the background
pressure. The disparity in the mass fractions of major species is much smaller than
for the minor species, and again these variations do not significantly change those
mixture properties (such as the gas constant) that are independent of the mixture
temperature.
Note that although this analysis has been illustrated with the example of an ar-
bitrarily chosen flamelet, the same conclusions are obtained for the entire S-shaped
curve (i.e., for different values of the scalar dissipation rate χ). In other words, prop-
erties that only depend on the mixture composition can be estimated with relatively
good accuracy with the flamelet solutions computed at the reference values. However,
the calculation of properties that also depend on the temperature (e.g., specific heat
ratio and viscosity) must account for temperature variations.
In order to validate Eqs. (3.8), (3.10), and (3.14), a flamelet library is generated at
reference conditions Tox “ 1550 K, Tf “ 300 K, and p “ 1.0 bar. This table is then
used to estimate different properties of a flamelet solution with significantly higher
temperature, Tox “ 2000 K, in order to test the ability of the perturbation functions
introduced above to approximate the S-shaped hypersurface. The temperature is
first computed using the internal energy of the perturbed flamelet and the mixture
composition from the flamelet library at the reference conditions. Figure 6.5a shows
that computing the temperature from Eq. (3.10) (red curve) leads to a maximum
relative error smaller than 3%. This approach, based on the analytical relation be-
tween temperature and internal energy, is then compared to the iterative solution of
Eq. (3.5). The difference is plotted as a function of the mixture fraction in Fig. 6.5b.

6.1. A PRIORI ANALYSIS OF COMPRESSIBILITY EFFECTS 83
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
Z
YH
2
O
T
f
= 100 K
Tf
= 300 K
T
f
= 500 K
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Z
Y
H
2
T
f
= 100 K
Tf
= 300 K
Tf
= 500 K
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
Z
YO
2
T
f
= 100 K
Tf
= 300 K
T
f
= 500 K
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
x 10
−3
Z
Y
H
Tf
= 100 K
T
f
= 300 K
T
f
= 500 K
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
0.02
0.025
Z
YOH
T
f
= 100 K
T
f
= 300 K
Tf
= 500 K
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
Z
Y
O
T
f
= 100 K
T
f
= 300 K
Tf
= 500 K
Figure 6.1: Species mass fractions 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 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.

0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
Z
Y
H
2
O
T
ox
= 1000 K
Tox
= 1550 K
T
ox
= 2000 K
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Z
Y
H
2
T
ox
= 1000 K
Tox
= 1550 K
Tox
= 2000 K
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
Z
YO
2
T
ox
= 1000 K
Tox
= 1550 K
T
ox
= 2000 K
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
x 10
−3
Z
Y
H
Tox
= 1000 K
T
ox
= 1550 K
T
ox
= 2000 K
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
0.02
0.025
0.03
Z
YOH
T
ox
= 1000 K
T
ox
= 1550 K
Tox
= 2000 K
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
Z
Y
O
T
ox
= 1000 K
Tox
= 1550 K
T
ox
= 2000 K
Figure 6.2: Species mass fraction of the major species (H2O, H2, O2) and important
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

0 0.2 0.4 0.6 0.8 1
0
1000
2000
3000
4000
5000
Z
R
Tox
= 1000 K
Tox
= 1550 K
T
ox
= 2000 K
Figure 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

0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
Z
Y
H
2
O
p = 0.5 bar
p = 1.0 bar
p = 2.0 bar
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Z
Y
H
2
p = 0.5 bar
p = 1.0 bar
p = 2.0 bar
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
Z
YO
2
p = 0.5 bar
p = 1.0 bar
p = 2.0 bar
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
x 10
−3
Z
Y
H
p = 0.5 bar
p = 1.0 bar
p = 2.0 bar
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
0.02
0.025
Z
YOH
p = 0.5 bar
p = 1.0 bar
p = 2.0 bar
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
0.02
Z
Y
O
p = 0.5 bar
p = 1.0 bar
p = 2.0 bar
Figure 6.4: Species mass fractions of the major species (H2O, H2, O2) and important
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

(a) Temperature (b) Temperature discrepancy
Figure 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.
The maximum difference is about two Kelvins, which shows the good accuracy of the
approximation (Eq. (3.10)). Using this approximation improves the computational
efficiency of the CFPV method, especially for LES computations where explicit tem-
poral integration is used. Similarly, the error committed using a linear expansion in
temperature for the specific heat ratio (Eq. (3.8)) leads to a maximum error that is
less than 0.3%. This is shown in Fig. 6.6.
The source term of the progress variable is extremely sensitive to perturbations in
temperature and pressure. Equation (3.14) is devised to model the effects of these
perturbations. The rescaling of the source term given by Eq. (3.14) is illustrated
in Fig. 6.7, which demonstrates that this rescaling performs quite well for relatively
strong perturbations in temperature and pressure that are often present in high-speed
flows due to shock waves and expansion fans. This simple a priori analysis indicates
that the important effects associated with compressibility are captured relatively well

Figure 6.6: Speciﬁc heat ratio as function of mixture fraction. 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 values at reference condition.
by the proposed model. Further validations are presented in the next two sections.

6.2. SENSITIVITY ANALYSIS 89
Figure 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).
6.2 Sensitivity analysis
The CFPV model represents a low dimensional manifold parameterized with rZ, ĄZ22,
and rC, and internal energy and pressure with a flamelet library evaluated at a ref-
erence internal energy and pressure and an expansion around this base solution in
terms of these parameters. At what condition the base solution is computed is to
some degree arbitrary, and, therefore, the sensitivity to this choice should be deter-
mined. If the sensitivity of the solution to this choice was large, the model accuracy
would depend on it. In other words, a perfect model would be totally insensitive
to the reference parameters. Also, if the model was sensitive to these values, more
accurate guidelines for this choice would have to be provided.
The only user-specified parameters in the CFPV model are the oxidizer temperature
Tr
ox, fuel temperature Tr
f , and a background pressure pr
tb. The flamelet equations are
then solved subject to these parameters, and necessary variables (e.g. T0, aγ, aρ,
and Ta) are computed and tabulated. Consequently, all variables are functions of
these input parameters, which are statistically independent. It should be emphasized
that pressure and temperature are actually extremely correlated in compressible flows.

However, Tr
ox, Tr
f , and pr
tb are not correlated; these quantities are reference parameters,
and are thus statistically independent.
The proposed CFPV model attempts to partly account for the compressibility ef-
fects and viscous heating through the use of compressibility corrections (see section
3.3). A comparison of Figs. 6.1 and 6.2 reveals that the flamelet solutions are much
more sensitive to the oxidizer temperature than the fuel temperature; therefore, only
variations in the oxidizer temperature and background pressure are considered here.
Since the results of an LES or RANS simulation using the CFPV model depend
on the choice of pr
tb and Tr
ox, these parameters shall now be treated as independent
random variables influencing the solution. To quantitatively assess the impact of the
choice of these parameters, the statistics of the solution with respect to variations
in the parameters can be computed and the resulting expectation is compared with
the standard deviation. The solution can be called insensitive to the choice of the
parameters, if the standard deviation is small compared to the expectation.
The statistical properties of any variable (at a given rZ, ĄZ22, and rC), can be com-
puted, if the joint probability density function (PDF) of the parameters (i.e, pr
tb,
Tr
ox) is known; sensitivity of any variable f pTr
ox, pr
tbq to the variations of the input
parameters can then be quantified by computing its expectation and variance,
µf “
ĳ
f pTr
ox, pr
tbq P pTr
ox, pr
tbq dTr
oxdpr
tb, (6.1)
σ2
f “
ĳ
rf pTr
ox, pr
tbq ´ µf s2
P pTr
ox, pr
tbq dTr
oxdpr
tb, (6.2)
where f could be any variable, µf is its expectation, and σ2
f its variance. P denotes
the joint PDF of the oxidizer temperature and background pressure, such that,
P pTr
ox, pr
tbq “ P pTr
oxq P ppr
tbq , (6.3)
because all the input parameters of the CFPV method are independent. Note that f
is a function of space and time; this is considered here, but the dependency has not
been explicitly denoted for simplicity. Moreover, the same notation has been used for

the sample space variable and stochastic variable.
Quadrature rules can be used to numerically compute the integrals in Eqs. (6.1) and
(6.2). The joint PDF in Eq. (6.3) is evaluated assuming a Gaussian distribution for the
oxidizer temperature and background pressure. Equation (6.1) is then approximated
using Gauss-Hermite quadrature,
µf »
1
π
N1ÿ
i“1
N2ÿ
j“1
ωiωjf
´
Tr
oxi, pr
tbj
¯
, (6.4)
where ωi are the Gauss-Hermite quadrature weights, and N1 and N2 are the number
of sample points in the oxidizer temperature and background pressure, respectively.
Equation (6.4) requires a total of N1N2 simulations of the problem, each with a
different flamelet library. Variances can then be computed using the same quadrature
rule,
σ2
f »
1
π
N1ÿ
i“1
N2ÿ
j“1
ωiωj
”
f
´
Tr
oxi, pr
tbj
¯
´ µf
ı2
. (6.5)
The mean and variance of the oxidizer temperature and pressure are estimated
using the LES results of a hydrogen jet in a supersonic cross-flow. Refer to section
7.2 for more details about this configuration. A total of nine simulations, three
sample points in each direction, are performed with different flamelet libraries. Each
circle in Fig. 6.8 represents one of these simulations where the horizontal and vertical
axes show the background pressure and oxidizer temperature of the flamelet library,
respectively. It should be mentioned that increasing the number of sample points
does not significantly decrease the truncation error in Eqs. (6.4) and (6.5), because
the Gauss-Hermite quadrature weights drop exponentially.
The expectation and standard deviation of the progress variable, using Eqs. (6.4)
and (6.5), are plotted in Fig. 6.9 for RANS simulations. The standard deviation is
at least one order of magnitude smaller than the expectation, which shows that the
CFPV method is not very sensitive to variations in the background pressure and
oxidizer temperature when the compressibility corrections introduced in section 3.3
are active.

Figure 6.8: Each blue circle represents a sample point of the reference condition used
to create the chemistry table based on the Gauss-Hermite quadrature points.
Figure 6.9: Expectation (top) and standard deviation (bottom) of the progress vari-
able computed using RANS and Gauss-Hermite quadrature.

Among these compressibility corrections, the rescaling of the source term of the
progress variable, Eq. (3.14), plays a crucial role in increasing the fidelity of the
CFPV method. To show the importance of this rescaling, we performed RANS sim-
ulations for all nine sample points in Fig. 6.8, where the rescaling with Eq. (3.14)
was deactivated. Figure 6.10 shows the expectation and standard deviation of the
progress variable without using Eq. (3.14). The standard deviation in this case is
almost of the same order of magnitude as the expectation. The comparison between
Figs. 6.9 and 6.10 demonstrates that the CFPV model can account for the major
part of the compressibility effects. To better observe the effects of Eq. (3.14), con-
tours of the progress variable are plotted in Fig. 6.11 for the case with Tr
ox “ 1180 K
and pr
tb “ 1.2 bar. When Eq. (3.14) is deactivated, the simulation does not show
any burning region. On the other hand, using the compressibility correction for the
source term of the progress variable leads to a flow-field which is consistent with
experimental results. If the correction is not used, large errors are observed in the
regions where compressibility is important, for example, where shock waves or expan-
sion fans interact with the flame or in the boundary layer where viscous heating and
heat losses are important.

Figure 6.10: Same as Fig. 6.9, but without the compressibility correction of the source
term of the progress variable (Eq. (3.14)).

Figure 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).

6.3 Validation of the CFPV model using DNS data
In this section, the CFPV model introduced in Chapter 3 is validated in an a priori
and a posteriori study using the DNS databases presented in Chapter 5.
6.3.1 A priori analysis
In the a priori analysis, the mixture fraction and progress variable from the DNS
databases of Chapter 5 are used to compute the quantities of interest by using the
CFPV approach and a chemical table at reference conditions Tf “ 500 K, Tox “
1500 K, and p “ 2 bar. These quantities are computed from the relationships derived
for the CFPV model in section 3.3, e.g. Eq. (3.10) for temperature and Eq. (3.8) for
specific heat ratio.
Figure 6.12 shows the contours of temperature for the highest Mach number case,
Mc “ 2.5, using Eq. (3.10), while T0, γ0, aγ, and e0 are interpolated from the flamelet
table. The computed temperature is very close to the corresponding temperature field
from the DNS database presented in Fig. 6.12. In order to compare these flow fields
more quantitatively, the mixture fraction conditioned PDFs of rT “ TCFPV {TDNS,
rγ “ γCFPV {γDNS, and rR “ RCFPV {RDNS are plotted in Fig. 6.13. If the two fields
were exactly identical, the PDFs would be a delta function at one. For comparison,
the conditional PDFs of these ratios computed using the FPV approach are also
plotted. It is observed in Fig. 6.13 that the PDFs using the CFPV approach are
indeed close to a delta function, whereas the PDFs using the FPV approach show
a larger departure. For instance, the conditional PDF of the temperature ratio at
Z « 0.65 is close to a delta function at 0.5 in the FPV approach (i.e. 50% error in
the predicted temperature). On the other hand, the error of the CFPV approach is
less than 10%. Also note that the PDFs of the gas constant ratio are identical in
both the CFPV and FPV approaches, because it is assumed that the mass fractions
are identical (see Eq. 3.3). It should be noted that, the conditional PDFs are only
computed for the mixture fraction larger than 0.01 and smaller than 0.99, to exclude
the regions of pure fuel and oxidizer.

6.3. VALIDATION OF THE CFPV MODEL USING DNS DATA 97
Figure 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.

Figure 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).
6.3.2 A posteriori analysis
In the a posteriori analysis, the reactive mixing layer is simulated using the CFPV
model with a flamelet library at the reference conditions Tf “ 500 K, Tox “ 1500 K,
and p “ 2 bar. The same computational mesh and parameters, as defined in section
5.1.2 for the Mc “ 2.5 case, are used in this study, and the simulation is continued
until the mixing layer reaches self-similarity.
Figure 6.14 compares the averaged CFPV results with the corresponding DNS data.
The profiles of the averaged density, turbulent kinetic energy, and mixture molecu-
lar weight show a good agreement with the DNS results, whereas the temperature
profile shows larger departure, albeit less than 8%. The maximum error in the Favre-
averaged temperature profile is observed near the flame location. It should be noted
that, this comparison is for the highest Mach number case, where compressibility ef-
fects are very strong. This study shows that even with strong compressibility effects
the CFPV approach is able to reasonably capture the interplay between chemistry
and compressibility.

6.3. VALIDATION OF THE CFPV MODEL USING DNS DATA 99
(a) Density (b) Turbulent kinetic energy
(c) Mixture molecular weight (d) Temperature
Figure 6.14: Comparison of the Favre averaged CFPV results (solid lines) and DNS
data (dashed lines) for the Mc “ 2.5 case.

Chapter 7
Application of CFPV Approach
In this chapter, the CFPV model developed in Chapter 3 and validated in previous
chapter is applied to more realistic configurations (i.e. an under expanded hydrogen
jet in a supersonic cross-flow and HIFiRE scramjet) in order to assess the model
performance. In the former configuration, instantaneous (LES) and time-averaged
(LES and RANS) flow fields are qualitatively compared with the experimental data
obtained through PLIF imaging of the OH radical. In the former configuration, LES
results are validated against the ground test experimental measurements at NASA’s
Arc-Heated Scramjet Test Facility.
7.1 Mathematical framework
The governing equations are presented in Chapter 2. They are summarized here,
including a transport equation for the residual mixture fraction variance. In the
following, we will discuss modeling of supersonic combustion both in the context of
RANS and LES. The governing equations have a similar form, and, therefore, will be
presented only once. The averaging operators appearing in these equations, ¨ and ˜¨,
refer to an ensemble average in RANS, and to a spatial filter in LES, where ¨ and ˜¨
denote direct and Favre averaging or filtering, respectively.
Transport equations for the Favre averaged or filtered variables ¯ρ, ¯ρrui, ¯ρ rE, ¯ρ rZ, ¯ρĄZ22,
and ¯ρ rC are solved in conservative form, where ¯ρ is the density, rui the components of
101

102 CHAPTER 7. APPLICATION OF CFPV APPROACH
the velocity vector, rE the total energy, including the chemical energy, rZ the mixture
fraction, ĄZ22 the variance of the mixture fraction, and rC a progress variable. Their
respective transport equations are
B¯ρ
Bt
`
B¯ρruj
Bxj
“ 0, (7.1)
B¯ρrui
Bt
`
B¯ρruiruj
Bxj
“ ´
B¯p
Bxi
`
B¯τji
Bxj
`
B¯τR
ji
Bxj
, (7.2)
B¯ρ rE
Bt
`
B¯ρruj
rE
Bxj
“
B
Bxj
«˜
Ăλ
cp
`
µt
Prt
¸
Brh
Bxj
ff
`
B
Bxj
«
Nÿ
k“1
˜
¯ρ rDk ´
Ăλ
cpk
¸
rhk
B rYk
Bxj
ff
`
B
Bxj
„ˆ
rµ `
µt
σk
˙
Bk
Bxj
´ ruj ¯p ` rui
`
¯τij ` ¯τR
ij
˘

,
(7.3)
B¯ρ rZ
Bt
`
B¯ρruj
rZ
Bxj
“
B
Bxj
«ˆ
¯ρD `
µt
Sct
˙
B rZ
Bxj
ff
, (7.4)
B¯ρĄZ22
Bt
`
B¯ρruj
ĄZ22
Bxj
“
B
Bxj
«ˆ
¯ρ rD `
µt
Sct
˙
BĄZ22
Bxj
ff
` 2
µt
Sct
B rZ
Bxj
B rZ
Bxj
´ ¯ρrχ, (7.5)
B¯ρ rC
Bt
`
B¯ρruj
rC
Bxj
“
B
Bxj
«ˆ
¯ρ rD `
µt
Sct
˙
B rC
Bxj
ff
` ¯9ωC, (7.6)
where
¯τR
ij “ ¯ρruiruj ´ ¯ρ Ąuiuj, (7.7)
is the Reynolds stress tensor, µ and µt are the laminar and turbulent viscosity, k is
the turbulent kinetic energy, λ the thermal diffusivity, cp the specific heat capacity
at constant pressure, Prt a turbulent Prandtl number, h the enthalpy including the
sensible and chemical contributions, Yk the species mass fractions, σk the turbulent
kinetic energy Schmidt number, τij the viscous stress tensor, D the diffusion coefficient

7.2. HYDROGEN JET IN A SUPERSONIC CROSS-FLOW 103
for the scalars, Sct a turbulent Schmidt number, χ the scalar dissipation rate, and
9ωC the source term for the progress variable. The molecular properties µ, λ, cp are
computed using mixing rules (Bird et al., 2007) to account for changes in composition
and temperature. The turbulent viscosity µt, kinetic energy k, Schmidt number
Sct,and Prandtl number Prt are computed in the RANS context using the SST model
(Menter, 1994) or based on the dynamic subgrid-scale model (Moin et al., 1991) in
the LES context.
Note that the summation on the right-hand side of Eq. (7.3) vanishes under the
unity Lewis number assumption so that the species mass fraction gradients do not
need to be explicitly computed. Although it is well-known that Lewis number effects
can be quite important in the case of hydrogen, it is assumed that turbulent diffusion
is much larger than laminar diffusion, so that this term can be neglected.
The mixture composition depends directly on rZ, ĄZ22, and rC. In order to close this
system, the pressure and the temperature need to be determined. This is achieved, as
explained in Chapter 2, by using an equation of state and the energy of the mixture.
For hydrogen chemistry, the progress variable is defined as the mass fraction of H2O.
7.2 Hydrogen jet in a supersonic cross-flow
The combustion model is tested for the case of a jet in a supersonic cross-flow (JICF)
and compared with the experimental measurements of Gamba et al. (2011a) and
Heltsley et al. (2007). The JICF configuration is a canonical flow often encountered
in supersonic combustion. It is of relative simplicity, yet it retains many features
of interest, such as three-dimensionality, separation and recirculation regions, wall
effects, and vortical flows. Furthermore, it maintains a high level of practical relevance
for engineering applications.
A wide body of work has been conducted to characterize and study this system, both
at low and high speeds, especially under non-reacting conditions. Reacting transverse
jets in the supersonic regime have, however, received much less attention, partially
due to the requirements and complexity to sustain combustion in a supersonic flow
(Ben-Yakar and Hanson, 1999; Heltsley et al., 2007; Lee et al., 1992; McMillin et al.,

1994).
Several numerical investigations have been conducted on the JICF configuration.
Direct numerical simulations at very low Mach number have been used to study jet
mixing, flow structures, and jet trajectories (Muppidi and Mahesh, 2005, 2007; Sau
and Mahesh, 2010). In recent years, several numerical simulations of under-expanded
jets in supersonic cross-flow have been performed. However, a vast majority of these
simulations have been devoted to inert flows (Boles et al., 2010; Génin and Menon,
2010; Kawai and Lele, 2010). For instance, Boles et al. (2010) have shown that in their
simulations, a hybrid LES/RANS method was more accurate than a simple RANS
approach; however, even in the hybrid approach, jet penetration and mixing showed
some discrepancy with experimental measurements. Kawai and Lele (2010) used
LES to study jet mixing showing good agreement with experimental data. On the
other hand, numerical studies of reacting jets in supersonic cross-flow are rare. Won
et al. (2010) used the detached eddy simulation (DES) approach with a finite-rate
chemistry model. Although their numerical results show overall good agreement with
experimental observations, there are some inconsistencies in quantitative comparisons,
e.g. lower jet penetration and narrower bandwidth. Also, the turbulent structures in
the jet shear layer were very diffusive, likely due to the dissipative nature of their
numerical scheme or their turbulence model.
7.2.1 Flow configuration
The experimental measurements used in the present study (Gamba et al., 2011a) have
been conducted in an expansion tube at the end of which a flat plate was mounted
parallel to the free stream direction. The final aero-thermodynamic state of the test
gas could be adjusted by changing the filling pressure of each of the three sections
of the expansion tube, allowing for a wide range of conditions to be explored. An
under-expanded hydrogen jet was injected perpendicularly into the free stream from
a contoured nozzle with diameter d “ 2 mm in the flat plate. The nozzle was located
at a distance of 66.7 mm downstream of the flat plate leading edge. Different diagnos-
tics techniques, such as Schlieren, OH˚
chemiluminescence, OH planar laser-induced

fluorescence (PLIF) imaging, and surface pressure measurements were used to inves-
tigate mixing, ignition, combustion, flame holding, and the stabilization processes of
a hydrogen flame in the supersonic regime.
A single free stream condition representative of the conditions in a scramjet engine
of a hypersonic vehicle at Mach 8 flight and 30 km altitude is used in this computa-
tional study. The corresponding nominal conditions of the free-stream flow, estimated
from direct observations on directly measurable quantities such as Mach number and
static pressure, are characterized by a static pressure P “ 40 kPa, Mach number
M “ 2.4, and static temperature T “ 1550 K, corresponding to a speed of U “ 1800
m/s. Indirect measurements indicated that the incoming boundary layer was laminar
in the upstream region of the jet. Because of the very short test time of the exper-
iment, i.e. 0.1 ´ 0.5 ms, where (quasi)-steady test gas conditions were observed, the
temperature of the plate is assumed to remain constant and an isothermal boundary
condition with Tw “ 300 K is applied.
Global characteristics of a JICF, such as penetration, have been found to primarily
depend on a single parameter, the jet-to-crossflow momentum flux ratio J defined as
J “
ρu2
H2
ρu2
air
. (7.8)
As experimental measurements have shown (see Gamba et al., 2011a), the combustion
characteristics are also strongly dependent on J. In this computational study, the
highest J value studied in the experiment (Gamba et al., 2011a), i.e. J “ 5.0, is
chosen. For this J value, the total pressure, total temperature, and Reynolds number
in the plenum of the injection nozzle are P0 “ 2024 kPa, T0 “ 300 K, and Red “
507400, respectively.
7.2.2 Numerical implementation
The proposed combustion model has been implemented in both Joe RANS and
Charles LES solvers. The parallel LES 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 cells where the ENO scheme should be applied. The sub-grid
terms in Eqs. (7.1)-(7.6) are modeled using the dynamic procedure (Moin et al., 1991).
The RANS solver used for the solution of the compressible Navier-Stokes equations
is based on a finite volume formulation and implicit time-integration on arbitrary
polyhedral meshes (Pecnik et al., 2012). The Euler fluxes are computed with a HLLC
(Batten et al., 1997; Toro et al., 1994) approximate Riemann solver. Second-order
accuracy is typically achieved by computing the states at each side of a given cell
face using second-order interpolation and then applying the same flux evaluation
scheme to the reconstructed states. Gradients are limited with a modified version of
Barth and Jespersen slope limiter (Barth and Jespersen, 1989; Pecnik et al., 2009;
Venkatakrishnan, 1995). The system of Eqs. (7.1)-(7.6) is solved fully coupled and
the resulting large sparse system (the Jacobian matrices are obtained using first-order
discretization) is solved with the generalized minimal residual method (GMRES)
using the freely available linear solver package PETSc (Satish et al., 2009). Different
turbulence models are implemented, but all results shown here are computed with
the two-equation turbulence model k-ω SST (Menter, 1994). The turbulent Schmidt
number for the scalars is set to Sct “ 0.5 and the turbulent Prandtl number is
Prt “ 0.5.
In both cases, a very accurate 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 28 reactions, and does not include nitrogen
chemistry.
7.2.3 Results
In this section, instantaneous LES flow fields are qualitatively compared to exper-
imental data obtained through PLIF imaging of the OH radical first, then average
LES fields are compared to RANS results and average experimental measurements.

It should be mentioned that the experiments only report the OH fluorescence sig-
nals, and, although, the experimental OH PLIF signal is correlated with the hydroxyl
radical concentration, it is a non-linear function of other parameters like mole fractions
of all species and more importantly pressure and temperature of the mixture. More-
over, the OH concentration in the LES results is interpolated from a pre-computed
flamelet library, which is constructed at a constant background pressure, oxidizer and
fuel temperatures. Even though compressibility corrections for the mixture properties
and source term of progress variable have been introduced in section 3.3, because of
the qualitative nature of the comparison, no compressibility correction is used for the
OH concentration flow fields presented in this section. As shown in Fig. 6.2, OH mass
fractions vary up to a factor of two due to compressibility effects. In this respect, the
following simulation results should be interpreted only in a qualitative sense.
Instantaneous flow-fields
Figure 7.1 shows a side view of the instantaneous OH distribution for the symmetry
plane in a comparison of LES results (top) and experimental measurements (bottom).
Two distinct regions of strong combustion are observed, i.e. at the upper shear layer
between the air and fuel streams and in the boundary layer. Notice that burning
starts in the recirculation region upstream of the jet exit. Length and height of this
recirculating zone, which are about 6d and 0.5d, respectively, are in good agreement
with the experimental data (Gamba et al., 2011a). In the experiment, burning is seen
in the boundary layer at a distance of about 5d downstream of the jet, and the LES
results also show higher concentration of OH in the lower shear layer around the same
location.
A very strong fuel cooling takes place just downstream of the jet as the fuel expands
out of the nozzle, explaining the weak burning in the boundary layer before x „ 5d. To
demonstrate the importance of the temperature correction (Eq. (3.14)), a simulation
has been performed without temperature rescaling of the progress variable source
term. Figure 7.2 displays higher concentration of OH in the lower shear layer, which
is inconsistent with the experiment and LES results of Fig. 7.1.
Computed instantaneous OH mass fraction distributions in different planes parallel

Figure 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).
Figure 7.2: Instantaneous OH mass fraction without applying temperature correction
for the source term of the progress variable (Eq. (3.14)). Contours are from rYOH “ 0
(blue) to rYOH “ 0.024 (white).

(a) y “ 0.25d (b) y “ 0.5d
(c) y “ d (d) y “ 3d
Figure 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).
to the plate are shown in Fig. 7.3. Burning is observed in a large part of the boundary
layer in Figs. 7.3a and 7.3b, in good agreement with the experimental data for y “ 0.5d
shown in Fig. 7.4 (Gamba et al., 2011a). While Fig. 7.3c still shows some traces of
burning in the boundary layer at y “ d, burning is only present in the mixing layer
at y “ 3d (Fig. 7.3d). Figure 7.4 shows the experimental OH PLIF signal in the
y “ 0.5d plane, which corresponds to the LES results shown in Fig. 7.3b. The
borders of the burning region in the boundary layer and the recirculation region are
in good agreement with the experimental data.
Discrepancies between LES and experiment are also observed. For instance, the

Figure 7.4: Experimental OH PLIF signal in the plane y “ 0.5d. Contours are from
blue to white.
comparison in Fig. 7.1 shows that the burning in the boundary layer downstream of
the jet exit seems to be closer to the wall in the experiment. This is also observed in
the time-averaged data in section 7.2.3. The two aspects discussed earlier in this sec-
tion, the diﬀerence between experimental OH PLIF signal and OH mass fraction and
the absence of the compressibility correction for the OH visualization of the LES re-
sults, could partly explain these discrepancies. However, other modeling assumptions
might also contribute to the discrepancies, e.g., closure assumptions in Eq. (7.5), and
the presumed PDF assumptions. Auto-ignition of the fuel jet in this region could be
another explanation, as it cannot be predicted accurately by the proposed method,
and needs additional modiﬁcations (Ihme et al., 2005). A more detailed analysis is,
however, required to investigate the combustion regime in this region.

Time-averaged flow-fields
In this section, statistics from LES, obtained by time-averaging instantaneous flow-
fields are compared to steady RANS results and experimental data averaged over
eleven single images. It should be noted that the number of samples that contributed
to the experimental average is insufficient to yield converged statistics. Nonetheless,
they provide a useful approximation of the general flow features.
The time-averaged LES and RANS results at the y “ 0.25d and the y “ 0.5d
planes are juxtaposed in Fig. 7.5. The LES results show a wide burning zone in the
boundary layer, which is in good agreement with the experimental observations shown
in Fig. 7.6. However, this burning zone is much narrower in the RANS case. This is
due to the smaller upstream recirculation zone predicted by the RANS simulations,
which can also be seen in Fig. 7.7. The smaller recirculation zone reduces the amount
of fuel transported around the jet in the boundary layer and, thereby, the extent of
the reactive region. Note that the smaller recirculation zone predicted by RANS is
partly due to the laminar character of the boundary layer upstream of the jet, which
is not correctly represented by the RANS simulations. Experimental measurements
with a trip wire to enforce a turbulent incoming boundary layer have shown a much
smaller recirculation zone (Gamba et al., 2011b).
Figure 7.7 also illustrates the discrepancies between LES and RANS results. Not
only the recirculation zone upstream of the jet is smaller in the RANS computation,
but also the mixing layer between fuel and air is less diffusive. In the LES result, the
mixing layer on the leeward side starts to burn around x „ 5d, which is consistent
with the experiment. Burning in the experiment is, however, closer to the wall, as
discussed above.
Another difference between RANS and LES is illustrated in Fig. 7.8, where the
time-averaged mixture fraction is compared. In particular, a much larger jet plume
is predicted by the RANS simulation, indicating deficiencies of the RANS mixing
model as already observed in Fig. 7.7. Stronger mixing could be achieved by reducing
the value of the turbulent Schmidt number, but such an approach would require a
problem-dependent calibration. To circumvent this issue, a more elaborate mixing
model would be required.

(a) Time averaged OH mass fraction (b) Time averaged mixture fraction
Figure 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).
Figure 7.6: Eleven-frame average of the experimental OH PLIF signal in the y “ 0.5d
plane.

Figure 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.

Figure 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).
(a) OH mass fraction (b) mixture fraction (c) water mass fraction
Figure 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.

7.3. HIFIRE SCRAMJET 115
Time averaged contours of the OH mass fraction, mixture fraction, and water mass
fraction (progress variable) at three different planes normal to the cross-flow at the
locations x “ 4d, x “ 8d, and x “ 12d, are shown in Fig. 7.9. Again, burning is
observed in the boundary layer and in the shear layer between the fuel and air streams,
while the core of the hydrogen jet does not burn. This figure further illustrates the
lower mixing rate demonstrated by the RANS simulations.
7.3 HIFiRE scramjet
Figure 7.10 shows the HIFiRE Flight 2 project (HF2) scramjet (Jackson et al., 2011).
The engine has a 0.2032 m long, constant-area isolator with the dimensions of 0.0254
m and 0.1016 m in the lateral and spanwise directions, respectively. At each stream-
wise fuel injection station shown in Fig. 7.10, there are four injectors (spaced equally
in the spanwise direction) on the body side (upper surface) and four injectors on the
cowl side (lower surface). The primary injectors have a diameter of 3.175 mm, and
are canted at 15˝
relative to the combustor wall. The secondary injectors are normal
to the combustor wall with a diameter of 2.3876 mm. For more details on the HF2
geometry refer to Storch et al. (2011).
Figure 7.10: Schematic of HIFiRE 2 scramjet.
The fuel is a JP-7 surrogate with the molar composition of 64% ethylene and 36%
methane. The total equivalence ratio is 1.0, where 40% of the fuel by mass is injected
from the primary injectors, and the balance from the secondary injectors (Storch
et al., 2011).
The experimental measurements (Cabell et al., 2011) of the full scale version of the
HIFiRE scramjet are conducted at NASA’s Arc-Heated Scramjet Test Facility (Guy
et al., 2012). In this study, the experimental results corresponding to the highest

Mach number are considered, where the Mach number at the isolator inlet is 3.46
(equivalent to the Mach 8 HF2 flight test).
7.3.1 Wall modeling
Wall-resolved LES of scramjets is not practical with the current computational re-
sources (Larsson, 2012), therefore, a wall model for the wall shear stress and heat
flux is applied. The unstructured LES grid is defined all the way to the wall and
is designed to resolve outer-layer scale motions of the boundary layer (Kawai and
Larsson, 2012). The wall model then solves the compressible equilibrium boundary
layer equations as described in Bodart and Larsson (2011)
d
dη
ˆ
pµ ` µt,wmq
du||
dη
˙
“ 0, (7.9)
d
dη
ˆ
pµ ` µt,wmq u||
du||
dη
` pλ ` λt,wmq
dT
dη
˙
“ 0, (7.10)
where u|| is the wall-parallel velocity component, η refers to the wall-normal coor-
dinate. These equations form a set of independent, one-dimensional ordinary dif-
ferential equations for each computational cell adjacent to a wall. In practice, a
one-dimensional structured grid with stretching is employed with the total height
δ based on the length scale of the corresponding LES wall-adjacent cell. No-slip
and adiabatic wall boundary conditions are utilized for the momentum and energy
equations, respectively,
u|| pη “ 0q “ 0 ,
dT
dη
ˇ
ˇ
ˇ
ˇ
η“0
“ 0. (7.11)
The manner of exchanging data between the LES and wall model has been modified
from Bodart and Larsson (2011) to use an integral constraint based on the wall-
adjacent LES cell. The following equations,
1
δ
ż δ
0
u||dη “ ru ,
1
δ
ż δ
0
Tdη “ rT, (7.12)

are solved iteratively at each wall-adjacent LES cell, ensuring that the average stream-
wise velocity and temperature from the wall model match the filtered values from the
LES. The wall-shear stress computed by the wall-model is applied as a flux in the
LES momentum equation. This integral approach also allows for the wall-modeled
viscous heating to be passed as a source term to the LES energy equation. In prior
wall-modeling implementations, themal feedback to the LES solution was limited to
the wall heat flux and was simply zero in the adiabatic case.
7.3.2 Boundary conditions
The LES is performed on a quarter of the HF2 domain, including the cowl-side surface
and one side wall, with symmetry boundary conditions applied at the mid-plane in
both the spanwise and lateral directions.
Specific inflow boundary conditions of the isolator, primary injectors, and secondary
injectors are shown in Table 7.1. At the isolator inlet, synthetic turbulence is gener-
ated using a digital filtering approach (di Mare et al., 2006; Touber and Sandham,
2009). Profiles of the Reynolds stress tensor components from the boundary layer
DNS data of Wu and Moin (2010) are rescaled and used as inputs. As the turbulent
conditions inside the HIFiRE isolator are not well characterized (see Jackson et al.,
2011; Storch et al., 2011), the objective of this approach is simply to develop a fully
turbulent boundary layer prior to the primary fuel injectors. This is similar to the
approach taken in Brès et al. (2013), where for practical design studies the synthetic
turbulence inside a jet nozzle is prescribed in a manner to ensure realistic turbulence
levels by the nozzle exit. All of the walls are assumed to be adiabatic due to the long
duration of the experiment (Cabell et al., 2011), and wall modeling is applied to all
surfaces. Characteristic boundary conditions are applied at the outflow boundary.

Table 7.1: Inflow boundary condition parameters.
Inflow boundary Pressure [kPa] Temperature [K] Mach number
Isolator 40.3 736.2 3.46
Primary injector 105.7 293.3 1.0
Secondary injetor 290.7 301.1 1.0
7.3.3 Numerical implementation
The unstructured finite volume LES solver “Chris” developed at Cascade Technologies,
Inc. (Brès et al., 2012) is employed in this study. Numerical stencils based on higher-
order polynomial expansions are used to minimize numerical dissipation. The spatial
discretization relies on a hybrid central/ENO method, in which a shock sensor is used
to identify the cells where the ENO scheme should be applied. An explicit third-
order Runge-Kutta scheme is used for temporal integration. All sub-grid turbulent
quantities are modeled using the dynamic procedure (Moin et al., 1991).
A body-fitted block refined unstructured mesh (Brès et al., 2012) is used for all of
the simulations in this study. Several embedded zones of refinement are defined, to
provide sufficient resolution in regions where gradients are large, e.g., in boundary
layers, fuel/air mixing layers, and the cavity. The simulations are performed using
a coarse mesh with 21 M cells, and a fine mesh with 96 M cells to evaluate the grid
independence of the solution.
7.3.4 Results
Figure 7.11 shows the LES wall pressure distribution in the symmetry plane normal
to the spanwise direction (centerline between the injectors). Experimental measure-
ments on the body and cowl sides are also plotted for comparison. The HF2 schematic
diagram is shown on the plot for reference. Comparisons with the experimental mea-
surements show relatively good agreements. Combustion heat-release causes a sharp
pressure rise beginning just upstream of the primary injectors. The pressure reaches
a plateau in the cavity, and rises again due to the compression on the ramp wall
of the cavity. Strong turbulence mixing produces a relatively homogeneous flow in

the cavity, and nearly constant pressure in this region. The LES under-predicts the
pressure in the cavity by about 4.5%. The pressure drops suddenly as the supersonic
flow expands at the end of the ramp wall. The pressure is also under-predicted in the
expansion region downstream of the secondary injectors; however, experimental un-
certainty is uncharacteristically high in this region for reasons that are not explained
in Cabell et al. (2011). For example, note the large variations in the experimental
measurements on the body and cowl sides around the secondary injectors. For this
reason, the LES results are believed to be in the range of experimental uncertainties.
It should be noted that the current combustion model uses a single mixture fraction
that cannot fully represents the interactions between the primary and secondary fuel
jets. Properly modeling this interaction could enhance the combustion around the
secondary injectors, and increase the pressure in this region. The addition of a second
mixture fraction and modeling of cross-dissipation effects in the flamelet equations
are the subjects of ongoing research.
Figure 7.12 shows the mixture fraction contours and the pressure rise on the com-
bustor walls (isolator wall is excluded for clarity). The primary jets are confined near
the wall by the relatively strong cross-flow. The secondary jets, which are injected
normal to the cross-flow and have a higher momentum flux ratio, penetrate more
deeply into the combustor interior. Figure 7.13 shows the iso-surface of the stoichio-
metric mixture fraction colored by the progress variable. Upstream of the cavity, the
primary jets burn mostly in the low speed boundary layer, and the mixing region
between the injectors. In fact, Figs. 7.13 and 7.12 indicate that the fuel streams
merge before entering the cavity. Mixing is enhanced in this region as the boundary
layer separates just prior to the cavity due to the strong adverse pressure gradient
(see Fig. 7.11). This can be clearly seen in Fig. 7.14, where a small recirculation zone
is visible between the primary injectors and the cavity step. The side wall boundary
layer similarly separates before entering the cavity.
Auxiliary simulations were also performed to independently evaluate the effect of
the source term correction outlined in Eq. 3.14, and the effects of the wall model.
Figure 7.11 compares the pressure distribution from the full CFPV model with a
second LES where the source term correction was not applied. It is clear from the

Figure 7.11: Wall pressure along the centerline between the injectors. LES results
(solid lines) are compared with the experimental measurements (symbols). 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.
delayed pressure response in the case with no correction term that Eq. 3.14 is an in-
dispensable step in the CFPV approach. Contours of the progress variable from these
two cases are plotted in Figs. 7.15a and 7.15b, which shows that combustion cannot
be sustained at the primary injectors without correcting the progress variable source
term for compressibility. A third LES simulation was performed that included the
source term correction but did not apply the wall model. Comparing Figs. 7.15a and

Figure 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).
Figure 7.13: Iso-surface of the stoichiometric mixture fraction colored by the progress
variable from C “ 0 (blue) to C “ 0.3 (red).

(a) u1 contours
(b) p contours
Figure 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).
7.15c shows that the viscous heating provided by the wall model is equally important
in stabilizing the flame at the primary injectors. The viscous heating in the isola-
tor boundary layer transfers the kinetic energy of the flow into sensible energy. The
corresponding increase in the fluid temperature enhances the reactions and stabilizes
the flame near the primary injectors. Equation 3.14 provides a mechanism to couple
these two phenomena in the CFPV model.

(a) with source term correction, with wall model
(b) without source term correction, with wall model
(c) with source term correction, without wall model
Figure 7.15: Contours of the progress variable from C “ 0 (black) to C “ 0.3 (white).

Chapter 8
Conclusions
High-fidelity direct numerical simulations of compressible temporal mixing layer are
performed using finite-rate chemistry. The compressible mixing layers presented in
this study evolve self-similarity after an initial transition. The self-similarity is ob-
served by plotting integral of dissipation rate and production of turbulent kinetic
energy, and by the collapse of Reynolds stress profiles. The Reynolds stress budget
shows similar behavior as low Mach number flows, i.e., the pressure-strain term ex-
tract energy from the streamwise Reynolds stress component, R11, which is injected in
the other components of Reynolds stress. At sufficiently high convective Mach num-
bers, eddy shocklets are observed in the flow. These shocklets could interact with the
flame, which results in regions where the the heat release rate is negative. This study
shows that important reactions proceed in the endothermic direction in the shock-
let/flame interaction regions making heat release rate strongly negative. Although
this negative heat release may indicate that shocklets tend to weaken combustion,
huge generation of OH radicals in these regions provides a pool of radicals which
have a significant impact on the combustion.
An efficient flamelet-based combustion model for compressible flows has been in-
troduced in this work. In this approach, only two or three additional scalar transport
equations need to be solved, independently of the complexity of the reaction mecha-
nism used in the simulations.
125

126 CHAPTER 8. CONCLUSIONS
The CEFT model is the extension of the low Mach number flamelet/progress vari-
able approach. The flamelet library in this approach includes all possible flamelet
solutions which can be represented by an S-shaped hypersurface. The memory re-
quirement of a five-dimensional flamelet library is, however, prohibitive. The CFPV
approach circumvents this by utilizing compressibility corrections around a flamelet
library at the reference condition, thus keeping the dimension of the flamelet library
the same as for the FPV model. Compressibility corrections have been introduced
considering the physical behavior of each parameter. These compressibility correc-
tions are validated in an a priori analysis. This analysis also reveals that mixture
properties which are not function of temperature (e.g. the gas constant) have very
small departure from the reference 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 sensitivity study of the CFPV approach confirms the importance of the com-
pressibility corrections, especially for the source term of the progress variable. This
study shows that deactivating the compressibility correction of the source term of
the progress variable, Eq. (3.14), produces a standard deviation which is of the same
order of magnitude as the expectation, whereas activating this correction results in a
standard deviation which is at least one order of magnitude smaller than the expec-
tation.
The model is then tested in both LES and RANS computations of an under-
expanded hydrogen jet in a supersonic cross-flow. Comparisons with experimental
measurements show relatively good agreement particularly for the LES results. Burn-
ing is observed upstream of the jet in the recirculation region, and in a big part of
the boundary layer downstream of the jet exit, consistent with the experimental mea-
surements. The burning recirculation region functions as a physical flame-holder,
and presumably renders a more stable flame; this could result into a smaller scram-
jet by promoting ignition. Time-averaged LES results also show that discrepancies
in the RANS results are likely due to deficiencies of the RANS mixing model. The
state of the incoming boundary layer also plays an important role in the size of the
recirculation zone.

127
The CFPV combustion model is validated in an a priori and a posteriori analyses
using DNS databases presented in Chapter 5. It is found that the CFPV model
perfumes accurately for relatively high convective Mach numbers. The maximum
error in temperature is observed around the location of flame which is around 8% for
the highest Mach number case based on the A posteriori analysis.
It is important to note, however, that the proposed models suffer the same limita-
tions of the low Mach number FPV approach. For instance, additional modifications
should be considered in order to differentiate between regions where auto-ignition or
unsteady effects are important. Also, the presumed PDF assumption of the mixture
fraction and reaction progress parameter and the closure assumptions in the transport
equation of the sub-grid mixture fraction variance have to be tested more rigorously.
Large eddy simulations using the CFPV combustion model are also reported for
the HIFiRE scramjet at Mach 8 flight conditions. Results are compared with experi-
mental measurements from the HIFiRE Direct-Connect Rig at NASA Langley. The
CFPV model utilizes functional corrections to a nominal flamelet solution to account
for compressibility, thus providing an efficient description of high-speed combustion
with a low memory footprint. Turbulent wall modeling of the viscous near-wall re-
gion is used to accurately predict the wall shear stress and heat flux in the LES. The
numerical results show good agreement with the experiment when both compressibil-
ity and wall effects are modeled. Neglecting either of these factors, however, leads to
reduced combustion and poor prediction of the pressure trends observed in the exper-
iment. Corrections to the progress variable source term are particularly important,
as they provide strong coupling between the combustion chemistry and compressible
phenomena such as shocks, expansions, and viscous heating. All of these effects must
be considered to stabilize combustion in the proper location.
It is important to note, however, that the proposed models suffer the same limita-
tions of the low Mach number FPV approach. For instance, additional modifications
should be considered in order to differentiate between regions where auto-ignition or
unsteady effects are important. Also, the presumed PDF assumption of the mixture
fraction and reaction progress parameter and the closure assumptions in the transport
equation of the sub-grid mixture fraction variance have to be tested more rigorously.

128 CHAPTER 8. CONCLUSIONS
Recommendations for future work
The work presented here addresses many aspects of compressible combustion. How-
ever, many open questions are remained. Some suggestions for expanding the knowl-
edge of supersonic combustion and improving the models are discussed below.
" Shock/flame interactions: Although this study sheds some light on the interplay
between compressibility and chemistry in the shocklet/flame interaction regions,
it cannot fully describe the interactions between a strong shock and flame.
The co-existence of OH production and strong negative heat release rate in
the shock/flame interaction regions gives rise to two possible scenarios: either
the pool of radicals generated in these regions enhances the reactions, or the
reactions are suppressed due to the negative heat release rate. DNS of a shock
wave impinged on a spatially evolving mixing layer could answer this important
question.
" DNS of the flame stabilization mechanism: This study, section 7.2, shows that
the recirculation region upstream of a jet in a cross-flow provides proper condi-
tions that promote combustion, and is thus responsible for stabilizing the flame.
The jet in cross-flow configuration is of relative simplicity yet retaining many
features of interest, such as three-dimensionality, flame stabilization mechanism,
separation and recirculation regions, wall effects, and vortical flows. DNS of this
configuration will provide a better understanding of the physics of supersonic
combustion, and guidelines for improving the models.
" CFPV model improvements: It is important to note that the proposed models
suffer the same limitations of the low Mach number FPV approach. For in-
stance, additional modifications should be considered in order to differentiate
between regions where auto-ignition or unsteady effects are important. Also,
the presumed PDF assumption of the mixture fraction and reaction progress pa-
rameter and the closure assumptions in the transport equation of the sub-grid
mixture fraction variance have to be tested more rigorously. The CFPV ap-
proach, presented in Chapter 3, relies on the compressibility corrections, which

129
can be improved.
" Application of the CEFT model: Although the memory requirement of the
CEFT approach, section 3.2, is prohibitive, LES of compressible reacting flows
using this approach could provide invaluable information for improving the
CFPV model. The memory issue could be alleviated by using kd-trees for
storing the five-dimensional table (see Shunn, 2009).
" Radiative heat transfer: Radiation and its coupling with turbulence and chem-
istry through the associated redistribution of heat contributes significantly to
the overall energy balance in scramjets. Estimates range from less than 1% for
a small hydrogen fueled scramjet (HyShot) to 5% for a larger hydrocarbon fu-
eled system (HIFiRE). Its importance will further increase as these devices are
scaled up in size and pressure for operational systems. As such, it is important
that modeling and simulation tools include the effects of radiation heat transfer
at an appropriate level of fidelity for these devices. A common practice for as-
sessing the effects of radiation is to completely decouple the radiation transport
from the computational fluid dynamics. This approach completely omits the
important effects of radiation on temperature and hence on combustion chem-
istry. Due to a lack of high quality heat transfer measurements, the effects of
this decoupled radiation modeling approach on simulations are unknown.

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