Large Eddy Simulation of
Supersonic Combustion with
Application to Scramjet Engines
Peter Alan Thomas Cocks
Corpus Christi College
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
March 2011
To my Mum and Dad...
Acknowledgements
I would like to thank my supervisor Professor Bill Dawes for his guidance and
support, both technically and financially, over the last few years. I would also like
to thank my advisor Professor Stewart Cant for the many technical discussions,
from which I learned a great deal, and for access to the SENGA code, which was
a significant aid in the development of PULSAR. I must also thank Dr Caleb
Dhanasekaran for his constant help with anything from computational resources
to software compilation. I am also thankful for the support of Mary Aldridge,
without whom the CFD Lab would struggle to operate.
My PhD has been funded by a Departmental EPSRC DTA studentship, for
which I am very grateful. I want to separately thank the EPSRC for awarding
to me a grant for time on the national supercomputer, HECToR, without which
many of the simulations discussed in this thesis would not have been possible. I
wish to thank Peter Benie and the University HPC support team for access to
resources such at the Stokes cluster and for general help in compiling and running
PULSAR on such multi-core machines. I must also thank the Department of
Engineering and Corpus Christi College for their financial support, which enabled
me to attend and present at two conferences.
I am very grateful to the German Research Training Group GRK 1095/2
for inviting me to multiple events throughout my PhD, from summer schools to
meetings, from which I have learnt a lot. This interaction with fellow researchers
in the scramjet field has been invaluable. I must also thank Professor Andrew
Cutler from George Washington University for providing experimental data for
the two test cases used in my work.
Many past and present members of the CFD Lab have helped me throughout
my PhD and I am grateful for their guidance and insightful technical discussions.
Last, but not least, I would like to thank my family for their constant support
and encouragement, which made all this possible.
Abstract
This work evaluates the capabilities of the RANS and LES techniques for the
simulation of high speed reacting flows. These methods are used to gain further
insight into the physics encountered and regimes present in supersonic combus-
tion. The target application of this research is the scramjet engine, a propulsion
system of great promise for efficient hypersonic flight. In order to conduct this
work a new highly parallelised code, PULSAR, is developed. PULSAR is capable
of simulating complex chemistry combustion in highly compressible flows, based
on a second order upwind method to provide a monotonic solution in the presence
of high gradient physics.
Through the simulation of a non-reacting supersonic coaxial helium jet the
RANS method is shown to be sensitive to constants involved in the modelling
process. The LES technique is more computationally demanding but is shown
to be much less sensitive to these model parameters. Nevertheless, LES results
are shown to be sensitive to the nature of turbulence at the inflow; however this
information can be experimentally obtained.
The SCHOLAR test case is used to validate the reacting aspects of PUL-
SAR. Comparing RANS results from laminar chemistry and assumed PDF com-
bustion model simulations, the influence of turbulence-chemistry interactions in
supersonic combustion is shown to be small. In the presence of reactions, the
RANS results are sensitive to inflow turbulence, due to its influence on mixing.
From complex chemistry simulations the combustion behaviour is evaluated to
sit between the flamelet and distributed reaction regimes. LES results allow an
evaluation of the physics involved, with a pair of coherent vortices identified as
the dominant influence on mixing for the oblique wall fuel injection method. It is
shown that inflow turbulence has a significant impact on the behaviour of these
vortices and hence it is vital for turbulence intensities and length scales to be
measured by experimentalists, in order for accurate simulations to be possible.
Statement
This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration with others. No parts of this disserta-
tion have been submitted for any other qualification. This dissertation contains
approximately 48,000 words and 135 figures.
Peter Cocks
March 2011
Contents
Contents v
List of Figures xi
Nomenclature xxviii
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . 1
1.2 Turbulent Combustion Modelling . . . . . . . . . . . . . . . . . . 4
1.2.1 Direct Numerical Simulation (DNS) . . . . . . . . . . . . . 4
1.2.2 Large Eddy Simulation (LES) . . . . . . . . . . . . . . . . 5
1.2.3 Reynolds Averaged Navier Stokes (RANS) . . . . . . . . . 5
1.3 Aims of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Turbulence Modelling 9
2.1 Introduction to Turbulence . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Reynolds Averaged Navier Stokes (RANS) . . . . . . . . . . . . . 15
2.3.1 Favre Averaging . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3.1 Mixing Length Model . . . . . . . . . . . . . . . 19
2.3.3.2 One Equation Models . . . . . . . . . . . . . . . 19
2.3.3.3 Two Equation Models . . . . . . . . . . . . . . . 20
v
CONTENTS
2.3.3.4 Reynolds Stress Model . . . . . . . . . . . . . . . 22
2.3.3.5 Model Constants . . . . . . . . . . . . . . . . . . 23
2.3.4 Unsteady RANS (URANS) . . . . . . . . . . . . . . . . . . 24
2.4 Large Eddy Simulation (LES) . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Favre Filtering . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 Explicit Subgrid-Scale (SGS) Models . . . . . . . . . . . . 28
2.4.2.1 Smagorinsky model . . . . . . . . . . . . . . . . . 29
2.4.2.2 Dynamic model . . . . . . . . . . . . . . . . . . . 30
2.4.3 Implicit LES . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.4 Wall Bounded LES . . . . . . . . . . . . . . . . . . . . . . 32
2.4.4.1 Wall Functions . . . . . . . . . . . . . . . . . . . 33
2.4.4.2 Two Layer Model . . . . . . . . . . . . . . . . . . 34
2.4.4.3 DES . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.4.4 DDES . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.4.5 Transition Region Turbulence . . . . . . . . . . . 38
3 Combustion Modelling 40
3.1 Supersonic Combustion . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Laminar Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Turbulence-Chemistry Interaction Modelling . . . . . . . . . . . . 47
3.3.1 Flamelet Model . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 Probability Density Function Approach . . . . . . . . . . . 49
3.3.2.1 Transported PDF . . . . . . . . . . . . . . . . . . 50
3.3.2.2 Assumed PDF . . . . . . . . . . . . . . . . . . . 51
3.3.3 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Combustion Model Influence . . . . . . . . . . . . . . . . . . . . . 54
3.5 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Numerical Methods and Implementation 60
4.1 Introduction to PULSAR . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
vi
CONTENTS
4.4 Inviscid Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.1 Desirable Properties . . . . . . . . . . . . . . . . . . . . . 64
4.4.1.1 Monotonicity . . . . . . . . . . . . . . . . . . . . 65
4.4.1.2 High Order of Accuracy . . . . . . . . . . . . . . 65
4.4.2 Discretisation Techniques . . . . . . . . . . . . . . . . . . 65
4.4.2.1 Centred Scheme . . . . . . . . . . . . . . . . . . 66
4.4.2.2 Test case: Discontinuity Capturing - Centred . . 67
4.4.2.3 AUSM+ . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.2.4 AUSM+UP . . . . . . . . . . . . . . . . . . . . . 72
4.4.2.5 Test case: Discontinuity Capturing - Upwind . . 74
4.4.3 Linear Interpolation . . . . . . . . . . . . . . . . . . . . . 75
4.4.3.1 Gradient Limiter . . . . . . . . . . . . . . . . . . 77
4.4.3.2 Test case: Discontinuity Capturing . . . . . . . . 78
4.4.3.3 Test case: Sod’s Shock Tube Problem . . . . . . 79
4.5 Viscous Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5.1 Gradient Reconstruction . . . . . . . . . . . . . . . . . . . 82
4.5.2 Discretisation Technique . . . . . . . . . . . . . . . . . . . 83
4.6 Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6.1 RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.6.2 DDES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6.3 Wall Distance . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6.4 Compressibility Corrections . . . . . . . . . . . . . . . . . 88
4.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7.1 Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7.1.1 Supersonic . . . . . . . . . . . . . . . . . . . . . 89
4.7.1.2 Subsonic . . . . . . . . . . . . . . . . . . . . . . . 90
4.7.1.3 Turbulence . . . . . . . . . . . . . . . . . . . . . 91
4.7.2 Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.7.3 Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.7.3.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . 92
4.7.3.2 Temperature . . . . . . . . . . . . . . . . . . . . 94
4.7.3.3 Turbulence . . . . . . . . . . . . . . . . . . . . . 95
4.8 Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
vii
CONTENTS
4.8.1 Thermo-chemistry . . . . . . . . . . . . . . . . . . . . . . 96
4.8.2 Assumed PDF Combustion Model . . . . . . . . . . . . . . 98
4.8.2.1 Clipped Gaussian PDF . . . . . . . . . . . . . . . 99
4.8.2.2 Multivariate Beta PDF . . . . . . . . . . . . . . 100
4.8.2.3 Mean Reaction Rate . . . . . . . . . . . . . . . . 102
4.8.2.4 Backward Reaction Rate Constant . . . . . . . . 102
4.9 Temporal Discretisation . . . . . . . . . . . . . . . . . . . . . . . 102
4.10 Turbulence Initialisation . . . . . . . . . . . . . . . . . . . . . . . 103
4.11 Parallel Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.11.1 Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.11.2 HECToR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.11.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 Supersonic Non-Reacting Coaxial Jet - RANS 108
5.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . 110
5.1.2 Computational Mesh . . . . . . . . . . . . . . . . . . . . . 112
5.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 114
5.1.4 Computational Methods . . . . . . . . . . . . . . . . . . . 114
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1 Turbulent Schmidt Number . . . . . . . . . . . . . . . . . 116
5.2.2 Compressibility Correction . . . . . . . . . . . . . . . . . . 119
5.2.3 Turbulent Prandtl Number . . . . . . . . . . . . . . . . . . 121
5.2.4 Velocity Distribution . . . . . . . . . . . . . . . . . . . . . 123
5.2.5 Mesh Convergence . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Supersonic Non-Reacting Coaxial Jet - LES 134
6.1 Delayed Detached Eddy Simulation . . . . . . . . . . . . . . . . . 135
6.1.1 Computational Mesh . . . . . . . . . . . . . . . . . . . . . 135
6.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 136
6.1.3 Computational Methods . . . . . . . . . . . . . . . . . . . 136
6.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
viii
CONTENTS
6.2 Turbulence Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2.1 Computational Approach . . . . . . . . . . . . . . . . . . . 140
6.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2.2.1 Helium Mass Fraction . . . . . . . . . . . . . . . 147
6.2.2.2 Pitot Pressure . . . . . . . . . . . . . . . . . . . 149
6.2.2.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . 151
6.2.2.4 Statistical Convergence . . . . . . . . . . . . . . 156
6.3 Parameter Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3.1 Computational Mesh . . . . . . . . . . . . . . . . . . . . . 161
6.3.2 Filter Width . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3.3 Reynolds Stresses . . . . . . . . . . . . . . . . . . . . . . . 167
6.3.4 Near Wall Scales . . . . . . . . . . . . . . . . . . . . . . . 170
6.3.5 Mesh Resolution . . . . . . . . . . . . . . . . . . . . . . . 173
6.4 Modified Turbulence Initialisation . . . . . . . . . . . . . . . . . . 176
7 SCHOLAR Test Case 184
7.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.1.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . 186
7.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 187
7.2 RANS Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.2.1 Computational Mesh . . . . . . . . . . . . . . . . . . . . . 188
7.2.2 Computational Methods . . . . . . . . . . . . . . . . . . . 190
7.2.3 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . 191
7.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.2.5 Regime of Supersonic Combustion . . . . . . . . . . . . . . 197
7.2.6 Inlet Turbulence Influence . . . . . . . . . . . . . . . . . . 201
7.2.7 Radicals Influence . . . . . . . . . . . . . . . . . . . . . . . 204
7.2.8 Mesh Convergence . . . . . . . . . . . . . . . . . . . . . . 206
7.2.9 Computational Cost . . . . . . . . . . . . . . . . . . . . . 206
7.3 DDES Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.3.1 Computational Mesh . . . . . . . . . . . . . . . . . . . . . 207
7.3.2 Computational Methods . . . . . . . . . . . . . . . . . . . 208
7.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
ix
CONTENTS
8 Conclusions 236
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.1.2 Non-Reacting Flow . . . . . . . . . . . . . . . . . . . . . . 237
8.1.3 Reacting Flow . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.1.4 Revealed Physics . . . . . . . . . . . . . . . . . . . . . . . 242
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.2.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . 244
8.2.2 Turbulence Modelling . . . . . . . . . . . . . . . . . . . . . 245
8.2.3 Turbulence Inflow . . . . . . . . . . . . . . . . . . . . . . . 246
8.2.4 Supersonic Combustion . . . . . . . . . . . . . . . . . . . . 247
Appendix A: Jachimowski Reaction Mechanisms 248
References 250
x
List of Figures
1.1 Normal (left) and oblique (right) wall injection [6] . . . . . . . . . 3
1.2 Schematic of strut based injection [8] . . . . . . . . . . . . . . . . 3
2.1 Generic energy spectrum, with energy E and frequency k. . . . . . 10
2.2 One dimensional energy spectra for different experimental cases,
showing problem dependent large scales and universal small scales
[24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Vorticity magnitude contours from a square cylinder URANS sim-
ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Vorticity magnitude contours from a square cylinder large eddy
simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Turbulence energy spectrum showing mesh and test filters . . . . 31
3.1 Regime diagram: Log-log plot of Damko¨hler number versus tur-
bulent Reynolds number . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Supersonic combustion regimes proposed by Balakrishnan andWilliams
(light blue) and Ingenito and Bruno (red) . . . . . . . . . . . . . . 44
4.1 Data storage locations. Cell-vertex (blue) and cell-centre (green). 63
4.2 Left and right states used to calculate face data for the centred
discretisation method. . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Disadvantages of the centred scheme on a non-regular mesh. Black
dot corresponds to half way between cell centres and red star cor-
responds to centre of face. . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Initial conditions for the mass fraction discontinuity test case . . . 68
xi
LIST OF FIGURES
4.5 Capability of the centred scheme for discontinuity capturing . . . 69
4.6 Nitrogen mass fraction profile when using the centred scheme . . . 69
4.7 Capability of the 1st order upwind scheme for discontinuity capturing 74
4.8 Linear interpolation approach . . . . . . . . . . . . . . . . . . . . 75
4.9 Resolution comparison for 1st order upwind (top) and second order
upwind with Van Rosendale gradient limiter (bottom), schemes . 78
4.10 Capabilities of the higher order upwind method for discontinuity
capturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.11 Initial conditions and analytical solution profile for Sod’s shock
tube problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.12 Pressure profile for Sod’s shock tube problem, after 1500 iterations 81
4.13 Temperature profile for Sod’s shock tube problem, after 1500 iter-
ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.14 Stencil for least-squares gradient reconstruction in shaded cell.
Anisotropic grid displayed. . . . . . . . . . . . . . . . . . . . . . . 83
4.15 Dimensionless profile for a flat plate zero pressure gradient bound-
ary layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.16 Strong scaling of PULSAR on the HECToR XT6 cluster, before
and after interconnect upgrade . . . . . . . . . . . . . . . . . . . . 107
5.1 Coaxial jet assembly [100] . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Mach number profile . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Helium mass fraction profile . . . . . . . . . . . . . . . . . . . . . 110
5.4 Velocity magnitude profile, m/s . . . . . . . . . . . . . . . . . . . 110
5.5 Pitot tube shock schematic . . . . . . . . . . . . . . . . . . . . . . 111
5.6 Slice through RANS mesh at z = 0, near the jet exit . . . . . . . 112
5.7 y+ values: Scaled to show distributions for the co-flow (top) and
central (bottom) jet surfaces . . . . . . . . . . . . . . . . . . . . . 113
5.8 Convergence studied through data from monitor point . . . . . . . 115
5.9 Radial (r) profiles of normalised helium mass fraction at several
axial (x) locations, with varying Schmidt number and a Prandtl
number of 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xii
LIST OF FIGURES
5.10 Radial (r) profiles of normalised pitot pressure at several axial (x)
locations, with varying Schmidt number and a Prandtl number of
0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.11 Radial (r) profiles of normalised helium mass fraction at several
axial (x) locations, with different compressibility corrections and
turbulent Schmidt and Prandtl numbers of 1.0 and 0.9, respectively120
5.12 Turbulence Mach number profile . . . . . . . . . . . . . . . . . . . 121
5.13 Radial (r) profiles of normalised pitot pressure at several axial
(x) locations, showing the influence of the Sarkar compressibility
correction and variation of the turbulent Prandtl number, with the
turbulence Schmidt number held constant at 1.0 . . . . . . . . . . 122
5.14 Radial (r) profiles of normalised helium mass fraction at several
axial (x) locations, with different turbulent Prandtl numbers and
the turbulent Schmidt equal to 1.0 . . . . . . . . . . . . . . . . . 124
5.15 Radial (r) profiles of mean axial velocity at several axial (x) locations.125
5.16 Radial (r) profiles of the axial rms velocity, comparing the Boussi-
nesq and turbulence kinetic energy approximations to experimental
data at several axial (x) locations. . . . . . . . . . . . . . . . . . . 127
5.17 Radial (r) profiles of the terms involved in the Boussinesq approx-
imation to the Reynolds stress. . . . . . . . . . . . . . . . . . . . 128
5.18 Turbulence kinetic energy profile . . . . . . . . . . . . . . . . . . 129
5.19 Radial (r) profiles of the transverse rms velocities, comparing the
Boussinesq (red and blue) and turbulence kinetic energy approxi-
mations at several axial (x) locations. . . . . . . . . . . . . . . . . 131
5.20 Helium mass fraction profiles confirming mesh convergence . . . . 132
5.21 Pitot pressure profiles confirming mesh convergence . . . . . . . . 133
6.1 y+ distribution for coaxial jet DDES . . . . . . . . . . . . . . . . 136
6.2 Fine DDES mesh, showing high aspect ratio cells in the shear layer
regions and isotropic cells elsewhere. . . . . . . . . . . . . . . . . 137
6.3 Instantaneous helium mass fraction distribution for coaxial jet DDES137
6.4 Experimental schlieren image [100], with angled shroud removed . 138
xiii
LIST OF FIGURES
6.5 Radial (r) profiles of normalised helium mass fraction at two axial
(x) locations for the DDES . . . . . . . . . . . . . . . . . . . . . . 138
6.6 Location of turbulence inlet plane and corresponding geometry
cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.7 Mesh used to generate turbulence inlet data. Half of the mesh
provided to show internal characteristics. . . . . . . . . . . . . . . 141
6.8 Generated inlet turbulence velocity data, in m/s. Half of the do-
main provided to show the internal profile. . . . . . . . . . . . . . 142
6.9 z-velocity component in m/s, showing injected coherent structures
and shear layer interactions. . . . . . . . . . . . . . . . . . . . . . 143
6.10 Instantaneous helium mass fraction profile for the coaxial jet DDES
simulation with imposed turbulent fluctuations. . . . . . . . . . . 144
6.11 Positive Q iso-surfaces coloured by helium mass fraction. . . . . . 145
6.12 Helium mass fraction time history (left) and mean convergence
(right) on the centreline at x=0.27m. . . . . . . . . . . . . . . . . 146
6.13 Turbulence energy spectrum for coaxial jet DDES . . . . . . . . . 147
6.14 Radial (r) profiles of mean normalised helium mass fraction at sev-
eral axial (x) locations, with varying Prandtl and Schmidt numbers 148
6.15 Comparison of ensemble averaged DDES (top) and RANS (bot-
tom) helium mass fraction profiles. . . . . . . . . . . . . . . . . . 149
6.16 Radial (r) profiles of mean normalised pitot pressure at several
axial (x) locations, with varying Prandtl and Schmidt numbers. . 150
6.17 Numerical schlieren showing interactions between outer and inner
shear layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.18 Radial (r) profiles of mean axial velocity at several axial (x) loca-
tions, with varying Prandtl and Schmidt numbers. . . . . . . . . . 153
6.19 Radial (r) profiles of rms axial velocity at several axial (x) loca-
tions, with varying Prandtl and Schmidt. . . . . . . . . . . . . . . 154
6.20 Radial (r) profiles of rms velocity components at several axial (x)
locations, from Simulation 2. . . . . . . . . . . . . . . . . . . . . . 155
6.21 x (left), y (middle) and z (right) velocity profiles on a sample slice
of the turbulence inlet data, in m/s. . . . . . . . . . . . . . . . . . 156
xiv
LIST OF FIGURES
6.22 Radial (r) profiles of mean normalised helium mass fraction at
several axial (x) locations, after 11 and 19.5 flow-through-times,
for Simulation 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.23 Radial (r) profiles of mean axial velocity at several axial (x) loca-
tions, after 11 and 19.5 flow-through-times, for Simulation 1. . . . 158
6.24 Radial (r) profiles of mean normalised pitot pressure at several
axial (x) locations, after 11 and 19.5 flow-through-times, for Sim-
ulation 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.25 Axial rms velocity convergence . . . . . . . . . . . . . . . . . . . . 160
6.26 Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, comparing
the cube root of volume and cell maximum length filter width
calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.27 Instantaneous SGS viscosity profiles after 630,000 iterations, com-
paring the cell maximum length (top) and cube root of volume
(bottom) filter width calculations. . . . . . . . . . . . . . . . . . . 164
6.28 Instantaneous helium mass fraction profiles after 630,000 itera-
tions, comparing the cell maximum length (top) and cube root
of volume (bottom) filter width calculations. . . . . . . . . . . . . 165
6.29 Instantaneous numerical schlieren images after 630,000 iterations,
comparing the cell maximum length (top) and cube root of volume
(bottom) filter width calculations. . . . . . . . . . . . . . . . . . . 166
6.30 RANS-LES blending function for the DDES method, with blue
and red corresponding to the RANS and LES regimes, respectively.
Maximum length (left) and cube root of volume (right) filter widths
are shown at the tip of the central jet nozzle. . . . . . . . . . . . . 167
6.31 Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, with vary-
ing inlet turbulence Reynolds stress, in m2s−2. . . . . . . . . . . . 168
6.32 Instantaneous numerical schlieren images after 810,000 iterations,
comparing inlet turbulence Reynolds stresses of 20m2s−2 (top) and
100m2s−2 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 169
xv
LIST OF FIGURES
6.33 Comparing near wall scaling coefficients, αw, of 0.2 (left), 0.4 (mid-
dle) and 0.6 (right) on a sample slice of the turbulence inlet data.
Velocity magnitude shown, with scale 0 to 10 m/s. . . . . . . . . . 170
6.34 Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, with vary-
ing near wall scaling coefficient. . . . . . . . . . . . . . . . . . . . 171
6.35 Instantaneous numerical schlieren images after 810,000 iterations,
comparing inlet turbulence near wall scaling coefficients of 0.2 (top)
and 0.6 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.36 Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, on coarse
and fine computational meshes. . . . . . . . . . . . . . . . . . . . 174
6.37 Instantaneous SGS viscosity profiles after 810,000 iterations on
coarse (top) and fine (bottom) computational meshes. . . . . . . . 175
6.38 Turbulence energy spectrum for coarse mesh using x-velocity (green),
y-velocity (blue) and z-velocity (red). . . . . . . . . . . . . . . . . 176
6.39 Showing influence of cell volume scaling on turbulence initialisation
with volume multiplication (left), no volume scaling (middle) and
volume division (right). Velocity magnitude shown, with scale 0
to 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.40 Instantaneous numerical schlieren image with no volume scaling
applied to the inlet turbulence initialisation. . . . . . . . . . . . . 178
6.41 Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, with orig-
inal and no volume scaling applied to the inlet turbulence initiali-
sation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.42 Radial (r) profiles of rms axial velocity at several axial (x) lo-
cations, with no volume scaling applied to the inlet turbulence
initialisation, after 90,000 and 180,000 iterations of rms calculations.181
6.43 Instantaneous numerical schlieren image with the co-flow inlet tur-
bulence removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
xvi
LIST OF FIGURES
7.1 Schematic of the SCHOLAR experiment [105] (a) Detail of noz-
zle and combustor (b) Detail of combustor entrance and injector
geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.2 Hydrogen mass fraction distribution on the combustor centreline . 185
7.3 Planes of temperature and oxygen mole fraction experimental data
down the axis of the SCHOLAR combustor . . . . . . . . . . . . . 186
7.4 Experimental pressure distributions for the upper (left) and lower
(right) walls of the combustor, after 10 and 24 seconds and the mean.187
7.5 RANS mesh for the SCHOLAR test case. Detail around the step
and injector shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.6 y+ distribution for the SCHOLAR test case RANS simulations . . 190
7.7 Convergence of the temperature and hydrogen mass fraction on
the combustor centreline at x = 0.777m . . . . . . . . . . . . . . . 190
7.8 Comparison of mean temperature profiles from each RANS simu-
lation to experimental data . . . . . . . . . . . . . . . . . . . . . 193
7.9 Shock induced ignition shown by contours of static pressure over
a static temperature profile . . . . . . . . . . . . . . . . . . . . . 193
7.10 Comparison of mean oxygen mole fraction profiles from each RANS
simulation to experimental data . . . . . . . . . . . . . . . . . . . 195
7.11 Comparison of upper (left) and lower (right) wall pressure distri-
butions to experimental data, for each RANS simulation . . . . . 195
7.12 Comparison of upper (left) and lower (right) wall centreline and
near wall pressure distributions, for Simulation 1 . . . . . . . . . . 196
7.13 Regime diagram showing data from 1-step chemistry post-processing,
at locations down the axis of the combustor . . . . . . . . . . . . 198
7.14 Damko¨hler profiles down the axis of the SCHOLAR combustor for
each complex chemistry RANS simulation . . . . . . . . . . . . . 199
7.15 Distribution of large eddy turnover time for Simulation 3 . . . . . 200
7.16 Influence of inflow turbulence on the upper (left) and lower (right)
wall pressure distributions for Simulations 1 and 2. . . . . . . . . 202
7.17 Influence of heat release on the upper (left) and lower (right) wall
pressure distributions and influence of turbulence inflow on results
from a non-reacting simulation . . . . . . . . . . . . . . . . . . . . 203
xvii
LIST OF FIGURES
7.18 Influence of turbulence inflow on temperature profiles down the
axis of the SCHOLAR combustor, showing reduced mechanism
results with (a) laminar chemistry and low turbulence, (b) assumed
PDF combustion model and low turbulence, (c) laminar chemistry
and high turbulence and (d) assumed PDF combustion model and
high turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.19 Influence of turbulence inflow on oxygen mole fraction profiles
down the axis of the SCHOLAR combustor, showing reduced mech-
anism results with (a) laminar chemistry and low turbulence, (b)
assumed PDF combustion model and low turbulence, (c) laminar
chemistry and high turbulence and (d) assumed PDF combustion
model and high turbulence . . . . . . . . . . . . . . . . . . . . . . 204
7.20 Influence of inlet radical species concentration on upper (left) and
lower (right) wall pressure distributions for Simulation 1 . . . . . 205
7.21 Centreline static temperature profiles for Simulation 1 with nozzle
reactions (top) and without (bottom) . . . . . . . . . . . . . . . . 205
7.22 Positive Q isosurfaces coloured by velocity magnitude in m/s, show-
ing the inflow turbulence profile . . . . . . . . . . . . . . . . . . . 210
7.23 Comparison of the ensemble averaged oxygen mole fraction to ex-
perimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.24 Comparison of the ensemble averaged temperature to experimental
data. Experimental data and (a) are scaled from 200K (blue) to
2200K (red), whilst (b) is scaled from 200K (blue) to 1700K (red).
(a) and (b) are the same computational data. . . . . . . . . . . . 212
7.25 Comparison of upper (left) and lower (right) wall pressure distri-
butions to experimental data, showing instantaneous and ensemble
averaged data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.26 Comparison of ensemble averaged oxygen mole fraction profiles for
simulations using (a) laminar chemistry and a turbulence inflow,
(b) laminar chemistry and no turbulence inflow and (c) the as-
sumed PDF combustion model and a turbulence inflow . . . . . . 214
xviii
LIST OF FIGURES
7.27 Comparison of ensemble averaged temperature profiles for simu-
lations using (a) laminar chemistry and a turbulence inflow, (b)
laminar chemistry and no turbulence inflow and (c) the assumed
PDF combustion model and a turbulence inflow . . . . . . . . . . 215
7.28 Comparison of upper (left) and lower (right) wall pressure distri-
butions for the three DDES simulations . . . . . . . . . . . . . . . 216
7.29 Convergence of ensemble averaged hydrogen mole fraction on the
centreline of the combustor at x=0.777m . . . . . . . . . . . . . . 216
7.30 Comparison of ensemble averaged oxygen mole fraction and tem-
perature profiles after 780,000 and 1,230,000 iterations for the lam-
inar chemistry simulation with a turbulence inflow . . . . . . . . . 217
7.31 Positive Q isosurfaces coloured by hydrogen mass fraction for the
three DDES simulations after 180,000 (top) and 780,000 (bottom)
iterations, for each case . . . . . . . . . . . . . . . . . . . . . . . . 218
7.32 Instantaneous temperature profiles for the (a) laminar chemistry
with a turbulence inflow, (b) laminar chemistry with no turbulence
inflow and (c) assumed PDF combustion model with a turbulence
inflow, simulations, at three axial locations and at three instances
in time. Scaled from 200K (blue) to 2200K (red). . . . . . . . . . 220
7.33 Instantaneous oxygen mole fraction profiles for the (a) laminar
chemistry with a turbulence inflow, (b) laminar chemistry with no
turbulence inflow and (c) assumed PDF combustion model with
a turbulence inflow, simulations, at three axial locations and at
three instances in time. Scaled from 0 (blue) to 0.24 (red). . . . . 221
7.34 Instantaneous hydrogen mass fraction profiles for the (a) laminar
chemistry with a turbulence inflow, (b) laminar chemistry with no
turbulence inflow and (c) assumed PDF combustion model with
a turbulence inflow, simulations, at three axial locations and at
three instances in time. Scaled from 0 (blue) to 0.7 (red). . . . . . 222
7.35 Positive Q isosurfaces coloured by hydrogen mass fraction after
300,000 (top), 600,000 (middle) and 900,000 (bottom) time steps,
for the laminar chemistry simulation with a turbulence inflow. . . 223
xix
LIST OF FIGURES
7.36 Profiles of vorticity magnitude down the axis of the combustor for
the laminar chemistry simulation with a turbulence inflow, after
1.2M iterations. The images on the first five rows are in 0.01m
steps, starting at x=0.16m and ending at x=0.35m. The images
on the last two rows are in 0.05m steps, from x=0.40m to x=0.75m.224
7.37 Positive Q isosurfaces around the region of injection coloured by
hydrogen mass fraction, showing views from an angle (top), the
side (middle) and above (bottom). . . . . . . . . . . . . . . . . . . 226
7.38 Profiles of positive Q (top) and velocity vectors coloured by vor-
ticity magnitude (bottom) at three axial locations, showing the
counter-rotating and secondary vortices. . . . . . . . . . . . . . . 227
7.39 Positive Q isosurfaces around the region of injection coloured by
static pressure, showing views from an angle (top), the side (mid-
dle) and above (bottom). . . . . . . . . . . . . . . . . . . . . . . . 229
7.40 Three dimensional schlieren image coloured by static pressure . . 230
7.41 Pressure contours coloured by static pressure either side of the
centreline (top and bottom) and on the centreline (middle). . . . . 231
7.42 Three-dimensional contours of: Temperature = 1900K, YOH =
0.015, YO = 0.008 and YH = 0.004, with a two-dimensional pres-
sure contour on the combustor centreline. . . . . . . . . . . . . . . 232
7.43 Side view of positive Q contours coloured by hydrogen mass fraction233
7.44 Instantaneous hydrogen mass fraction profile on the centreline after
1.2M iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.45 Centreline σY profile from Simulation 2 of the RANS study. . . . 234
7.46 Turbulence energy spectrum using data for the y-velocity from
225,000 to 990,000 iterations on the centreline of the combustor at
x=0.777m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
xx
Nomenclature
Roman Symbols
a
(l)
α,k Mass based coefficients of thermochemistry polynomial
a¯
(l)
α,k Molar based coefficients of thermochemistry polynomial
Ar Pre-exponential factor for reaction r
A Area
a Speed of sound
B Beta function parameter
CDES DES model constant
Cµ k − ε model constant
cβ Concentration of species β
cp Mixture specific heat capacity at constant pressure
Cs Smagorinsky constant
CT , CY Assumed PDF combustion model constants = 2.0
cv Mixture specific heat capacity at constant volume
D Diffusion term
Da Damko¨hler number
xxi
Nomenclature
Dα Diffusion coefficient for species α
d˜ Spalart-Allmaras DES model length scale
e Specific internal energy
Er Activation energy for reaction r
F1, F2 Menter SST model blending functions
fa AUSM+UP low Mach number scaling term
fd DDES RANS/LES blending function
G Filter function
g¯α Molar Gibbs function for species α
h Specific enthalpy
hα Specific enthalpy of species α
H Heaviside step function
I Turbulence intensity
IσT Temperature variance intensity
k Turbulence kinetic energy
kfr, kbr Forward and backward reaction rate constant for reaction r,
respectively
Kp AUSM+UP model constant
Ku AUSM+UP model constant
L Characteristic length
Leα Lewis number for species α
Lij Leonard stresses
xxii
Nomenclature
lij Distance from centre of cell i to centre of cell j
lm Mixing length
Lmax Maximum turbulence length scale
l˜ Menter SST DES model length scale
M Mach number, or third body in the context of chemistry
M AUSM Mach function
M∞ AUSM+UP free-stream Mach number
Mo AUSM+UP reference Mach number
Mt Turbulence Mach number
Mα Chemical symbol for species α
m˙ Mass flux
Mp AUSM+UP pressure diffusion term
~n Outward pointing unit normal vector
Np Degree of thermochemistry polynomial
Nr Number of reactions
nr Temperature power constant for reaction r
Ns Number of species
p Static pressure
P− Destruction term
P Probability density function
P+ Production term
Pr Prandtl number
xxiii
Nomenclature
P AUSM pressure function
pu AUSM+UP velocity diffusion term
qj Heat flux vector
Rα Specific gas constant for species α
rd DDES RANS/LES blending parameter
Re Reynolds number
Rij Reynolds stress tensor
Rm Mixture specific gas constant
R0 Universal gas constant
~rL, ~rR Vectors from centre of left and right cells, respectively, to centre
of common face
~rij Vector from centre of cell i to centre of cell j
S Strain invariant
Sc Schmidt number
Sij Strain rate tensor
τc Chemical time scale
τK Kolmogorov time scale
lc Chemical length scale
T Temperature
Tij Sub-test-filter stresses
~tij Unit vector from centre of cell i to centre of cell j
t Time
xxiv
Nomenclature
T+ Dimensionless temperature
U Characteristic velocity
uj Velocity vector component in direction j
uK Kolmogorov velocity scale
u+ Dimensionless velocity
utang Near wall tangential velocity
uτ Near wall shear velocity
Vα,j Diffusion velocity vector for species α
V Volume
Wα Molecular weight of species α
w Weight for use in gradient limiter
xj Spatial vector component in direction j
Yα Mass fraction of species α
y+ Dimensionless wall distance
yw Distance to nearest viscous wall
Z Mixture fraction
Greek Symbols
αw Near wall eddy scaling coefficient
βγ Beta function parameter for species γ
βk L2 norm of gradient in cell k
β⋆ Menter SST model constant = 0.09
χ Scalar dissipation rate
xxv
Nomenclature
∆ Filter width
δij Kronecker delta
∆t Time step
δ Delta function
∆x Cell width
ε Turbulence dissipation rate
ηK Kolmogorov length scale
ηβ,M Third body efficiency of species β for third body M
Γ Gamma function or diffusion coefficient
γ Ratio of specific heats
κ Ka´rma´n constant
λ Mixture thermal conductivity
µ Dynamic viscosity
ν Kinematic viscosity
ν
′
αr, ν
′′
αr Molar reactant and product stoichiometric coefficients for species
α in reaction r, respectively
ω Turbulence frequency
ωf Fluid mechanics time scale
ψf , ψg Flux and gradient limiters, respectively
ω˙α Production rate of species α
ρ Density
σT Temperature variance
xxvi
Nomenclature
σY Sum of species mass fraction variances
τij Viscous stress tensor
τK Kolmogorov time scale
τw Wall shear stress
Subscripts
0 Stagnation property
f Face quantity
i, j, k Vector components
in Inlet parameter
K Kolmogorov parameter
l Molecular/laminar
L State in left hand cell
R State in right hand cell
t Turbulent
w Viscous wall boundary condition
Accents
¯ Averaged (RANS) or filtered (LES) quantity
′
Fluctuating quantity
ˆ Sample space variable
˜ Favre averaged (RANS) or favre filtered (LES) quantity
~ Vector
xxvii
Nomenclature
Acronyms
AUSM Advection Upstream Splitting Method
CARS Coherent Anti-Stokes Raman Spectroscopy
CFD Computational Fluid Dynamics
CFL Courant-Friedrichs-Lewy number
CV Control Volume
DDES Delayed Detached Eddy Simulation
DES Detached Eddy Simulation
DNS Direct Numerical Simulation
ILES Implicit Large Eddy Simulation
LES Large Eddy Simulation
PDF Probability density function
PULSAR Parallel Unstructured Large-eddy Simulation And Reactions
RANS Reynolds Averaged Navier Stokes
RSM Reynolds Stress Model
SGS Subgrid-scale
TVD Total Variation Diminishing
URANS Unsteady Reynolds Averaged Navier Stokes
xxviii
Chapter 1
Introduction
1.1 Background and Motivation
The desire for hypersonic flight has motivated engineers for decades to satisfy
demands for high speed transport and economical access to space. In order to
meet these needs and ambitions, significant advancements in propulsion technol-
ogy are required. The gas turbine engines commonly employed in subsonic and
low supersonic aircraft are not practical above flight Mach numbers of approxi-
mately three. Rocket engines are currently the propulsion system of choice for
hypersonic flight, but suffer from poor efficiency [1] and low levels of safety. One
device that holds great promise in the area of hypersonic air-breathing propulsion
is the supersonic combustion ramjet (scramjet) engine.
The standard ramjet engine slows the incoming flow down to subsonic speeds
in order to allow efficient mixing of air and fuel and subsequent combustion.
However, as the free stream Mach number increases, the removal of a greater
amount of kinetic energy from the flow incurs significant losses. Therefore, at
high free stream Mach numbers the flow inside the combustion chamber is allowed
to remain supersonic, leading to the creation of the supersonic combustion ramjet
(scramjet).
The scramjet engine is a mechanically simple system due to its lack of moving
parts. Rather than employing the active compression system found in gas turbine
engines, scramjet propulsion relies on the technique of ram compression to slow
1
the incoming air and increase its pressure, pre-combustion. This ram compression
is facilitated through a combination of the forward speed of the aircraft and
the oblique shock system present at the engine inlet. However, this method of
compression requires the incoming air speed to be supersonic, which is a major
disadvantage for the technology in question. Flight tests to-date have used rocket
propulsion to accelerate to the speeds required, but current research [2] is looking
at using scramjet propulsion in combined cycle engines to provide continual thrust
from takeoff to landing.
Although scramjet engines are mechanically simple, the flow physics inherent
in such devices is complex and not well understood. The main challenge with
scramjet propulsion is the supersonic combustion itself, since sustaining reactions
in such a high velocity flow is by no means a simple task. The residence time of
reactants within the combustion chamber is of the order of one millisecond [3],
during which time fuel needs to be injected, efficiently mixed, fully reacted and
then expanded back to free-stream conditions. Rapid mixing and combustion
are important in order to reduce the required length and hence weight of the
combustion chamber. However, this rapid mixing must be done as efficiently as
possible, minimising losses in total pressure. Mixing is influenced by the fuel
injection method employed, for which there are currently two main approaches
[4]; injection at the combustor wall or injection in the core region of the combustor
through the use of struts.
Wall injection can be facilitated through the use of transverse or oblique jets,
as shown in Figure 1.1. Although wall normal injection achieves efficient mixing
[5], significant losses in stagnation pressure are caused due to the presence of
strong shocks [4]. The strength of these shocks can be reduced through the use
of angled wall injection [6], with a component of the injectant momentum also
contributing to the thrust. Fuel can also be injected parallel to the wall through
the use of the ramp injectors, but can suffer from low levels of penetration into
the combustor [4].
An example of strut injection is shown in Figure 1.2, which directly injects fuel
into the core of the combustor airstream [4]. Due to the limited mixing capabilities
inherent in streamwise injection, the generation of streamwise vorticity is required
[7] in order for complete mixing to be achieved within the combustor length.
2
Figure 1.1: Normal (left) and oblique (right) wall injection [6]
Figure 1.2: Schematic of strut based injection [8]
Due to the severe aerothermal environment in a scramjet engine and the
additional drag losses introduced with strut injection of fuel, wall injection is
the method of choice [9] in the flight vehicles of today and hence will be the focus
of the thesis.
The first hypersonic flight of an air-breathing vehicle occurred in March,
2004, when NASA’s scramjet powered, hydrogen fuelled, X-43A research vehi-
cle reached a flight Mach number of 6.8 [10]. This was followed in November of
the same year by a Mach 9.6 flight of another X-43A demonstrator [10], currently
holding the Mach number record for a flight powered by an air-breathing engine.
The scramjet powered phase of these flights only lasted on the order of 10 sec-
onds, but in May, 2010, the X-51A WaveRider vehicle, powered by a Pratt &
Whitney Rocketdyne hydrocarbon fuelled scramjet engine, sustained supersonic
combustion at a flight Mach number of 5 for over 200 seconds [11].
Although flight tests are perhaps the best way to conduct scramjet research,
they are a hugely expensive activity which to-date have resulted in vehicle de-
struction. Ground-based investigations are far more cost effective and constitute
3
the majority of supersonic combustion research carried out today. Since it is
difficult to replicate the extreme conditions of hypersonic flight in ground-based
facilities [12], computational methods have begun to play a major role in scram-
jet research and design. Accurate simulation and prediction of the complex flow
physics in question presents a significant challenge to the computational methods
in use today.
1.2 Turbulent Combustion Modelling
The simulation of low Mach number turbulent reacting flows is a relatively ma-
ture area of research, but the extension of the models and techniques involved
to the supersonic reacting regime is still at an early stage. The instantaneous
equations governing the physics of a turbulent reacting flow will be presented
in Chapter 2. Techniques of varying fidelity are available to find a solution to
these equations, with the most common being the DNS, LES and RANS methods
briefly introduced here.
1.2.1 Direct Numerical Simulation (DNS)
DNS directly solves the instantaneous system of governing equations and is the
simplest method available for the simulation of turbulent reacting flows, since no
modelling contributions are required. However, for the absence of modelling to be
valid DNS must capture all physical scales in the simulation, from the turbulence
integral scales, right down to the small Kolmogorov and chemical scales [13]. This
requirement is enabled through mesh resolution, where the cell size must be at
least as small as the smallest length scale to be captured.
The mesh resolution requirements are the main obstacle to the widespread use
of DNS, since there are limitations on total cell count due to the capabilities of
computational resources available today. It will be shown in Chapter 2 that the
Kolmogorov scales are inversely proportional to the turbulent Reynolds number
to the power 3
4
. This means that DNS is only affordable for low Reynolds number
academic research and is not suitable for the simulation of practical systems of
interest.
4
1.2.2 Large Eddy Simulation (LES)
In order to reduce the computational expense of DNS, the mesh resolution re-
quirements can be relaxed by directly capturing only the large scale turbulent
eddies in the flow, leading to the method of large eddy simulation (LES). The
minimum eddy size that can be directly captured is defined by the mesh spacing,
which acts as a filter, removing the smallest scales from the solution. However,
these small scales still have an influence on the large scale structures, which is
included through the use of a subgrid-scale (SGS) model. These models obtain
their name from the fact that they provide for the turbulent scales whose size is
below (sub) the resolution of the computational grid.
Since combustion occurs at the smallest scales and since these scales are now
only modelled, instantaneous reaction rates are no longer applicable and models
must also be introduced to handle the physics involved in turbulence-chemistry
interactions.
The size and behaviour of large scale eddies is dependent on the boundary con-
ditions or geometrical configuration of the problem under consideration, whereas
it can be shown [14] that the behaviour of subgrid-scales is more universal. The
strength of LES is that the problem dependent large scales are directly captured
and only the small scales, which are more suitable for a universal mathemati-
cal description, require modelling. It is the limitations of the subgrid-scale and
turbulent combustion modelling which lead to limitations for the LES method.
Since the filter width should lie within the inertial range of the turbulence
spectrum [14], mesh resolution requirements are still significant for LES but
greatly reduced when compared to DNS. With the increase in computational
power over the last decade, LES is now affordable for industrial applications.
1.2.3 Reynolds Averaged Navier Stokes (RANS)
RANS is the most traditional simulation method and is historically the most
mature. Until recently, RANS was the only viable option for the modelling of
practical systems due to the efficient (coarse) computational grids it employs. The
RANS equations are obtained through time averaging the system of instantaneous
equations, effectively creating a system which can be solved to find mean values
5
for the primary variables involved. However, the averaging process introduces
a series of extra terms into the system of equations which require the use of
turbulence models to obtain closure. Modelling is also required for the species
reaction rate in order to include the effects of turbulence-chemistry interactions.
Since the full turbulence spectrum is subjected to this averaging procedure,
turbulence models are very case sensitive due to the inclusion of the case depen-
dent large scale structures. Model constants are therefore built into the turbulence
model formulation, which require tuning to accurately simulate the non-universal
large scales of the flow. Since tuning is required to reproduce the experimental
data with which these models are validated, their extension to flight conditions or
use in research where experimental data is not available, can then lead to uncer-
tainties in the accuracy of results. However, RANS methods can often produce
accurate trends and are therefore suitable for use in the initial design process in
an industrial environment.
1.3 Aims of this Work
The extension of turbulence and combustion modelling to the supersonic combus-
tion regime presents a significant challenge since the physics involved is not well
understood. RANS methods are capable of providing reasonable agreement with
experimental data but suffer from sensitivities to model constants such as the
turbulent Prandtl and Schmidt numbers [15]. Since modelling is more universal
in the LES approach and since this modelling plays a less significant role due to
the direct capture of the dominant structures in the flow, LES may be capable of
improving over the RANS technique commonly employed.
The aim of this research therefore is to compare the capabilities of the RANS
and LES methods for the simulation of supersonic combustion, investigating the
strengths and weaknesses of each approach.
Through the application of these simulation techniques, it is aimed to un-
derstand better the physics involved in supersonic combustion and to therefore
understand better the efforts required to improve turbulence and combustion
modelling in order to obtain accurate computational predictions in this regime.
Particular attention is paid to oblique wall injection in order to understand better
6
the physics involved in mixing under such conditions.
These comparisons and modifications of approaches for the simulation of su-
personic combustion require access to the source code of computational fluid
dynamics software, ruling out any of the widely used commercial packages. Since
no in-house capabilities existed for such simulations, the development of a new
code, PULSAR, also forms a significant aim of this research and the numerical
methods employed will be discussed in detail.
1.4 Thesis Outline
Chapter 2 reviews the field of turbulence modelling, introducing the instantaneous
system of equations and the additional terms resulting from the averaging and
filtering procedures used for RANS and LES, respectively. A range of turbulence
and subgrid-scale models are described and the hybrid RANS-LES approach for
the simulation of high Reynolds number flows is introduced.
The regime of supersonic combustion and the models available to account for
turbulence-chemistry interactions are presented in Chapter 3, along with details
of chemical kinetics and experimental studies of the physics involved.
The numerical methods employed in PULSAR are described in Chapter 4
and details of turbulence and combustion model implementation are provided.
Methods for the discretisation of convective fluxes in supersonic reacting flows
are compared, where the focus is on developing the capability of obtaining a
monotonic solution. The parallel capabilities of PULSAR are also analysed.
Validation of PULSAR’s mixing prediction capability is provided in Chapter
5 through the simulation of a supersonic coaxial helium jet. Capabilities of the
RANS approach are evaluated and sensitivities of the results to turbulent Prandtl
and Schmidt numbers are presented.
Chapter 6 demonstrates the insensitivity of the LES technique to model con-
stants, through simulation of the coaxial jet used in Chapter 5, but evaluates the
important influence of inlet turbulence parameters on the results obtained.
The SCHOLAR scramjet test case is presented in Chapter 7. The influence
of turbulence-chemistry interaction modelling and chemical kinetics are inves-
tigated through RANS simulations, with computational and experimental data
7
being compared. An investigation into the regime of supersonic combustion is
conducted and the effects of inflow turbulence and radical species concentration
are evaluated. The LES method is employed to study the physics involved and
sensitivities to combustion modelling and inflow turbulence are presented.
Conclusions are drawn in Chapter 8 and areas of future work are suggested.
8
Chapter 2
Turbulence Modelling
2.1 Introduction to Turbulence
Because of the length scales, flow rates and pressures of interest, it is known that
the flow within a scramjet engine is of a turbulent nature [16]. This turbulence
can play an important role in the mixing and combustion process [4] and hence
accurate turbulence simulation is vital.
Turbulent flow fields are highly disordered in nature and hence notoriously
difficult to accurately model. However, disordered behaviour is not the only
prerequisite to a flow being defined as turbulent. All turbulent flows are rotational
and exhibit high levels of vorticity, greatly enhancing the transport of mass,
momentum and energy [14].
If we consider a turbulent flow of characteristic length (L) and velocity (U),
we can calculate the Reynolds number, Re:
Re =
UL
ν
(2.1)
where ν is the kinematic viscosity. For a supersonic combustor Re ≫ 1. Due
to its rotational nature, the turbulent flow is characterised by a distribution of
eddies of varying scale, where the large scale eddies are usually smaller than, but
of comparable size to [17], the scale of the mean basic flow (L). The large scales
are the most energetic and due to their high turbulent Reynolds number, given
by:
9
Ret =
UtLt
ν
(2.2)
viscous forces are assumed negligible. The integral length (Lt) and velocity (Ut)
scales are representative of the large scales of the flow. However, the large scale
motions are unstable and break into smaller eddies, which take energy with them
[17]. This process continues to smaller and smaller scales until they are small
enough for the Reynolds number, based on their characteristic length and velocity,
to be of the order of unity [14]. This Reynolds number is too low to cause eddy
instabilities and hence these are the smallest scales possible in the flow, called
the Kolmogorov length scale. At these scales, viscous forces dominate and the
transfer of energy is through this dissipative mechanism [17].
Figure 2.1: Generic energy spectrum, with energy E and frequency k.
This process is called the energy cascade and Figure 2.1 shows a generic energy
spectrum. The large (low frequency) scales contain the most energy and this
decreases down to the small dissipative eddies. One distinguishing feature of the
energy spectrum is the inertial range, with a gradient of k−5/3 which is obtained
through dimensional analysis [18]. Through the introduction of density into this
10
dimensional analysis Ingenito and Bruno [19] suggested the rate of decay in a
supersonic reacting flow is higher, leading to a steeper gradient of k−8/3. This
faster decay is said to be brought about by baroclinic and dilatational effects
[19] and it is suggested [20] that the baroclinic effects also play a role in the
enhancement of fuel-air mixing. Due to the steeper decay of the inertial range
in compressible flows, the smallest scales are found to be larger than would be
expected under incompressible conditions [19].
Although the mean effect of the small scales is the dissipation of energy away
from the large scales, the intermittent transfer of energy from the small to the
large scales is also possible [21] and is known as backscatter. The backscatter
magnitude can be significant and can therefore play a significant role in turbulent
flow behaviour.
The following relations [22] can be used to find the Kolmogorov length (ηK),
time (τK) and velocity (uK) scales from the large eddy (integral scale) properties:
ηK = LtRe
− 3
4
t (2.3)
τK = TtRe
− 1
2
t (2.4)
uK = UtRe
− 1
4
t (2.5)
where Tt is the large eddy turnover time. It can be seen from the above relations
that, for the same large eddy properties, an increase in the turbulent Reynolds
number results in a reduction of the Kolmogorov length, time and velocity scales
[22]. However, the turbulent Reynolds number does not affect the structure of
the large or small scales; it just defines the size ratio between them [17].
It was therefore hypothesised by Kolmogorov [23], that the structure of the
smallest scales is independent of the turbulent Reynolds number and can be
thought of as universal in all flows. The large scales are highly dependent on
both system geometry and boundary conditions; hence large variations in eddy
structure can occur [14; 17]. This is well shown by the experimental study of
Saddoughi and Veeravalli [24] presented in Figure 2.2. It can clearly be seen that
11
the large scales are highly dependent on the boundary conditions of the flow,
whereas data for the small scales (high wave numbers) collapses nicely for all
cases.
Figure 2.2: One dimensional energy spectra for different experimental cases,
showing problem dependent large scales and universal small scales [24].
12
2.2 Governing Equations
A system of equations exists which governs the physics of a chemically reactive
fluid, namely the instantaneous continuity, momentum, energy and species trans-
port equations. Since the application in question is a high Mach number flow
with heat release, the compressible form of the equations is used.
The continuity equation handles the conservation of mass and is defined as:
∂ρ
∂t
+
∂ρuj
∂xj
= 0 (2.6)
where ρ is the fluid density, t is time, uj is the velocity in direction j and xj is
the spatial vector. The momentum equation is defined as:
∂ρui
∂t
+
∂ρuiuj
∂xj
= − ∂p
∂xi
+
∂τij
∂xj
(2.7)
where p is the static pressure and τij is the viscous stress tensor, given by:
τij = 2µl
(
Sij − 1
3
∂uk
∂xk
δij
)
(2.8)
where µl is the molecular viscosity and δij is the Kronecker delta. The strain rate
is defined as:
Sij =
1
2
(
∂ui
∂xj
+
∂uj
∂xi
)
(2.9)
The energy equation is given by:
∂ρe0
∂t
+
∂ρujh0
∂xj
= − ∂qj
∂xj
+
∂τijui
∂xj
(2.10)
where qj is the heat flux vector. The specific stagnation internal energy, e0, and
specific stagnation enthalpy, h0, are defined as:
e0 = h− p
ρ
+
1
2
uiui (2.11)
h0 = e0 +
p
ρ
(2.12)
13
with the specific static enthalpy, h, given by:
h =
Ns∑
α=1
Yαhα (2.13)
where Yα and hα are the mass fraction and specific enthalpy of species α, respec-
tively, and Ns is the number of chemical species. The calculation of hα and other
thermodynamic properties is to be discussed in chapter 4.
The species transport equation is defined using the species mass fraction:
∂ρYα
∂t
+
∂ρujYα
∂xj
= ω˙α − ∂ρVα,jYα
∂xj
, α = 1, . . . , Ns − 1 (2.14)
where ω˙α is the reaction rate for species α. Vα,j is the j component of the diffusion
velocity for species α, which can be described using Fick’s law [25]:
Vα,jYα = −Dα∂Yα
∂xj
(2.15)
where the species diffusion coefficient, Dα, is defined as:
Dα =
λ
ρcpLeα
(2.16)
with the Lewis number, Leα, assumed to be constant for each species. λ and cp
are the thermal conductivity and specific heat capacity at constant pressure of
the mixture, respectively.
Only Ns−1 species transport equations are required due to the compatibility
condition:
Ns∑
α=1
Yα = 1 (2.17)
meaning YNs can be found from:
YNs = 1−
Ns−1∑
α=1
Yα (2.18)
It is this set of instantaneous equations (2.6, 2.7, 2.10 and 2.14) which could
be solved using the Direct Numerical Simulation (DNS) technique, with all phys-
14
ical scales captured and no modelling contributions required. However, the mesh
requirements for such a simulation are very high, with required cell size to be
of the order of the Kolmogorov length scale, restricting such simulations to low
Reynolds number flows. For reacting cases mesh resolution requirements are even
higher since the cell size is required to be of the order of the smallest chemical
length scale [13]. Methods exist to reduce the computational expense involved
in solving the system of governing equations, with the most popular being the
averaging and filtering techniques used in the RANS and LES approaches, re-
spectively.
2.3 Reynolds Averaged Navier Stokes (RANS)
For many engineering applications it is unnecessary to resolve all physical length
scales, since it is often only the mean properties of the flow which are of interest.
Therefore, in order to significantly reduce the computational time required for a
DNS calculation, the instantaneous governing equations can be averaged in time
to produce the Reynolds Averaged Navies Stokes (RANS) equations. The time
averaging of a quantity φ is defined as:
φ¯ =
1
T
∫ t+T
t
φ (τ) dτ (2.19)
where T is a time scale much greater than any physical time scales in the flow. The
Reynolds Averaged Navier Stokes equations are obtained when this averaging is
applied to the instantaneous system of equations and they describe the evolution
of mean quantities [14].
2.3.1 Favre Averaging
The averaging process introduces extra terms into the system of equations, which
will be discussed in detail shortly. In order to avoid the introduction of extra
terms into the continuity equation the standard Reynolds averaging of Equation
2.19 can be replaced by Favre averaging [26; 27], defined as:
15
φ˜ =
ρφ
ρ¯
(2.20)
where the overbar and tilde indicate Reynolds averaged and Favre averaged quan-
tities, respectively. An instantaneous quantity can be decomposed into a mean
and fluctuating contribution:
φ = φ˜+ φ
′
(2.21)
with the fluctuations denoted by
′
. Although the averaged equations describe the
evolution of mean quantities and although φ˜′ = 0 [25], when the averaging process
is applied to the product of more than one variable, extra terms are introduced
into the system of equations since it can be shown that:
φ˜θ = φ˜θ˜ + φ˜′θ′ (2.22)
where, φ˜′θ′ is not equal to zero [25]. Applying Favre averaging to the instan-
taneous system of equations (2.6, 2.7, 2.10 and 2.14), the compressible RANS
equations can be obtained:
∂ρ¯
∂t
+
∂ρ¯u˜j
∂xj
= 0 (2.23)
∂ρ¯u˜i
∂t
+
∂ρ¯u˜iu˜j
∂xj
= − ∂p¯
∂xi
+
∂τ¯ij
∂xj
− ∂ρ¯u˜
′
iu
′
j
∂xj
(2.24)
∂ρ¯e˜0
∂t
+
∂ρ¯u˜jh˜0
∂xj
=
∂
∂xj
(
τ¯ij u˜ij + τiju
′
i − q¯j − ρu′jh′0
)
(2.25)
∂ρ¯Y˜α
∂t
+
∂ρ¯u˜jY˜α
∂xj
= ¯˙ωα − ∂
∂xj
(
ρVα,jYα + ρ¯u˜
′
jY
′
α
)
(2.26)
Although it may be the mean properties of the flow which are of interest, the
influence of turbulent fluctuations on this mean flow is still of great importance.
This influence arises in the system of equations through the extra terms generated
by the averaging process and they require modelling to obtain system closure.
16
2.3.2 Closure
The turbulent Reynolds stress term, ρ¯u˜
′
iu
′
j, in the momentum equation is often
modelled using the concept proposed by Boussinesq [28], where the turbulent
stresses are proportional to gradients in mean velocity:
− ρ¯u˜′iu′j = 2µt
(
S˜ij − 1
3
∂u˜k
∂xk
δij
)
− 2
3
ρ¯kδij (2.27)
with k being the mean turbulence kinetic energy per unit mass, defined as:
k =
1
2
u˜
′
iu
′
i (2.28)
The 2
3
ρkδij term is often neglected, but may be non-negligible in high speed flows
[29], so is included here.
The mean stagnation energy is defined as:
e˜0 = h˜− p˜
ρ¯
+
1
2
u˜iu˜i + k (2.29)
where, again, the turbulence kinetic energy is included.
The turbulence kinetic energy gradient can can be used to approximate the
molecular diffusion term in the time averaged energy equation [30]:
∂τiju
′
i
∂xj
≈ − ∂
∂xj
(
µl
∂k˜
∂xj
)
(2.30)
and the mean heat flux vector is defined as:
q¯j = −λ ∂T
∂xj
+
Ns∑
α=1
ρVα,jYαhα (2.31)
where the heat conduction and energy flux due to gradients in species mass frac-
tion can respectively be approximated as:
λ
∂T
∂xj
≈ λ ∂T˜
∂xj
(2.32)
17
Ns∑
α=1
ρVα,jYαhα ≈ −
Ns∑
α=1
ρ¯Dαh˜α
∂Y˜α
∂xj
(2.33)
In order to model the turbulent flux of stagnation enthalpy, ρu
′
jh
′
0, the fluc-
tuating stagnation enthalpy, h
′
0, needs to be defined. Subtracting Equation 2.29
from Equation 2.12, the following is obtained:
h
′
0 = h
′
+ u˜iu
′
i + k
′
(2.34)
giving
ρ¯u˜
′
jh
′
0 = ρ¯u˜
′
jh
′ + ρ¯u˜iu˜
′
iu
′
j + ρ¯u˜
′
jk
′ (2.35)
The second term can easily be found once the turbulent Reynolds stresses are
known. The turbulent enthalpy flux and turbulent kinetic energy flux, along with
the turbulent diffusion term from the time-average species transport equation, can
all be modelled using the gradient diffusion approximation [30]:
ρ¯u˜
′
jh
′ = −µtcp
Prt
∂T˜
∂xj
(2.36)
ρ¯u˜
′
jk
′ = −µt
σk
∂k˜
∂xj
(2.37)
ρ¯u˜
′
jY
′
α = −
µt
Sct
∂Y˜α
∂xj
(2.38)
where σk is a model constant whilst Prt and Sct and the turbulent Prandtl and
Schmidt numbers, respectively. Terms involving µt will usually dominate over
the laminar terms in the mean transport equations and sensitivities of RANS to
the choice of Prt and Sct will be investigated in Chapter 5.
Modelling of the mean reaction rate, ¯˙ω, will be the subject of Chapter 3 since
a wide range of methods are available.
Studying the approximations presented for modelling the additional terms,
other than model constants, it is just the turbulent viscosity which is unknown.
It is the job of the turbulence model to find its value and obtain closure for the
system of equations.
18
2.3.3 Turbulence Models
Turbulence models of varying degrees of complexity exist, from simple mixing
length approaches to full Reynolds stress formulations. Central to most turbu-
lence models is the Boussinesq approximation of Equation 2.27, but the Reynolds
stress model attempts to overcome the limitations of this concept.
2.3.3.1 Mixing Length Model
A mixing length turbulence model consists of a simple algebraic expression defin-
ing the turbulent viscosity. Prandtl’s mixing length model can be defined as
[31]:
µt = ρ¯l
2
m
∣∣∣∣(∂u˜i∂xj + ∂u˜j∂xi
)
∂u˜i
∂xj
∣∣∣∣ (2.39)
where lm is the mixing length. Simple algebraic formulae can be used to describe
the mixing length and the mixing length model can be applied with good success
to basic flows such as the plane mixing layer. However, it is difficult to use this
low order model in more complex flows where the definition of the required mixing
length is not straightforward. Although the mixing length model does provide a
spatially varying turbulent viscosity due to proportionality to the mean velocity
gradient, more sophisticated turbulence models have been developed which handle
the transport of turbulence properties. Differential equations handle the effects
of transport through convection and diffusion and the effects of production and
destruction of these turbulence properties can also be modelled.
2.3.3.2 One Equation Models
Models exist which provide these capabilities through use of a single differential
equation. The Spalart-Allmaras model [32] is one example of this, which solves
a transport equation for the kinematic eddy viscosity (ν˜):
∂ρν˜
∂t
+
∂ρuj ν˜
∂xj
= Dν˜ + P
+
ν˜ − P−ν˜ (2.40)
where Dν˜ , P
+
ν˜ and P
−
ν˜ are the terms modelling diffusion, production and destruc-
19
tion, respectively. The turbulent viscosity can be found from:
µt = ρν˜fν1 (2.41)
where fν1 is a wall damping function. Away from the wall this damping function
is unity and the kinematic eddy viscosity (ν˜) is simply equal to the kinematic
viscosity (νt). The Spalart-Allmaras turbulence model performs well in external
aerodynamic and boundary layer flows [33] but, similarly to the mixing length
model, one equation turbulence models require an algebraic expression for the
turbulence length scale. This means they can perform poorly in complex flows,
where the turbulence length scale can vary and is harder to define. The Spalart-
Allmaras turbulence model has also been shown to perform poorly in round jets
and round wakes [34], but is robust and does not require any low Reynolds number
modifications.
2.3.3.3 Two Equation Models
Two equation turbulence models can be used to remove the requirements for an
algebraic expression for the turbulence length scale. The turbulence length scale
can then be calculated on the fly, from a combination of the two variables for
which differential transport equations are provided. Various turbulence models
of this sort are available; the three most common are briefly discussed here.
k − ε
The most widely used and validated two equation turbulence model is the k − ε
model [35; 36], where transport equations for the turbulence kinetic energy (k)
and the turbulence dissipation rate (ε) are employed:
∂ρk
∂t
+
∂ρujk
∂xj
= Dk + P
+
k − P−k (2.42)
∂ρε
∂t
+
∂ρujε
∂xj
= Dε + P
+
ε − P−ε (2.43)
The turbulent viscosity is given by:
20
µt = ρCµ
k2
ε
(2.44)
where Cµ is a model constant. The turbulence length scale can now be calculated
using [33]:
Lt =
k
3
2
ε
(2.45)
Although the k− ε turbulence model is capable of producing good results for
free shear layer flows [33] it is not capable of accurately predicting separation or
the viscous region near a wall [37]. For boundary layer flows, the addition of low
Reynolds number damping functions (such as that by Chien [38]) is required, but
these make the differential equations numerically stiff through the introduction
of additional source terms, and hence difficult to solve.
k − ω
The k − ω turbulence model of Wilcox [37] was developed to improve over the
k − ε model for the calculation of boundary layers in adverse pressure gradient
flows. For the k − ω turbulence model, a transport equation for the turbulence
frequency (ω) is used instead of for the turbulence dissipation rate:
∂ρω
∂t
+
∂ρujω
∂xj
= Dω + P
+
ω − P−ω (2.46)
The k equation (2.42) is only modified with a change of variables in the destruc-
tion term (P−k ), through use of the simple relation between k, ε and ω:
β⋆ω =
ε
k
(2.47)
where β⋆ is a model constant. The turbulent viscosity is given by:
µt =
ρk
ω
(2.48)
The k − ω turbulence model is numerically robust, since the addition of low
Reynolds number damping functions is not required. The model is able to ac-
21
curately predict mildly separated flows, but can prove sensitive to free-stream
values of ω (i.e. inlet boundary condition values). However, these sensitivities
have been improved by recent modifications to the model [39].
Menter SST
The Menter SST turbulence model [40] attempts to improve upon the k−ε and
k−ω formulations by combining the two, using the strengths of the k−ω model
near the wall and the k − ε model in the free-stream. Still only two transport
equations are required, due to the simple relationship between the three variables
(Equation 2.47), but blending functions are used to smoothly transition between
the two turbulence models outside of the boundary layer. The turbulent viscosity
is given by:
µt =
ρk
max(a1ω, S˜F2)
(2.49)
where F2 is one such blending function and S is the strain invariant, defined as:
S =
√
2SijSij (2.50)
2.3.3.4 Reynolds Stress Model
Two-equation turbulence models have been found to perform well for many indus-
trially relevant flows [33] and so have been widely used and validated. However,
such models require the presence of velocity gradients in order to generate eddy
viscosity, meaning away from shear layers no Reynolds stresses are calculated.
Also, when significant velocity gradients exist in more than one direction the
two-equation approach may be unsuitable, and led to the development of the
Reynolds-stress model (RSM) [41].
Rather than employing the Boussinesq assumptions of Equation 2.27, the
RSM employs a transport equation for each Reynolds stress. Since the turbulent
stress tensor is symmetric, six additional transport equations are required for u˜
′2
1 ,
u˜
′2
2 , u˜
′2
3 , u˜
′
12, u˜
′
13 and u˜
′
23:
22
∂ρu˜
′
iu
′
j
∂t
+
∂ρuku˜
′
iu
′
j
∂xk
= Dij + P
+
ij − P−ij +Θij (2.51)
where, subscript 1 corresponds to the x-direction (u-velocity), 2 to the y-direction
(v-velocity) and 3 to the z-direction (w-velocity). Dij, P
+
ij , P
−
ij and Θij are the
diffusion, production, dissipation and pressure-strain terms [41], respectively. Due
to the increased complexity of this approach and due to the attempt at directly
modelling each Reynolds-stress, the RSM is often referred to as a second order
closure for the RANS equations.
Models are required for the diffusion, dissipation and pressure strain terms.
An additional transport equation, for the turbulence dissipation rate (ε), is re-
quired for this modelling, bringing the total number of additional differential
equations to seven. This means the Reynolds-stress model is very computation-
ally expensive compared to the zero, one and two equation turbulence models
previously discussed.
In order to model the turbulent fluxes arising in the energy and species trans-
port equations, the turbulent viscosity can be used, as before. The turbulence
kinetic energy can be calculated from the three normal stresses using Equation
2.28 and this, in addition to the availability of the turbulence dissipation rate,
means the turbulence viscosity can be calculated as for the k−ε turbulence model,
from Equation 2.44.
2.3.3.5 Model Constants
Since the turbulence model is required to account for all physical length scales
in a RANS environment, empirically tuned constants are employed to deal with
the problem dependent large scales of the flow. The eddy viscosity will dwarf the
molecular viscosity and hence it is the turbulent transport terms in the RANS
system of equations which will play the dominant role. Constants such as the
turbulent Prandtl and Schmidt numbers therefore have a significant influence on
the computational results obtained.
The Prandtl number is the ratio of momentum (viscous) and heat transport
and is linked to the species Lewis number through the relation:
23
Scα = PrLeα =
µl
ρDα
(2.52)
where Scα is the species Schmidt number, which is the ratio of momentum and
molecular diffusion. The turbulent Prandtl and Schmidt numbers are therefore
the turbulent equivalent, involving turbulent transport and diffusion rather than
the molecular properties defined here. The turbulent Schmidt number will be
assumed constant for all species.
Significant tuning of such constants is therefore required in order to accurately
simulate the problem dependent large scales of the flow. This can be successful if
experimental data is available to aid the tuning process, but it can often be time
consuming. For cases where experimental data is not available, it is difficult to
place a great deal of trust into the accuracy of the results obtained. It is therefore
difficult to use the RANS method in certain areas of research and extrapolation of
numerical methods from lab based experiments to full flight conditions can pro-
vide unreliable results. The sensitivities of RANS to the modification of certain
model constants will be demonstrated in Chapter 5.
However, the RANS method is computationally efficient and is capable of
predicting trends in the physics of a given problem.
2.3.4 Unsteady RANS (URANS)
The time step, ∆t, used to iterate the numerical methods employed is defined by
the CFL number:
∆t =
CFL ·∆x
|u|+ a (2.53)
where ∆x is the minimum length of the cell, u the flow velocity and a the speed
of sound. This approach ensures a hypothetical particle cannot move more than
∆x in a single time step. A maximum CFL number will exist to define stability,
which is dependent upon the method employed for time integration.
Although the time-dependent terms are present in the RANS equations (2.23-
2.26), they are not solved in a time accurate manner when a steady state solution
is required. In order to accelerate convergence to a steady state, variable time
24
steps are often employed whereby each computational cell uses a time step de-
pendent on its own minimum size and flow properties. However, steady RANS
methods struggle to predict flows involving massive separation [42]. If the physics
involved in a particular simulation are not of a steady nature, a time-accurate
simulation is required to capture the transient behaviour, where a global time step
is used for every computational cell, leading to the Unsteady RANS (URANS)
formulation.
It is intended that the URANS approach mimics the large scale dynamics of
the flow [42]. Such simulations can be ensemble averaged to provide reasonable
predictions for mean flow variables, however rms predictions are often in poor
agreement with experimental data.
The behaviour of the URANS technique can be demonstrated through the
simulation of the flow around a square cylinder [43]. Figure 2.3 displays the
vorticity magnitude, clearly showing the unsteady vortical structure is captured
in the simulation. However, it can be seen that the solution is smooth since, due
to the averaging process, only a single frequency is captured in the solution and
hence there is a lack of turbulent fluctuating data in the results.
Figure 2.3: Vorticity magnitude contours from a square cylinder URANS simu-
lation
The steady RANS approach is not suitable for the simulation of the square
cylinder test case, due to its transient behaviour. The resulting solution is depen-
dent upon the disparity in free-stream cell size and a steady state solution may
never be reached.
25
2.4 Large Eddy Simulation (LES)
Large eddy simulation aims to improve over the RANS technique by attempting
to directly capture some of the turbulent scales present in a given flow. It can be
considered as an intermediate between RANS and DNS, since similarly to DNS
direct resolution of turbulent structures is attempted; however, DNS of high
Reynolds number flows is not currently possible, so a coarser mesh is employed.
This mesh acts as a low-pass filter in Fourier space, removing the high wave
number (small length scale) eddies from the solution. These smaller eddies still
play an important role in the physics, providing overall dissipation to the large
scale structures, and this must now be modelled. In RANS, models are used to
account for all turbulent scales in the flow; now they are only used to account for
the smaller turbulent structures. Due to the more universal behaviour of these
small scales, the development of widely applicable models is more straightforward.
LES can be compared to the Reynolds averaged approach through the simu-
lation of the square cylinder test case presented in the previous section. It can be
seen from Figure 2.4 that the turbulent nature of the wake is far better captured
with the LES technique, with a range of turbulent scales present.
Figure 2.4: Vorticity magnitude contours from a square cylinder large eddy sim-
ulation
26
2.4.1 Favre Filtering
Since the mesh acts as a filter on the turbulence, the equations for LES can be
obtained by applying a filter to the instantaneous governing equations. A filtered
variable is defined as:
φ¯(x) =
∫
D
φ(x⋆)G(x− x⋆)dx⋆ (2.54)
where G(x − x⋆) is the filter function. Finite volume LES can be thought of as
using a top-hat filter in physical space. The top-hat filter is a sharp cutoff filter,
where the turbulent structures are either big enough to be captured by the mesh,
or they are not [14]:
G(x) =
{
1/∆ if |x| ≤ ∆/2
0 otherwise
(2.55)
where ∆ is the filter width. In order to be able to capture the large scale eddies
of the flow on a given computational mesh, the filter width must lie within the
inertial range of the turbulence spectrum.
Similarly to Favre averaging, Favre filtering is used to obtain the compressible
form of the governing equations, in order to avoid the introduction of additional
terms into the continuity equation. Equation 2.20 is also used to define a Favre
filtered variable (φ˜), where the overbar now corresponds to the filtering defined
in Equation 2.54.
Since LES directly resolves the large scales of the flow, which are random
in nature by definition, statistical methods are required to analyse the resulting
data. The mean and rms values can respectively be defined as:
φ˜mean =
1
Nmean
Nmean∑
i=1
φ˜i φ˜rms =
√√√√ 1
Nrms
Nrms∑
j=1
(φ˜− φ˜mean)2 (2.56)
where Nmean and Nrms are the number of iterations over which the mean and rms
values are calculated, respectively. The number of iterations needs to be large
enough to allow the statistical data to reach a steady state.
27
Although Favre filtering is conceptually different to the Favre averaging pro-
cess and the resulting system of equations has a different physical meaning, they
are visually identical to those presented in Section 2.3 and so will not be repeated
here. This makes it relatively straightforward to convert a URANS code into one
capable of conducting large eddy simulations. The only modification required is
the way in which the turbulent viscosity is calculated, with this parameter now
referred to as the subgrid-scale (SGS) viscosity (µSGS). This subgrid viscosity is
responsible for modelling the dissipation that the small (subgrid) scales provide
to the large scale turbulence.
There are currently two main approaches to modelling this subgrid dissipation,
whereby the subgrid viscosity can either be explicitly calculated using a subgrid-
scale model or it can be neglected, with the required dissipation being implicitly
provided by the numerical dissipation of an upwind scheme.
2.4.2 Explicit Subgrid-Scale (SGS) Models
Subgrid-scale models of varying degrees of complexity are available to provide the
dissipation required. Rather than explicitly filtering the solution, as implied by
Equation 2.54, implicit filtering is applied by the computational grid. The filter
width provided by a given computational grid therefore plays a significant role
in subgrid-scale modelling; the smaller the filter width, the larger the range of
scales captured by the mesh and the lower the requirements for subgrid viscosity,
but the higher the computational cost.
The filtering operation of Equation 2.54 assumes a constant filter width so
that it commutes with the operation of differentiation [44]. However, in complex
geometries where unstructured grids are employed it is difficult to keep the mesh
spacing and hence filter width constant. There are two main methods for calcu-
lating a spatially varying filter width, with the cube root of the cell volume often
being used:
∆ = 3
√
δxδyδz =
3
√
volume (2.57)
where, δx, δy and δz are the cell dimensions in the x, y and z directions, respec-
tively. However, for a non-isotropic grid, it is reasonable to state that the smallest
28
turbulent eddy that a computational grid is capable of capturing, is that with
size greater than the maximum dimension of a cell [45]:
∆ = max(δx, δy, δz) (2.58)
This would provide the same filter width as Equation 2.57 on a regular structured
grid, but significantly different for a grid containing elongated cells. Although
Equation 2.58 has a physical background, it can generate significantly higher
levels of subgrid viscosity and the use of 2.57 can aid the growth of shear layer
instabilities, encouraging the transition to turbulence [46] when high aspect ratio
cells are present. The influence on results of the choice of filter width will be
investigated in Chapter 6.
2.4.2.1 Smagorinsky model
The most straightforward and also most widely used subgrid-scale model, is that
developed by Smagorinsky [47]:
µSGS = ρ (Cs∆)
2 |S˜| (2.59)
where, Cs is a model constant. This model was obtained by assuming the subgrid-
scales are in equilibrium, meaning the production and destruction of energy is
balanced. This assumption is used in the detached eddy simulation approach,
to be discussed in Section 2.4.4.3, in order to obtain a Smagorinsky like eddy
viscosity model.
Due to the model’s simplicity, it is easy to implement and is computationally
efficient. However, results can be heavily dependent on the model constant used
[48], with values in the approximate range 0.1-0.2 found to be most appropriate,
depending on the application in question. In complex flows, where different flow
physics may be encountered in different regions of the computational domain, it
is restrictive to require Cs to remain constant in space.
The Smagorinsky model is also purely dissipative and therefore is not capable
of modelling backscatter.
29
2.4.2.2 Dynamic model
The range of values found for the Smagorinsky constant implies the subgrid scales
are not as universal as initially thought. This led to the development of the dy-
namic SGS eddy viscosity model by Germano et al. [48], where the Smagorinsky
constant is calculated on-the-fly using information from the smallest resolved
turbulent eddies, and is allowed to vary in space and time.
Moin et al. [49] modified the dynamic model for compressible flows. The
methodology employed to calculate the dynamic Smagorinsky constant is briefly
discussed here. It can be shown that the subgrid stress can be defined as:
τij = ρ¯(u˜iuj − u˜iu˜j) (2.60)
with the tilde and bar here representing Favre and spatially filtered variables,
respectively. Expanding these Favre filtered expressions gives:
τij = ρuiuj − ρui ρuj
ρ¯
(2.61)
The dynamic model applies a second filter to the resolved scales in order to
use spectral data from the resolved field. This second filter is referred to as a test
filter and the sub-test-filter stresses can be defined as:
Tij = ρ̂uiuj − ρ̂ui ρ̂ujˆ¯ρ (2.62)
where ̂ refers to a filtered variable at the test-filter width, ∆̂, which is larger
than the mesh filter width, ∆.
Now, the stresses for the scales between the test and grid filter (denoted by the
shaded region in Figure 2.5), called the Leonard stresses, Lij , are calculated as
the difference between the sub-test-filter stresses and test-filtered subgrid stresses:
Lij ≡ Tij − τˆij (2.63)
which, through reverting back to Favre filtered variables, is given by:
Tij − τˆij = ̂¯ρu˜iu˜j − 1ˆ¯ρ(
̂¯ρu˜i ̂¯ρu˜j) (2.64)
30
Using a Smagorinsky model for both the sub-test-filter stresses, Tij , and the
subgrid stresses, τˆij , it can be shown that [49] Equation 2.64 can provide a for-
mulation for the dynamic constant. All data on the right hand side is readily
available from the resolved data. The spatially filtered density and Favre filtered
velocity data is provided by the grid filtered data. The test-filtered data can
then be found by averaging neighbouring values of grid filtered data, effectively
applying a larger spatial filter.
The dynamic model is capable of generating negative values for this constant
and so allowing for backscatter.
Figure 2.5: Turbulence energy spectrum showing mesh and test filters
2.4.3 Implicit LES
It will be presented in Chapter 4 that for the accurate simulation of supersonic
combustion a high order upwind scheme is required to capture flow discontinuities
in a monotonic fashion. Garnier et al. [50] investigated the effect of using a SGS
model in combination with a shock capturing scheme, for the LES of compressible
flows. It was concluded that the addition of a SGS model to shock-capturing
31
methods is unnecessary and inconvenient, since the dissipation provided by the
shock-capturing schemes is on its own greater than that provided by the SGS
model. This type of SGS model free LES leads into the realm of implicit LES
(ILES), where the numerical dissipation needed to model the sub-grid scales is
provided by the numerical methods employed.
Implicit LES can be thought of as under-resolved DNS, since it is the in-
stantaneous set of governing equations which are solved on a grid too coarse to
capture the smallest scales of the flow. No modelling is provided for the subgrid
scales since it is claimed [51] that the intrinsic dissipation of an upwind scheme
can mimic the effect of a SGS model. Fureby and Grinstein [52] and Fureby
[53] compared the implicit and explicit LES approaches through the simulation
of both a free shear layer and rearward facing step. It was concluded that the
concept of LES is virtually independent of the method employed, as long as grid
resolution is high enough to ensure that the minimum resolved scales are in the
inertial range of the turbulence spectrum.
Despite encouraging results using the implicit LES approach, its use is often
the topic of debate because the dissipation employed lacks the complex physical
meaning provided by explicit SGS models. It is also not well understood how well
the implicit approach handles complex combustion, since only the instantaneous
reaction rate (ω˙) is solved and no modelling for turbulence chemistry interaction is
provided. In order for turbulence chemistry interactions to be modelled, subgrid
properties such as the turbulence length scale would be needed (see Chapter 3)
for which an explicit SGS model is required.
2.4.4 Wall Bounded LES
Since the application of this work is to be the simulation of a supersonic com-
bustor, which is a wall bounded internal aerodynamics problem, it is important
to study the modelling requirements for the LES of boundary layer flows. As
previously stated, the resolution of a computational mesh must be such that the
cutoff wave number should be somewhere in the inertial range of the turbulence
spectrum. For this to be the case, it can be shown [54] that, far from a wall
in the free-stream, the mesh resolution requirements (number of cells, Nfree) are
32
proportional to the turbulent Reynolds number as:
Nfree ∝ Re0.6t (2.65)
However, near the wall, where the effects of viscosity are important, the required
resolution for a LES is much more demanding [55] since important near wall
vortical structures scale with wall units. It is therefore not only in the wall
normal directions that significant resolution is required (as would be the case in
RANS), but also in the two wall parallel directions, significantly increasing the
mesh resolution requirements. The number of cells required to accurately resolve
this region (Nwall) is proportional to the turbulent Reynolds number [54] as:
Nwall ∝ Re2.4t (2.66)
This places significant restrictions on the affordability of fully resolved wall bounded
LES, especially for the very high Reynolds number flows that are to be studied.
It is therefore necessary to investigate ways to relax these restrictions, by not
resolving the near wall turbulent structures but by applying models which pro-
vide the correct boundary layer behaviour (e.g. separation) and apply the correct
shear stresses to the free-stream flow.
2.4.4.1 Wall Functions
Early large eddy simulations [56; 57] of wall bounded high Reynolds number
flows employed the wall functions often used with the RANS equations, in order to
reduce the near wall resolution requirements. Wall functions assume the presence
of a near wall logarithmic layer which can be used to calculate the shear stress
to be applied to the fluid in a near wall cell. However, in practical engineering
applications the assumption that a logarithmic layer exists often does not hold
[54] due to the presence of severe favourable and adverse pressure gradients, along
with regions of separation.
33
2.4.4.2 Two Layer Model
The two layer model is a zonal approach originally proposed by Balaras and
Benocci [58], which solves the Reynolds averaged boundary layer equations on
a fine wall normal mesh, which is embedded on a coarser LES mesh, reducing
the computational demands of a fully resolved large eddy simulation. The main
assumption is a weak coupling between the near wall and outer layer regions,
where the LES solution provides velocity values to the inner layer calculation
and the inner layer provides shear stresses for the outer layer flow. Although the
two layer model has been applied with success and can improve significantly over
the wall function approach, the location at which the inner layer communicates
with the LES zone must be specified by the user and requires prior knowledge of
the final solution, which may not be available.
2.4.4.3 DES
It is also possible to solve the full system of Reynolds averaged Navier Stokes
equations in the near wall region, dramatically reducing the computational cost
of fully resolved LES by relaxing the requirements on mesh spacing, particularly
in the directions parallel to the surface. Standard RANS turbulence models can
therefore be used, with a transition to LES occurring away from the surface.
There is a stronger coupling between the RANS and LES regions than in the
two layer approach, since all required fluxes are calculated across this boundary.
There can however be a discontinuous eddy viscosity profile due to the transition
from RANS to SGS turbulence models. However, the severity of this transition
can be controlled through the choice of blending between the two regions.
One choice can be to apply an explicit blending function between a RANS
turbulence model with strong near wall modelling capabilities and a separate
SGS model for the free-stream, as was done by Baurle and Edwards [15] for a
supersonic coaxial jet. The choice of blending function and the location at which
this blending occurs, can have a significant impact on the accuracy of the near
wall modelling and resultant skin friction values.
Another approach for blending the near wall and free-stream regions is to
transition the length scale in the RANS model to one based on mesh spacing.
34
This effectively transitions the RANS turbulence model to an SGS model for
use in LES. This length scale modification is used in detached eddy simulation
and will be the focus here. Details for the whole spectrum of wall modelling
approaches are available in the comprehensive reviews of Piomelli [54], Piomelli
and Balaras [59] and Cabot and Moin [55].
Detached eddy simulation (DES) was developed by Spalart et al. [60] and is
a hybrid RANS-LES method which makes use of a single turbulence model for
both the near wall RANS regions and the free-stream LES calculations. DES
was originally formulated with the Spalart-Allmaras turbulence model due to its
robust nature and suitability for modelling boundary layer flows. It is possible to
transform this turbulence model into a Smagorinsky type SGS model by balancing
the production and destruction terms in the transport equation for kinematic
eddy viscosity (2.40), giving:
cb1S˜ν˜ = cw1
(
ν˜
d
)2
(2.67)
where, cb1 and cw1 are model constants and, critically, d is the RANS turbu-
lence length scale given by the distance to the nearest wall, yw. Through some
manipulation and use of Equation 2.41 this becomes:
µt ∝ ρS˜d2 (2.68)
Through comparison to Equation 2.59 it is straightforward to see that a
Smagorinsky type model can be achieved through alteration of the turbulence
length scale from d in the RANS region to CDES∆ in the free-stream, where
CDES is a model constant. The transition between these length scales is made
possible through the introduction of a new variable d˜, in place of d, in the kine-
matic eddy viscosity transport equation, defined as:
d˜ = min (yw, CDES∆) (2.69)
This assumes that the production and destruction terms are dominant in an
equilibrium flow (i.e. the diffusion term is negligible) and so they can be equated
to form the Smagorinsky like model.
35
This new length scale acts as a sharp switching function between the RANS
and LES approaches and is dependent upon mesh resolution and the method used
to calculate the filter width. Special care needs to be taken when designing the
grid on which a DES is to be conducted [61] in order to ensure transition to LES
occurs outside of the boundary layer. The recommended filter width [45] is that
given by Equation 2.58.
DES is not restricted to use with the Spalart-Allmaras turbulence model and
Strelets [45] formulated a version which uses the Menter SST turbulence model,
citing its good performance in boundary layer calculations and separation pre-
diction as to why it would make an ideal candidate for RANS modelling in the
near wall region. Equations 2.45 and 2.47, can be combined to define the RANS
length scale as:
lk−ω =
k
1
2
β⋆ω
(2.70)
Again, it is required that for an equilibrium free-stream flow the k and ω
transport equations are modified such that the turbulent viscosity of the Menter
SST turbulence model reduces to a Smagorinsky like form. For simplicity, Strelets
[45] suggested a modification to just the diffusion term in the k equation (2.42),
from
Dk = ρβ
⋆kω =
ρk
3
2
lk−ω
(2.71)
to
Dk,DES =
ρk
3
2
l˜
(2.72)
where, l˜ is the DES turbulence length scale, given as:
l˜ = min (lk−ω, CDES∆) (2.73)
Since the Menter SST turbulence model transitions between the k − ε and
k − ω turbulence models using a blending function F1, it is possible the subgrid
viscosity could be calculated using either model, since the transition between
36
models may not coincide with the transition to LES. For this reason, there exists
a CDES constant for each model, and that used in the calculation for l˜ must be
found using the blending function:
CDES = (1− F1)Ck−εDES + F1Ck−ωDES (2.74)
However, use of the k − ε model is more likely since the k − ω turbulence model
is only active near the wall.
These model constants were tuned by simulating the decay of homogeneous
isotropic turbulence and were found to be Ck−εDES = 0.61 and C
k−ω
DES = 0.78. For
DES with the Spalart-Allmaras turbulence model CDES was tuned to a value of
0.65.
2.4.4.4 DDES
The need to design the computational mesh to ensure switching from RANS
to LES occurs outside of the boundary layer can often be problematic, since the
required mesh spacing may not be possible due to geometric constraints, required
free-stream resolution or simply because knowledge of the flow field is not known
prior to mesh generation. If the wall parallel mesh spacing is small enough to
cause transition from RANS to LES inside the boundary layer, but is not fine
enough to resolve the turbulent streaks, a reduction in the Reynolds stresses are
obtained. This is because the reduction in turbulent viscosity from the SGS
model is not replaced by resolved shear stresses due to lack of mesh resolution
[62]. This leads to an erroneous prediction of the skin friction at the wall.
This reliance on mesh design lead to the birth of the delayed detached eddy
simulation (DDES) method [62], which modifies the switching position through
use of flow field quantities such as velocity gradients and the turbulent viscos-
ity. Concentrating on the Menter SST form of DES, the turbulence length scale
(Equation 2.73) can be modified through the introduction of the function fd:
l˜ = lk−ω − fdmax (0, lk−ω − CDES∆) (2.75)
which can be defined as:
37
fd = 1− tanh
(
[8rd]
3) (2.76)
where rd is a function of viscosity and velocity gradients:
rd =
µt + µl
ρ
√
∂Ui
∂xj
∂Ui
∂xj
κ2y2w
(2.77)
with κ being the Ka´rma´n constant. The blending function fd is designed to be
equal to 1 in the LES region and 0 elsewhere. As well as ensuring the transition
from RANS to LES occurs outside of the boundary layer, this blending function
also provides a smoother transition, removing the discontinuity in eddy viscosity.
Introducing a blending function based on flow variables also means the switch-
ing position is less sensitive to the filter width than is the case with DES. The
strict requirement to use Equation 2.58 for the filter width can therefore be re-
laxed and the more conventional filter width given by Equation 2.57 can also be
considered. The effects of using these two filter widths will be investigated in
Chapter 6.
2.4.4.5 Transition Region Turbulence
Although the methods mentioned above make the large eddy simulation of high
Reynolds number wall bounded flows affordable, there are significant drawbacks
to under-resolving the near wall region. In areas of separation, the turbulent
structures from the outer region of the boundary layer, which should be shed into
the free-stream, will not be present since they are not resolved. This can cause a
delay in the transition to turbulence and development of eddy structures in the
LES regime, which for high Mach number flows could therefore occur significantly
further downstream of separation than experienced in practise; this behaviour will
be discussed in more detail in Chapter 6.
Reynolds stresses in the near wall region are supported by the modelling effort
applied there and the Reynolds stresses in the free-stream are supported by the
presence of turbulent eddies [54]. However, in the transition region between the
two, where these turbulent eddies may not exist, errors can occur in the Reynolds
stress calculation. It is therefore important to try and assist the development of
38
these turbulent eddies in the transition region between the modelled near wall
region and LES free-stream domain.
Various methods for accelerating the development of these eddies have been
developed [54]. If the flow does not contain massively separated regions which
can themselves introduce instabilities that will naturally grow with the solution,
some sort of perturbation is required in regions of mild separation in order to
encourage eddy development. For hybrid RANS-LES techniques, Keating and
Piomelli [63] introduced stochastic forcing in the transition region in order to
generate small-scale fluctuations. Turbulent fluctuations applied to the inlet of
the solution domain are also possible, with the method of Kempf et al. [64]
capable of generating coherent structures on unstructured grids.
39
Chapter 3
Combustion Modelling
The averaging and filtering processes applied to the system of governing equa-
tions generates the reaction rate source term ω˙. Unfortunately, the instantaneous
reaction rate, ω˙, cannot be employed for this averaged or filtered quantity since
ω˙(ρ, T, Y ) 6= ω˙(ρ¯, T˜ , Y˜ ) (3.1)
due to its non-linear nature [25]. Therefore, a model is required in order to handle
the turbulence-chemistry interactions and provide ω˙(ρ, T, Y ).
3.1 Supersonic Combustion
In order to aid combustion model selection, it is important to try and understand
the regime in which supersonic combustion occurs. However, due to the coupling
of reactions, turbulence and shock waves, the flow physics in a scramjet combustor
is complex and hence not well understood.
Balakrishnan and Williams [65] declared the turbulent Reynolds number, Ret,
and Damko¨hler number, Da, the most significant parameters for defining the su-
personic combustion regime. The turbulent Reynolds number is defined by Equa-
tion 2.2 using integral scale properties, which can be calculated from turbulence
model variables. The Damko¨hler number is the ratio of turbulence time scale, Tt,
to chemical time scale, τc:
40
Da =
Tt
τc
(3.2)
where the turbulence time scale is defined as the integral scale turnover time,
which can be calculated from turbulence model parameters:
Tt =
1
β⋆ω
(3.3)
Since the instantaneous reaction rate, ω˙α, is the production rate of ρYα, the
chemical time scale, τc, can be defined as:
τc =
ρ
ω˙α
(3.4)
where the averaged or filtered quantities, ρ¯ and ω˙α, should be used in RANS and
LES, respectively.
Two limits on the Damko¨hler number are Da ≫ 1 and Da ≪ 1, which cor-
respond to fast and slow chemistry compared to mixing, respectively. Simplifi-
cations to reaction rate modelling can be made in both limiting situations, with
infinitely fast chemistry applicable for high Damko¨hler numbers and well stirred
reactor modelling used for those much less than unity.
The diagram shown in Figure 3.1 is often used to try and describe the regime
in which supersonic combustion occurs. The separate regimes displayed are to be
discussed.
A Damko¨hler number, DaK , based on the Kolmogorov time scale, τK , can
also be defined, which is given by:
DaK =
τK
τc
(3.5)
and using the definition of τK given in Equation 2.4 it can be shown that:
DaK =
TtRe
− 1
2
t
τc
= DaRe
− 1
2
t (3.6)
41
Figure 3.1: Regime diagram: Log-log plot of Damko¨hler number versus turbulent
Reynolds number
Alternative definitions are available for the Kolmogorov length and time scales
[22] in terms of the turbulence dissipation rate, ε:
ηK =
(
ν3
ε
) 1
4
(3.7)
τK =
(ν
ε
) 1
2
(3.8)
Combining Equations 3.7 and 3.8 gives:
DaK =
η2K
ντc
(3.9)
42
and defining the chemical length scale as [66] lc = (ντc)
1
2 , the square root of
the Kolmogorov Damko¨hler number can be shown to be equal to the ratio of
Kolmogorov to chemical length scales:
√
DaK =
ηK
lc
(3.10)
Through use of Equation 2.3, the ratio of the integral and chemical length scales
can also be shown to be:
Lt
lc
=
√
DaRet (3.11)
Lines of unity Damko¨hler number, Kolmogorov Damko¨hler number and integral
to chemical length scale ratio are shown in Figure 3.1, which define some of the
different reaction regimes possible. Above the unity DaK line, all turbulence
length scales are larger than the chemical length scale, resulting in the reaction
sheets regime where the turbulent eddies cannot enter and disrupt the reaction
zone, but only wrinkle it. The region between the Da = 1 and DaK = 1 lines
can be described as the broken flamelet regime [66], where the reaction sheets
may be wrapped around the large eddies used to define Da, but extinguished
in the small eddies. Between the Da = 1 and (DaRet)
1/2 = 1 lines the degree
to which turbulence structures can enter the reaction zone increases, until the
(DaRet)
1/2 = 1 line is reached, beyond which all turbulence length scales are
smaller than the chemical length scale, causing distributed reactions to occur.
Figure 3.2 places the supersonic combustion regimes proposed by Balakrish-
nan and Williams [65] and Ingenito and Bruno [19] on the diagram described.
Balakrishnan and Williams analysed the conditions in a possible scramjet engine
flight envelope, showing high turbulent Reynolds numbers are most likely with
chemical reactions proceeding at a rate faster than the large scale mixing. In-
genito and Bruno extracted data from a large eddy simulation of the SCHOLAR
scramjet test case using 1-step chemistry. The possibility of lower turbulent
Reynolds and Damko¨hler numbers than predicted by Balakrishnan and Williams
is suggested, indicating finite reaction rates may be important in the supersonic
combustion regime.
43
Figure 3.2: Supersonic combustion regimes proposed by Balakrishnan and
Williams (light blue) and Ingenito and Bruno (red)
Flight Mach number and geometrical configuration would influence the regimes
proposed; Balakrishnan and Williams investigated flight Mach numbers up to 25
and the conditions in the SCHOLAR scramjet test case corresponding to a flight
Mach number of 7. The Mach number would influence the levels of stagnation
enthalpy and hence reaction rates, along with combustor residence times and tur-
bulent Reynolds numbers. Engine geometry and method of fuel injection would
also influence mixing, leading to changes in the Damko¨hler number through al-
teration of the corresponding turbulence time scales and reaction rates.
Both studies suggest the broken flamelets regime is the most likely, but the
wide range of Damko¨hler numbers proposed by Ingenito and Bruno, coupled with
the additional influence of compressibility effects, makes this conclusion uncertain.
Berglund et al. have applied LES to two different strut injector scramjet test
44
cases, for which it was concluded from experimental data that combustion occurs
in the flamelet regime for one [67] and with distributed reactions well outside
the flamelet regime in another [68], suggesting test case geometry and boundary
conditions can play a significant role in the regime achieved.
Through analysis of high Mach number compressibility effects on the rate of
reaction, Ingenito and Bruno suggest a modification to the position of the DaK =
1 and (DaRet)
1/2 = 1 lines on the regime diagram, shifting them both upwards
to higher Damko¨hler numbers. However, this appears somewhat contradictory
to the compressibility effects on turbulence that are discussed in Section 2.1. If
the smallest scales in supersonic combustion are indeed larger than those defined
by the Kolmogorov length scale, the reaction sheet regime would be applicable
at lower Damko¨hler numbers than implied in Figure 3.1, since just below the
DaK = 1 line there would be no small scales to enter and disrupt the reaction
zone.
Effects such as compressibility and the need to model finite rate reactions
due to low residence times, even if the Damko¨hler numbers do not support such
requirements, significantly complicate the physics involved in supersonic combus-
tion. Modelling of such physics may not be as straightforward as the extension
of knowledge and models from the subsonic regime and a deeper understanding
of the interactions involved is required.
3.2 Laminar Chemistry
Finite rate reactions can be simulated through the use of multi-step chemical
systems of the form:
Ns∑
α=1
ν
′
αrMα ⇔
Ns∑
α=1
ν
′′
αrMα, r = 1, . . . , Nr (3.12)
where Nr and Ns are the number of reactions and species in the chemical mech-
anism, respectively. Mα is the chemical symbol for species α with ν
′
αr and ν
′′
αr
being the molar reactant and product stoichiometric coefficients for species α in
reaction r, respectively. The ⇔ symbol implies a reversible reaction.
Using such chemical reactions, the instantaneous reaction rate for species α,
45
ω˙, which appears as a source term in the instantaneous species transport equation
(2.14), can be defined as:
ω˙α = Wα
Nr∑
r=1
[(
ν
′′
αr − ν
′
αr
)(
kfr
Ns∏
β=1
c
ν
′
βr
β − kbr
Ns∏
β=1
c
ν
′′
βr
β
)]
(3.13)
where Wα is the molecular weight of species α and cβ is the species concentration,
given by:
cβ =
ρYβ
Wβ
. (3.14)
kfr and kbr are the forward and backward reaction rate constants for reaction r,
respectively, for which an Arrhenius expression is used:
kfr = ArT
nr exp
(
− Er
R0T
)
(3.15)
where Ar is the pre-exponential factor and Er is the activation energy, for reaction
r. It is the exponential term in this equation which causes the significant non-
linearity of the system, which makes turbulent reacting flows so difficult to model
[22].
It should be noted that the process of radical recombination generates suffi-
cient energy to cause the reaction product to decompose back into the original
radical species [69]. The addition of a third body,M , to such reactions is therefore
required in order to absorb the excess energy, giving:
Ns∑
α=1
ν
′
αrMα +M ⇔
Ns∑
α=1
ν
′′
αrMα +M, r = 1, . . . , Nr (3.16)
which leads to a modified reaction rate equation given by:
ω˙α =Wα
Nr∑
r=1
[(
ν
′′
αr − ν
′
αr
)(
kfrcM
Ns∏
β=1
c
ν
′
βr
β − kbrcM
Ns∏
β=1
c
ν
′′
βr
β
)]
(3.17)
with the third body concentration, cM , defined as:
46
cM =
Ns∑
β=1
ηβ,Mcβ (3.18)
where ηβ,M is the third body efficiency of species β for third body M . The
modified equation for the instantaneous reaction rate can be universally used,
with cM set to unity for steps not involving a third body reaction.
It is this instantaneous reaction rate, ω˙α, which will be referred to as lam-
inar chemistry. The instantaneous reaction rate can directly be employed in
DNS for calculation of the reaction rate term in the species transport equation,
since the influence of all turbulence structures on the flame is directly captured.
Unfortunately the picture is significantly more complex for the mean and filtered
systems of equations, where a model is required in order to handle the turbulence-
chemistry interactions and calculate ω˙(ρ, T, Y ).
3.3 Turbulence-Chemistry InteractionModelling
Since the physics involved in supersonic combustion is still not well understood the
current trend for reaction rate modelling is to attempt the extension of methods
designed for subsonic combustion to the supersonic combustion regime. Although
the regime diagram of Figure 3.2 implies the flamelet regime is most likely, as
discussed in Section 3.1 further regimes may also be applicable.
Central to many combustion models is the use of the probability density func-
tion (PDF) to find mean variables from the instantaneous values:
φ =
∫
φP (φˆ)dφˆ (3.19)
where P (φˆ) is the PDF for the variable φ and the integration is carried out
over the sample space variable φˆ. A PDF is defined so that the probability that
φˆ < φ < φˆ+ dφˆ is P (φˆ)dφˆ [22].
47
3.3.1 Flamelet Model
The flamelet model has been applied [8; 67; 70] to the simulation of super-
sonic combustion because of the assumptions made on reactions occurring in the
flamelet regime. The flamelet approach simplifies combustion modelling through
use of geometrical concepts; assuming the flame is thin compared to other scales
of the flow.
Supersonic combustion in a scramjet engine is non-premixed in nature, due
to the injection of pure fuel directly into the combustion chamber. The mixture
fraction Z is introduced for the flamelet model, which is a conserved scalar taking
the value of 0 in regions of pure oxidiser and 1 in regions of pure fuel. Because
it is a conserved scalar, the mixture fraction denotes the mass fraction of fluid
originating in the fuel stream of the flow [22], at any point in the domain. The
flame is assumed to take the shape of the iso-contour of the stoichiometric value
for the mixture fraction, Zst.
Instantaneous mass fractions, temperatures and densities are set as functions
of the mixture fraction. Their mean values can therefore be calculated using the
PDF method in Equation 3.19, where the mixture fraction is set as the sample
space variable:
φ =
∫
φ(Z)P (Zˆ)dZˆ (3.20)
In order to take finite rate chemistry effects into account, Zheng and Bray
[70] introduced a strain rate parameter, whilst the scalar dissipation rate has
also been used [67]. The instantaneous properties can then be set as functions
of both the mixture fraction and rate parameter variables, with flamelet libraries
correspondingly defined. The mean variables can now be calculated from:
φ =
∫
φ(Z, χ)P (Zˆ, χˆ)dZˆdχˆ (3.21)
where χ is the rate parameter in question. However, in order to carry out the
integration, statistical independence between the two variables of the joint PDF
is assumed:
48
P (Z, χ) = P (Z)P (χ) (3.22)
where the shape of each PDF is presumed.
Zheng and Bray confirmed the importance of finite rate effects in supersonic
combustion through comparison of the joint PDF method with that in Equation
3.20. It was concluded that the combustor residence times are so low due to the
high velocities present that the reactions may fail to reach completion [70]. It
was also shown that the Kolmogorov based Damko¨hler number was below unity
throughout the reaction zone for the test case simulated. Recalling the regime
diagram in Figure 3.1, this suggests assumptions of flamelet type behaviour for
the Evans et al. [71] test case are reasonable.
So, in summary; the flamelet model employs transport equations for non-
reacting scalars and uses the PDF method to obtain the mean variables of the
flow rather than directly attempting to evaluate the mean reaction rate in the
transport equation of a reactive scalar. Assumptions are made on the structure
of the flame using a geometrical approach, assuming that it is thin. However,
since other regimes are possible in supersonic reacting flows the flamelet model
may be limited in its applicability.
3.3.2 Probability Density Function Approach
Rather than employing the PDF method to evaluate the mean properties of the
flow through use of the mixture fraction and flamelet libraries, the PDF method
can be used to directly evaluate the mean reaction rate, using a joint PDF of
temperature and composition:
¯˙ω =
∫
ω˙
(
Tˆ , cˆ1, . . . , cˆNs
)
P
(
Tˆ , cˆ1, . . . , cˆNs
)
dTˆdcˆ1, . . . , dcˆNs (3.23)
This approach does not make use of the mixture fraction variable and makes
no assumptions on the regime in which combustion occurs. Two methods exist
to evaluate the joint PDF, either by evaluation through the solution of a PDF
transport equation or by making assumptions on the PDF shape.
49
3.3.2.1 Transported PDF
An evolution equation for the joint PDF of thermochemical scalars, such as the
species mass fraction and enthalpy of the flow, can be derived. This equation
has the significant advantage of including the species reaction rate in closed form
[72]. Modelling for this term is therefore no longer required and the instanta-
neous reaction rate in Equation 3.17 can be used. However, an unclosed term
for mixing does arise, for which a model is required. Due to the high dimen-
sionality of a transport equation for the joint PDF of multiple scalars, the use
of finite difference methods for its solution is very expensive [73]. Lagrangian
particle methods however scale linearly with the dimensionality of the PDF [73]
and Pope first introduced [74] a Monte Carlo method for the efficient solution of
the PDF transport equations.
Since the PDF can be used to contain statistical information for any flow
variable, transport equations for the joint PDF of velocity can also be derived.
In a similar fashion to the reaction rate calculations, this equation circumvents
the gradient-diffusion assumption for turbulence transport modelling [73], since
these transport processes appear closed in the PDF evolution equation.
Mo¨bus et al. [75] employed a Monte Carlo method for the solution of a trans-
port equation for the joint PDF of velocity, enthalpy and species mass fraction,
for the simulation of a high speed reacting flow. Results were compared to a sim-
ilar method employing a joint PDF of just enthalpy and species mass fraction,
using standard turbulence modelling to obtain the required Reynolds stresses.
Good agreement with experimental data was obtained, with the joint PDF in-
cluding velocity providing the best results. However, the computational cost is
very high when compared to conventional finite volume approaches, particularly
when the dimensionality of the PDF is increased further through inclusion of
the velocity variables. Although the Lagrangian particle method provides an ef-
ficient approach for the solution of the PDF evolution equation, on the order
of 100 particles are required per computational cell [75] in order to obtain the
required statistics. Therefore, for practical problems of interest where compu-
tational meshes containing millions of cells are required, the transported PDF
method can be prohibitively expensive despite the significant advantage of the
50
reaction rate appearing in closed form. This cost is exacerbated in LES, due to
the high mesh resolution requirements.
3.3.2.2 Assumed PDF
In an attempt to reduce the high computational costs of the transported PDF
method, which allows the PDF to evolve in time and space, the shape of the
PDF can be assumed. However, it is difficult to form a mathematical definition
for the joint PDF of temperature and composition so statistical independence is
presumed between the temperature, species mass fraction and density:
P
(
ρˆ, Tˆ , Yˆ1, . . . , YˆNs
)
= PT (Tˆ )PY (Yˆ1, . . . , YˆNs)δ(ρˆ− ρ¯) (3.24)
The assumed PDF combustion model is the most widely applied for the sim-
ulation of supersonic combustion [76; 77; 78], where a clipped Gaussian profile
is employed for the temperature PDF and the multivariate β-PDF of Girimaji
[79; 80] is used for species mass fractions. To define each of these assumed PDFs,
both the mean variables (temperature and species mass fractions) and a higher
order moment are required. The higher order moments are obtained through the
introduction of two transport equations [77] for the temperature variance, σT ,
and sum of species mass fraction variances, σY :
∂ρ¯σT
∂t
+
∂ρ¯u˜jσT
∂xj
=
∂
∂xj
[(
γ
µ
Pr
+
µt
Prt
)
∂σT
∂xj
]
+ 2
µt
Prt
(
∂T˜
∂xj
)2
− CTγρ¯σTωf − 2 (γ − 1) ρ¯σT ∂u˜j
∂xj
+
1
cv
2T
′
Nk∑
α=1
ω˙αhfα (3.25)
∂ρ¯σY
∂t
+
∂ρ¯u˜jσY
∂xj
=
∂
∂xj
[
(D +Dt)
∂σY
∂xj
]
+ 2
Nk∑
α=1
ρ¯Dt
∂Y˜α
∂xj
∂Y˜α
∂xj
− CY ρ¯σY ωf + 2
Nk∑
α=1
ω˙αY
′
α (3.26)
51
where,
σT = T˜
′2, σY =
Ns∑
α=1
Y˜ ′2 (3.27)
ωf is the fluid mechanics timescale and the model constants CT and CY are set
to 2.0, as in [77]. It is the occurrence of the fluid mechanics timescale which
requires the use of a two equation turbulence model, which is capable of defining
such flow properties through use of:
ωf =
1
β⋆ω
(3.28)
The source terms at the end of each variance transport equation, which contain
the species reaction rate variables, could also be modelled using an assumed PDF
approach. However, highly erroneous results are obtained if this is done [77] and
so they are usually neglected.
The multivariate β-PDF of Girimaji is used to reduce the computational cost,
since it combines all the mass fraction variances into a single variable, σY . If a
PDF was defined for every single species, an additional transport equation would
be required for the variance of each mass fraction, leading to a large computational
cost when complex chemical mechanisms are employed.
Baurle and Girimaji [78] attempted to model temperature-composition cor-
relations which are neglected through the assumption of statistical independence
in Equation 3.24, but little improvement over the method presented here was
obtained.
The assumed PDF approach is also used in the flamelet model, as described
in Section 3.3.1, but rather than applying it to calculate the mean reaction rate
the mean species mass fractions are obtained from the mixture fraction variable.
The assumption of a certain shape for the mixture fraction PDF is somewhat
robust, since the mixture fraction is a non-reacting transported scalar. A large
body of knowledge is available for such fluid processes, showing the assumption
of such PDFs to be reasonable [22]. However, the application of assumed PDFs
to reacting scalars is less robust and particularly in supersonic combustion there
is no evidence known to the author of such PDFs arising in practice.
52
However, the assumed PDF approach for calculating the mean reaction rate
is still widely applied in supersonic combustion research due to its increased com-
putational efficiency over the transported PDF method and lack of assumptions
on flame structure and reaction rates. This model is even applicable to both pre-
mixed and non-premixed combustion [81]. It has been shown [77] that accurate
results can be obtained for both the mean quantities of the flow and the rms
values calculated from the variance transport equations, for a supersonic reacting
coaxial hydrogen jet [82].
3.3.3 Other Models
Other models which have been applied to the simulation of supersonic combustion
to handle the turbulence-chemistry interactions are the linear eddy mixing (LEM)
model [83] and the partially stirred reactor (PaSR) model [68], which have both
been employed under an LES framework. Neither makes assumptions on the
shape of the reaction zone, as is done in the flamelet model, but the PaSR model
does assume that Kolmogorov length scales are present in order to model the
subgrid mixing.
In the LEM model an unfiltered species transport equation is solved on a fine
one-dimensional sub-mesh grid. Due to the one-dimensionality, subgrid stirring
is simulated through a stochastic procedure called triplet mapping. The big
advantage of the LEM model for reacting flows is that the species reaction rate
appears in closed form in the one-dimensional equation to be solved. The LEM
model has been applied to the simulation of supersonic mixing by Sankaran and
Menon [84] and Ge´nin and Menon [85] and to reacting flow by Ge´nin et al. [86]
and Ghodke et al. [83], with success. However, since the resolution of the sub-
mesh grid must be fine enough to directly capture all scales of the flow (as in
DNS), this approach is very computationally expensive. Methods to reduce the
computational cost of such simulations are being attempted [83].
53
3.4 Combustion Model Influence
Since supersonic combustion is not well understood, the role of turbulence-chemistry
interactions under such conditions is unknown. In order to investigate such
physics, the laminar chemistry approach can be used to neglect any interactions
not captured by the computational mesh and results compared to those from a
combustion model designed to provide for such interactions.
This has been done by Gerlinger et al. [76] for the simulation of a supersonic
laboratory hydrogen jet flame, through comparison of the laminar chemistry ap-
proach to the assumed PDF combustion model. It was found that very compa-
rable results are achieved from both approaches, suggesting that the influence of
turbulence-chemistry interactions is minimal.
Mo¨bus et al. [75] compared a finite volume laminar chemistry simulation to
the results from two transported PDF calculations, for two supersonic hydro-
gen jet flames. Again, it was found that only small differences in the results
were obtained, in particular for the radial profiles of pressure and species mass
fraction. Although some improvements were gained through the use of a joint
PDF of velocity, enthalpy and composition, minimal improvements over lami-
nar chemistry were obtained when employing a standard turbulence model in
combination with a joint PDF of only enthalpy and composition, suggesting the
method chosen for turbulence modelling may have a more significant influence
over the results obtained than the approach employed to handle the turbulence-
chemistry interactions. Results from all three simulations were however in rea-
sonable agreement with the experimental data, again suggesting the influence of
turbulence-chemistry interactions is minimal.
The influence of these interactions in a realistic scramjet combustor geometry
was investigated by Keistler et al. [87], where results from laminar chemistry
and a variable Prandtl and Schmidt number combustion model were compared.
It was concluded that turbulence-chemistry interactions play an important role
in supersonic combustion, however the results from the combustion model em-
ployed are in far worse agreement with experimental data than those from laminar
chemistry, for which a reasonable comparison is obtained. Further work on the
importance of these interactions in realistic geometries is required.
54
Due to the finite volume laminar chemistry simulations somewhat surprisingly
providing results in good agreement with experimental data, the modelling of
turbulence chemistry interactions is often neglected [88; 89; 90].
3.5 Chemical Kinetics
Since finite rate effects play an important role in supersonic combustion, either
from the short residence times of reactants within the combustor or the possibility
of low Damko¨hler numbers, it is important to evaluate the influence of chemical
kinetics on computational results. The computational cost introduced through
the use of complex mechanisms with a large number of species and reaction steps
must also be taken into account when a selection of the chemical kinetics model
is to be made.
The most widely used mechanisms for the simulation of supersonic hydrogen
combustion are those of Jachimowski, with two mechanisms consisting of 9 species
and 19 reactions from 1988 [91; 92] and 1992 [93] and a reduced mechanism [94]
of just 7 species and 7 reactions, commonly employed. The complex mechanism
of O’Conaire et al. [95], which consists of 9 species and 21 reactions, is perhaps
considered the most comprehensive description of hydrogen chemical kinetics due
to its wide range of validation and inclusion of pressure effects through the use
of Troe parameters. More computationally efficient descriptions of the chemistry
are also available, with the 4 species single-step mechanism of Marinov et al.
[96] and the 5 species 2-step mechanism of Rogers and Chinitz [97]. It should
be noted that the number of species listed for each mechanism includes N2, for
which reactions are not included.
Gerlinger et al. [98] conducted a comprehensive comparison of the prediction
capabilities for all mechanisms discussed above, apart from the 2-step model. A
supersonic lifted flame was simulated with the ignition delays calculated for each
mechanism and radial profiles of the temperature and mass fraction statistics
evaluated. It was concluded that, due to the neglect of radical species, both the
temperatures and ignition delay times predicted by the 1-step mechanism are
in poor agreement with experimental data and hence it should not be used. It
was found that as long as conditions are not close to the ignition limit then the
55
reduced Jachimowski mechanism is in good agreement with both the 1988 and
1992 full Jachimowski mechanisms, which are both in turn in good agreement with
the more comprehensive mechanism of O’Conaire. This suggests that significant
savings in computational cost can be made by using the reduced mechanism. It
should be noted that results were sensitive to the concentration of radical species
at the inflow to the domain, so studies should be conducted to evaluate this
whenever employing detailed chemistry.
Berglund et al. [68] compared results from 1-step, 2-step and 7-step chemical
mechanisms for the large eddy simulation of strut injection into a realistic scram-
jet combustor geometry. The combustion for this test case was experimentally
evaluated to be in the low Damko¨hler number distributed reactions regime. It
was found that the induction time (defined as the time taken for the temperature
to increase 100K from the initial state) for the 1-step mechanism is much shorter
than the turbulence time scales, resulting in a corrugated flame which tends not to
interact with the turbulence scales of the flow, corresponding to high Damko¨hler
number combustion. The poor agreement of the 1-step mechanism corresponds
to the short ignition delay times found by Gerlinger et al., as discussed above.
However, for the 2-step and 7-step mechanisms the induction times were found
to be of the same order of magnitude as the time scales of the flow, leading
to a flame that can be wrinkled by the vortical structures present. It is also
noted that the behaviour of the more complex mechanisms are in much better
agreement with a 19-step chemical kinetics model, with respect to the induction
times calculated. When studying computational results for the distribution of
the OH radical and wall pressure distributions Berglund et al. found the 1-step
mechanism (OH distribution was calculated using chemical equilibrium) to again
be in poor agreement with experimental data, whilst the 7-step mechanism pro-
vided the best agreement. The 2-step mechanism provided results in reasonable
qualitative and quantitative agreement.
The choice of chemical mechanism can obviously have a significant impact on
the reaction rate calculations and hence it is important to study such influences
to ensure accurate computational results are obtained.
56
3.6 Experimental Studies
Detailed experimental studies of supersonic combustion are limited, due to both
the difficulties and costs involved in carrying out such research. The high stagna-
tion enthalpy conditions experienced in hypersonic flight are difficult to replicate
in ground based facilities and run times can be limited. Although limited in detail
and number, since supersonic combustion research is an active field in countries
such as the US, Australia, Japan, Germany and France, to name a few, a range
of experimental studies have been conducted. A basic overview is given here of
the most widely used sets of experimental data, but this review is by no means
exhaustive.
The supersonic reacting hydrogen coaxial jet of Evans et al. [71] has been used
by Zheng and Bray [70] and Gerlinger et al. [75; 76] for computational studies
into supersonic reacting flows. Radial profiles of mean pitot pressure and species
mass fractions (H2O, H2, O2 and N2) are available at several locations down the
axis of the jet. This is a simple test case which can provide useful validation
for computational techniques, but unknowns about it’s geometrical configuration
and setup provide a degree of uncertainty.
More detailed experimental data is available for the supersonic reacting hydro-
gen coaxial jet of Cheng et al. [82], which has been used by a number of authors
for the validation of computational methods [75; 76; 77; 78]. Simultaneous mea-
surements of the major species (H2O, H2, O2 and N2), OH radical concentrations
and temperature were made, providing both mean and variance data. An exper-
iment conducted on the same burner by Dancey [99] provides measurements for
the mean and fluctuating velocities in two directions. Unfortunately, geometrical
information for the burner is limited to sketches in the relevant papers, introduc-
ing significant uncertainties into computational studies when the whole burner
is to be simulated. There is also limited information about the radical concen-
trations in the jet co-flow, which Gerlinger et al. [98] found to be a significant
uncertainty in the computational setup.
A supersonic jet test case for which both geometrical data and detailed mea-
surements are available is that by Cutler et al. [100]. However, this is a non-
reacting coaxial jet, using helium as the test gas to simulate the supersonic com-
57
pressible shear layer mixing experienced in a scramjet combustor fuelled by hy-
drogen. Probes are used to measure the radial profiles of total temperature, pitot
pressure and composition at several locations down the axis of the jet. RELIEF
velocimetry is used to measure mean and fluctuating axial velocity profiles and
schlieren visualisation is employed. Detailed co-ordinates for the contours which
make up the central jet and co-flow surfaces are available.
Another non-reacting test case is that of wall injection into a supersonic cross-
flow, where the injected fluid is air. Both normal and oblique injection experi-
ments have been conducted, with the most widely used being the wall normal test
case, with velocity measurements available from Santiago et al. [101] and scalar
mixing data from [102]. Such test cases can be employed to study the physics
of injection into a scramjet combustor and to analyse the capabilities of numeri-
cal methods for their simulation. Reacting experiments of wall normal injection
are also available, such as that by Ben-Yakar et al. [103], for which OH-PLIF
measurements are available.
As well as supersonic coaxial jet and wall injection experiments, measure-
ments of the reacting flow within realistic scramjet combustor geometries are
also available. The SCHOLAR test case is perhaps the most widely used, with
data available for both normal wall injection [104] and angled injection at 30
degrees to the horizontal [105]. A dual-pump CARS technique is employed to
simultaneously measure composition and temperature, with static pressure mea-
surements made on all four walls of the combustor. However, not enough data
is collected using the CARS technique for reliable mean values to be evaluated,
so profiles are obtained using best-fit techniques. Also, no information on the
concentration of radical species entering the combustor is provided.
There is a distinct need for detailed measurements of a supersonic combus-
tion experiment, providing reliable profiles of the mean and fluctuating values
of composition, temperature and velocity components, along with geometry, in
order to evaluate the ability of turbulence and combustion models to capture
the physics involved. However, it is difficult to make such measurements in a
scramjet combustor due to its wall bounded nature. Laboratory jet flames such
as the Evans et al. and Cheng et al. experiments are attractive for their simplic-
ity, aiding both the experimental data collection and affordability for numerical
58
simulations. Cutler et al. [106] are working on such an experiment, using the
dual-pump CARS technique to obtain the required temperature and composi-
tion statistical data. An interferometric Rayleigh scattering technique is being
built into the dual-pump CARS method [107], for simultaneous measurements of
velocity. However, results are not currently available for this test case.
It should be noted that measurements of turbulence length and velocity scales
were not made in any of the experiments discussed. The importance of such data
is to be demonstrated throughout this thesis.
59
Chapter 4
Numerical Methods and
Implementation
4.1 Introduction to PULSAR
In order to carry out computational research into supersonic combustion a parallel
code capable of simulating high speed flows and finite rate complex chemistry
combustion is required. It is possible various commercial codes could provide
these capabilities, but the lack of detailed information on model implementation
and the ability for model development is prohibitive. When conducting LES
studies, a significant number of commercial licenses would be required for large
parallel computations and hence the use of such software is not practical from
the point of view of cost.
Various in-house codes were in existence at the start of this research but
none were suitable for the application in question. A parallel code (pNEWT) for
non-reacting turbo-machinery simulations and a serial code (McUNNEWT) for
premixed combustion research were the prominent tools in the CFD Lab at the
time.
Therefore, development of a new in-house code was required for the simulation
of supersonic combustion. This code has been named PULSAR (Parallel Unstruc-
tured Large-eddy Simulation And Reactions). The starting point for PULSAR
was the pNEWT in-house code developed by Professor Bill Dawes. Significant
60
modifications were made to every element of the solver, the development of which
is the topic of discussion for this chapter.
4.2 Finite Volume Method
In CFD, the governing equations of fluid flow are solved over the domain in
question using a distribution of control volumes known as a mesh. The governing
equations presented in Chapter 2 can be solved in an iterative manner on this
computational mesh through use of the finite volume method. The equations can
be described using a partial differential equation of the form:
∂ρφ
∂t
+
∂ρujφ
∂xj
=
∂
∂xj
(
Γ
∂φ
∂xj
)
+ Sφ (4.1)
where φ becomes 1, ui, h0 and Yα for the continuity, momentum, energy and
species transport equations, respectively. In order as displayed, the four terms
in the equation 4.1 are the transient term, convective term, diffusive term and
source term, where Γ corresponds to a diffusion coefficient. The starting point
for the finite volume method is the integral form of this equation:
∫
CV
∂ρφ
∂t
dV +
∫
CV
∂ρujφ
∂xj
dV =
∫
CV
∂
∂xj
(
Γ
∂φ
∂xj
)
dV +
∫
CV
SφdV (4.2)
where the integration is carried out over each control volume (CV ). Through the
application of Gauss’s divergence theorem:∫
CV
∂βj
∂xj
dV =
∮
(~n.~β)dA (4.3)
volume integrals can be re-cast as integrals over the surface of the control volume,
where ~n is the outward pointing unit normal vector to a cell face of area A, giving:
∂
∂t
∫
CV
ρφdV +
∮
~n.(ρ~uφ)dA =
∮
~n.
(
Γ
∂φ
∂xj
)
dA+
∫
CV
SφdV (4.4)
Assuming infinitesimal values for the time step, cell volumes and face areas,
61
this can be written as the flux sum over all cell faces to define the change in
conserved variable over the corresponding time step:
∆(ρφ)CV =
∆t
∆VCV
(
nfaces∑
f=1
~nf . (Γ∇φ)f ∆Af + (Sφ∆V )CV (4.5)
−
nfaces∑
f=1
~nf .(ρ~uφ)f∆Af
)
and an iterative method can then be used to solve for the conserved variables
over multiple time steps, n:
φn+1 =
(ρφ)n +∆(ρnφn)
ρn+1
(4.6)
where ∆(ρnφn) is the change in ρφ calculated using data from time step n.
Geometric data for the normal unit vectors and areas for the faces of each cell
in the computational mesh are therefore required. For clarity later, the inviscid
and viscous fluxes through face f are respectively defined as:
~nf .(ρφ~u)f∆Af (4.7)
~nf . (Γ∇φ)f ∆Af (4.8)
where f corresponds to a quantity at the centre of a cell face. Calculation of the
(ρφ~u)f and (Γ∇φ)f face quantities is the job of the specific discretisation scheme
employed. Determination of the ∂p
∂xi
source term in the momentum equation can
be absorbed into the inviscid flux calculation.
4.3 Data Structure
When the computational domain is divided into finite volumes (cells) there is the
choice to store the conserved variables at either the vertices of the cell or the cell
centre (see Figure 4.1), leading to the cell-vertex and cell-centred data structures,
62
respectively. The implementation of numerical methods differs between each
approach and their respective advantages and disadvantages are often debated.
A comprehensive comparison of the two techniques was recently presented in
[108; 109].
Figure 4.1: Data storage locations. Cell-vertex (blue) and cell-centre (green).
PULSAR employs a cell-centred data structure but pNEWT uses cell-vertex
storage for the conserved variables and hence significant modifications were made
to the data structure to convert PULSAR to the cell-centred equivalent.
When a computational domain is meshed into finite volumes, the cell size is
dependent upon the length scales of the flow physics to be captured and most
mesh generators used today provide a choice for cell topology. For a given length
scale, it is much more efficient to discretise a domain using hexahedral cells from
the point of view of cell count, with savings of the order of four times made
compared to their tetrahedral counterparts. However, the vertex count is com-
parable when using both hexahedral and tetrahedral cell topologies. Therefore,
when moving from a cell-vertex to a cell-centred data structure it is very impor-
tant from the point of efficiency to ensure the code is capable of handling meshes
containing hexahedral elements. pNEWT operates on solely tetrahedral meshes
and so significant effort was put into ensuring PULSAR could operate on mixed
grids, containing any mix of tetrahedral, hexahedral, prismatic and pyramidal
cell topologies.
The commercial package ANSYS ICEM CFD is used for mesh generation,
63
and although PULSAR does have mixed grid capabilities all meshes used for test
cases presented throughout this thesis are purely hexahedral.
The data structure modification was further complicated by the parallel ca-
pabilities of the code, which are further discussed in Section 4.11.
4.4 Inviscid Fluxes
The inviscid terms in the governing equations are those describing convection:
∂ρuj
∂xj
∂ρuiuj
∂xj
∂ρujh0
∂xj
∂ρujYα
∂xj
(4.9)
As discussed in Section 4.2, through application of Gauss’s divergence theorem
to the integral form of these terms there is the need to calculate face fluxes
containing face values (ρφ~u)f , where the following will be assumed:
(ρφ~u)f = (ρφ)f~uf (4.10)
and ~nf .∆Af can be defined as an area vector ~Af which splits the face area into
area projections in the three component directions (Ax,f , Ay,f , Az,f). The (ρ~u)f ·
~Af term which is common to all convective flux terms can be defined as the face
mass flux, m˙f :
(ρ~u)f . ~Af = m˙f (4.11)
There are many discretisation methods available to calculate the face data
required, with each method having its own numerical properties. It is therefore
important to understand the physics involved in the problem to be simulated in
order to select a discretisation method which provides the numerical properties
desired.
4.4.1 Desirable Properties
The simulation of supersonic combustion is computationally challenging due to
the presence of discontinuities provided by shock waves, flame fronts and species
64
transport. These flow phenomena result in high gradient regions and a computa-
tional method is required which can accurately predict this in a stable fashion.
4.4.1.1 Monotonicity
The most desirable property of a numerical scheme to be used for the calculation
of the inviscid fluxes is that of monotonicity. As will be shown, certain discretisa-
tion techniques can cause spurious oscillations in the presence of discontinuities.
These spurious oscillations can pollute the solution and cause numerical instabili-
ties. Oscillations in temperature can also cause non-physical ignition to occur. A
numerical scheme is required which can accurately capture high gradient regions
in a monotone (smooth) fashion.
The total variation diminishing (TVD) criterion [33] is often applied to the
development of numerical methods in order to obtain a monotonic solution. How-
ever, this can often be an over-restrictive requirement, proving detrimental to the
numerical accuracy.
4.4.1.2 High Order of Accuracy
High order accurate methods are desirable in order to obtain sharp resolution of
the flow physics encountered. Low dissipation methods are also extremely im-
portant when employing the LES technique. Since the large turbulent structures
are directly captured, low levels of numerical dissipation are required to prevent
these structures from being prematurely destroyed.
However, since unstructured computational meshes are employed the use of
extended stencils to achieve higher order of accuracy is not possible. Unstructured
meshes are used in order to handle complex geometries with an industrial focus,
for which computational cost is of primary concern. Therefore, no attempt has
been made to increase the accuracy of computational methods in PULSAR above
second order.
4.4.2 Discretisation Techniques
The performance of an upwind scheme for the simulation of discontinuous flows
will be investigated and compared to the centred discretisation approach. Test
65
cases are chosen which replicate a discontinuous flow field, which are characteristic
of the physics encountered in a scramjet combustor.
4.4.2.1 Centred Scheme
The pNEWT solver uses a centred scheme for the discretisation of the inviscid
fluxes, which is inherently unstable and therefore requires the addition of artificial
dissipation [110] in order for convergence to be obtained.
The face quantities required for the flux calculations are found through a
simple average of the neighbouring cell centred variables:
φf =
φL + φR
2
(4.12)
where L and R correspond to the respective left and right states, as shown in
Figure 4.2.
Figure 4.2: Left and right states used to calculate face data for the centred
discretisation method.
The mass flux can therefore be defined as:
m˙f =
3∑
i=1
(
ρLuiL + ρRuiR
2
Ai
)
(4.13)
with the complete discretisation of Equation 4.7, given by:
fφf = m˙φf (4.14)
The pressure gradient source term for the i momentum equation can be included
through the addition of:
66
(
pL + pR
2
)
Ai (4.15)
to the i momentum face flux calculations.
Figure 4.3: Disadvantages of the centred scheme on a non-regular mesh. Black
dot corresponds to half way between cell centres and red star corresponds to
centre of face.
This method is second order accurate on regular grids. However, if the mesh
is irregular and the half way point between the neighbouring cell centres does
not coincide with the centre of the face, as shown by the stretched and skewed
cells in Figure 4.3, the order of accuracy of the simple average in Equation 4.12
is reduced.
4.4.2.2 Test case: Discontinuity Capturing - Centred
In order to assess the monotonic properties of discretisation schemes, a propa-
gating discontinuity can be simulated. Often, a discontinuous pressure field is
employed in order to replicate a shock wave, however a discontinuous species
mass fraction field is to be used here to show that the monotonic solution of any
discontinuous conserved variable is of great importance.
Figure 4.4 shows the initial state of the test case, where an axial velocity of
150 ms−1 propagates a discontinuity of pure nitrogen into a field of pure oxygen.
The domain is 0.5m in length, 0.05m in height and 0.0015m in depth, discretised
by regular hexahedral elements of 0.5mm, totalling 300,000 cells. A time step
of 4 × 10−7 seconds is used and the computational results are compared to the
analytical solution after 1500 iterations.
67
Figure 4.4: Initial conditions for the mass fraction discontinuity test case
It can be seen from Figures 4.5 and 4.6 that the centred scheme causes spuri-
ous oscillations near the mass fraction discontinuity. These numerical oscillations
are known as Gibb’s phenomenon [111] and can cause unphysical predictions to
occur or can grow and destabilise the solution. For this test case these oscilla-
tions cause the prediction of erroneous values for species mass fraction, but near
discontinuities in a scramjet combustor these oscillations could arise for any con-
served variable. Oscillations in density lead to oscillations in temperature, as will
be shown in section 4.4.3.3, which could cause non-physical ignition to occur in
a reacting simulation.
4.4.2.3 AUSM+
In order to avoid the production of spurious oscillations in discontinuous regions
of the flow an upwind method can be used. The Roe scheme [112] is commonly
employed, which belongs to the flux-difference splitting family of upwind meth-
ods. However, the Roe scheme requires the formulation of a Roe matrix which
can become impractically complicated [113] when an undefined number of ad-
ditional transport equations are introduced for the species mass fractions. The
flux-vector splitting family of schemes are computationally less expensive than
their flux-difference splitting counterparts and do not require the formulation of
such matrices, but often provide a reduced level accuracy [114].
An advection upstream splitting method (AUSM) family of schemes has been
developed by M.-S. Liou, where the AUSM+ scheme [113] has been shown to
provide comparable accuracy to the Roe scheme, for a reduced computational
expense. AUSM+ is an improvement on the original AUSM method [114].
68
Figure 4.5: Capability of the centred scheme for discontinuity capturing
Figure 4.6: Nitrogen mass fraction profile when using the centred scheme
For simplicity, the AUSM+ approach is to be briefly presented by assuming
a one dimensional system of governing equations, but is easily extendable to the
fully three dimensional system presented in Chapter 2. The flux-vector splitting
definition corresponds to the separation of the inviscid fluxes into convective and
pressure terms:
69
~Finvis = ~Fc + ~P (4.16)
with
~Fc = ρMa

1
u
h0
Yα
 = ρMa~φ, ~P =

0
p
0
0
 (4.17)
where M is the Mach number and a is the speed of sound, since
M =
u
a
(4.18)
The numerical convective face flux can therefore be defined as:
fφf = ρfMfafφfAf + pfAf (4.19)
where ρfMfafAf is the mass flux, m˙.
It is the definition of the face values for Mach number and pressure which
make up the core of the AUSM family of schemes. The face Mach number is
defined as:
Mf = M
+(ML) + M
−(MR) (4.20)
where ML and MR correspond to the Mach numbers for the left and right cells,
defined using the face value for the speed of sound:
ML/R =
uL/R
af
(4.21)
The M function is defined as:
M
±(M) =
{
1
2
(M ± |M |) if |M | ≥ 1
M
±
β (M) otherwise
(4.22)
with
70
M
±
β (M) = ±
1
4
(M ± 1)2 ± β(M2 − 1)2, − 1
16
≤ β ≤ 1
2
(4.23)
The face pressure is calculated using:
pf = P
+(ML)pL + P
−(MR)pR (4.24)
and the P function is defined as:
P
±(M) =
{
1
2M
(M ± |M |) if |M | ≥ 1
P±α (M) otherwise
(4.25)
with
P
±
α (M) =
1
4
(M ± 1)2(2∓M)± αM(M2 − 1)2, −3
4
≤ α ≤ 3
16
(4.26)
It should be noted that Equation 4.23 differs slightly from that in reference
[113], due to an error in the original paper. That presented here is correct [115].
The terms involving α and β correspond to the higher order extension of the
original AUSM scheme. For simplicity, the numerical speed of sound will be
calculated as a simple average of the left and right states:
af =
aL + aR
2
(4.27)
where the speed of sound for a cell is defined as:
aL/R =
√
γL/R
pL/R
ρL/R
(4.28)
The face value of the conserved variables is defined through simple upwinding:
φf =
{
φL if Mf ≥ 0
φR otherwise
(4.29)
The use of neighbouring cell centred data for the left and right states cor-
responds to a first order upwind method. Extension to higher orders will be
discussed in Section 4.4.3.
71
4.4.2.4 AUSM+UP
The AUSM family of upwind schemes was developed for use in density based
solvers for the solution of high Mach number flows. However, it is often impor-
tant to ensure a numerical method is applicable over a wide range of flow regimes
[116]. Not only is it useful for a compressible flow solver to be applicable to the
solution of low speed problems, but high Mach number flows may also contain
low speed regions. Scramjet combustors may contain regions of separation to aid
flame-holding or the simulation of converging diverging nozzles used in experi-
mental apparatus to accelerate the flow to supersonic speeds may be required.
Computational techniques are therefore needed which can simultaneously handle
the simulation of these high and low speed regimes.
The AUSM and AUSM+ schemes are known to produce oscillations in pres-
sure under low Mach number conditions [116]. This may be due to the presence of
a low speed region or may simply occur in directions of corresponding low velocity,
even if the speed of the bulk flow is high (i.e. velocity component perpendicular
to main flow direction may be small).
This behaviour led to the development of the AUSM+UP scheme, which is
applicable at all speeds. Essentially, this scheme was developed through applica-
tion of a low Mach number series expansion to the governing system of equations.
Modified equations for Mf and pf were created, which effectively lead to the ad-
dition of diffusion terms to damp oscillations under low Mach number conditions.
The new face Mach number is expressed as:
Mf = M
+(ML) + M
−(MR) +Mp (4.30)
which is identical to Equation 4.20 with the addition of a pressure diffusion term,
Mp, defined as:
Mp = −Kp
fa
max(1− σM¯2, 0)pR − pL
ρfa2f
(4.31)
where Kp and σ are model constants with bounds 0 ≤ Kp ≤ 1 and σ ≤ 1. This
diffusion term is only activated in low Mach number regions of the flow through
the max(1− σM¯2, 0) term, where M¯ can be interpreted as the mean local Mach
72
number:
M¯2 =
u2L + u
2
R
2a2f
(4.32)
The term fa is introduced to properly scale the diffusion with flow speed. It is
defined in terms of a reference Mach number, Mo:
fa(Mo) = Mo(2−Mo) (4.33)
where
M2o = min(1,max(M¯
2,M2∞)) (4.34)
M∞ is the free-stream, or cut-off, Mach number, used to avoid fa tending to zero.
Since the test cases to be presented in this thesis will consist of supersonic core
flows, fa will usually become unity, providing zero scaling contribution.
The face pressure is modified in a similar manner, with the addition of a
velocity diffusion term, pu, to Equation 4.24:
pf = P
+(ML)pL + P
−(MR)pR + pu (4.35)
with
pu = −KuP+(ML)P−(MR)(ρL + ρR)(faaf)(uR − uL) (4.36)
where Ku is a model constant. The pu diffusion term is only active in subsonic
regions of the flow, since at Mach numbers above unity either P+(ML) or P
−(MR)
becomes zero to obtain a one sided approximation.
It is the addition of the velocity and pressure diffusion terms, pu and Mp, to
the AUSM+ scheme which gives AUSM+UP its name. The values employed for
the AUSM+UP model constants are presented in Table 4.1.
Since regions of subsonic flow exist in test cases to be presented in Chapters 5
6 and 7, and since the diffusion terms do not play a detrimental role in supersonic
flow regimes due to the switching employed, the AUSM+UP scheme is the upwind
method of choice.
73
Variable Value
α 3
16
β 1
8
Kp
1
4
Ku
3
4
σ 1.0
Table 4.1: Model constants used for the AUSM family of schemes [116].
4.4.2.5 Test case: Discontinuity Capturing - Upwind
The test case of Section 4.4.2.2 is again simulated, now using a first order AUSM+UP
upwind scheme. It can be seen from Figure 4.7 that the introduction of an upwind
scheme prevents the generation of spurious oscillations and provides a monotonic
solution for the discontinuity in question.
Figure 4.7: Capability of the 1st order upwind scheme for discontinuity capturing
74
However, it can be seen that the first order upwind method is very dissipative,
severely smoothing the discontinuity. Higher order methods are required in order
for a sharper resolution to be obtained. This first order method would not be
suitable for a large eddy simulation since it would rapidly dissipate the large scale
structures of the flow.
4.4.3 Linear Interpolation
In order to obtain a more accurate solution, it is possible to extend the upwind
scheme to higher orders using interpolation of cell-centred data to cell faces. The
extension of the AUSM+UP scheme to second order of accuracy using a linear
interpolation method is to be demonstrated.
As discussed, it is the face data that is required for the inviscid flux cal-
culations. The centred scheme approximates this through a simple average of
neighbouring cell-centred data and the first order upwind method simply sets the
face data equal to the cell-centred data in the upwind cell. In order to increase
the accuracy of the AUSM+UP scheme, the cell-centred data can be interpolated
to the centre of the face using gradient information for the variables concerned.
As can be seen from Figure 4.8, this interpolation generates new values for
the left and right states, which are simply used in place of the left and right
cell-centred data of the first order method.
Figure 4.8: Linear interpolation approach
The interpolation procedure for the new left and right values can be mathe-
matically described as:
75
φL = φi +∇φi · ~rL
φR = φj +∇φj · ~rR
(4.37)
where ~rL and ~rR are the vectors to the face centre from the left and right cell
centres, respectively, and ∇φi is the gradient of variable φ at cell centre i. The
weighted least squares method described later in Section 4.5.1 is employed for the
gradient calculation.
However, in high gradient regions this interpolation can produce erroneous
results, with the generation of spurious oscillations. A limiter is therefore required
to control the addition of the interpolated data to the cell centred values. Either
a flux limiter, ψf , which directly scales the interpolation:
φL = φi + ψf (∇φi · ~rL)
φR = φj + ψf (∇φj · ~rR)
(4.38)
or a gradient limiter, ψg, which modifies the gradient data used in the interpola-
tion:
φL = φi + ψg(∇φi) · ~rL
φR = φj + ψg(∇φj) · ~rR
(4.39)
can be used. A flux limiter simply varies between 1 in smooth regions of the flow
and 0 in high gradient regions, resorting back to the first order upwind method
near discontinuities. The gradient limiter modifies the gradient data through
a weighted average of neighbouring gradient values. Since gradient limiters are
capable of achieving second order accuracy right up to a discontinuity [117], they
will be the focus here.
When increasing the order of accuracy through use of the linear interpolation
approach, the computational cost is significantly increased over the first order up-
wind method. Additional calculations are required in order to obtain the gradient
data and then to conduct the limiting and interpolation procedures. However,
for a viscous simulation, where the gradient data is also required for the viscous
flux calculations (see Section 4.5), the additional cost of the increased order of
accuracy is not so severe, as shown in Table 4.2.
76
% increase
Inviscid 222
Viscous 54
Table 4.2: Percentage increase in computational cost of inviscid and viscous flow
calculations for the second order upwind method with Van Rosendale gradient
limiter, over the 1st order upwind scheme.
4.4.3.1 Gradient Limiter
It is possible to modify the gradient employed in the interpolation calculation in
such a way as to avoid the generation of spurious interpolated data. This is known
as gradient limiting and the gradient limiter presented here is based on that of
Van Rosendale [117]. The limited gradient for cell i, ∇φi|l, is calculated through
a weighted sum of all unlimited neighbouring cell-centred gradient values:
ψg(∇φi) = ∇φi|l =
Ni∑
j=1
wi,j∇φj (4.40)
where Ni is the number of neighbouring cells (e.g. 6 for a hexahedral cell) plus
one:
Ni = Nneighbours + 1 (4.41)
The plus one is present to ensure the unlimited gradient value for cell i is also
included in the calculation.
The weight wi,j, applied to the gradient in cell j in the calculation of the
reconstructed gradient value for cell i, is given by:
wi,j =
∏Ni
k 6=j βk + ǫ∑Ni
l=1
∏Ni
k 6=l βk +Niǫ
(4.42)
where βk is the square of the L2-norm of the gradient in question:
βk = ‖∇φk‖2 =
(
∂φ
∂x
∣∣∣∣
k
)2
+
(
∂φ
∂y
∣∣∣∣
k
)2
+
(
∂φ
∂z
∣∣∣∣
k
)2
(4.43)
and the small number ǫ avoids division by zero, with the Ni factor in the denom-
inator satisfying the constraint that:
77
Ni∑
j=1
wi,j = 1 (4.44)
The weights are calculated in this way so that if a cell-centred gradient in cell
j is significantly larger than those in other neighbouring cells, the corresponding∏Ni
k 6=j βk term will be relatively small, since it does not include the large squared
L2-norm for this gradient. This results in a small weighting, wi,j, and result-
ing small contribution from cell j to the limited gradient for use in the cell i
interpolation.
This procedure has the property that near discontinuities the reconstructed
gradient uses information just from one side of the discontinuity, avoiding the
generation of spurious oscillations whilst still achieving second order of accuracy
[117]. This gradient limiter therefore behaves in a similar fashion to the more
expensive essentially non-oscillatory (ENO) scheme [118].
4.4.3.2 Test case: Discontinuity Capturing
The test case of Section 4.4.2.2 is re-used to demonstrate the increase in accuracy
obtained through use of the linear interpolation method with the Van Rosendale
gradient limiter. It can be seen from both Figure 4.9 and Figure 4.10 that the
introduction of linear interpolation provides a sharper resolution of the mass
fraction discontinuity when compared to the first order upwind method, whilst
still maintaining a monotonic solution.
Figure 4.9: Resolution comparison for 1st order upwind (top) and second order
upwind with Van Rosendale gradient limiter (bottom), schemes
78
Figure 4.10: Capabilities of the higher order upwind method for discontinuity
capturing
4.4.3.3 Test case: Sod’s Shock Tube Problem
Another test case, which shows some important features of the Van Rosendale
gradient limiter and provides the capability to compare the higher order upwind
and central discretisations in subsonic smooth regions of the flow, is Sod’s shock
tube problem.
This test case consists of a fully enclosed domain with discontinuous initial
conditions. As is shown in Figure 4.11, the initial profile consists of two regions
separated by a hypothetical diaphragm. Table 4.3 provides the initial conditions
for this test case showing that region 1 and region 4 are high and low pressure
regions, respectively. The test gas is pure air.
Initiation of a simulation with such initial conditions corresponds to this di-
aphragm bursting and the characteristics of the resulting flow, for which an an-
alytical solution is available, are also shown in Figure 4.11. An expansion fan
separates regions 1 and 2 and initially propagates to the left, and a contact dis-
79
Figure 4.11: Initial conditions and analytical solution profile for Sod’s shock tube
problem
Region 1 Region 4
Pressure (bar) 10 1
Temperature (K) 520 416
Velocity (m/s) 0 0
Table 4.3: Initial conditions for Sod’s shock tube problem.
continuity separating regions 2 and 3 and a shock wave separating regions 3 and
4 initially propagate to the right.
The same mesh as used for the mass fraction discontinuity test case is em-
ployed here, but inviscid wall boundary conditions are used in place of the inlet
and outlet. A time step of 1.6× 10−7 seconds is now employed and the computa-
tional and analytical solutions are compared after 1500 iterations, which is before
the shock wave has reached the end wall. The AUSM+UP and central schemes
are employed for discretisation of the inviscid fluxes.
Figure 4.12 shows the resulting pressure distribution, where the monotonic
improvement over the centred scheme, through use of upwind methods, can again
clearly be seen at the shock location. Two regions are highlighted on this plot,
namely the top and bottom of the expansion fan. At the top of the expansion
fan it can be seen that the AUSM+UP upwind method with linear interpolation
80
and the Van Rosendale gradient limiter, is more accurate than the second order
centred scheme. However, at the bottom of the expansion fan, for the higher order
upwind scheme, a small undershoot can be seen in the solution. This undershoot
is due to the fact that the Van Rosendale gradient limiter does not obey the strict
TVD criterion. However, the magnitude of this oscillation is small and is deemed
acceptable.
Figure 4.12: Pressure profile for Sod’s shock tube problem, after 1500 iterations
As an aside, to further demonstrate why the centred scheme is unsuitable for
discretisation of the inviscid fluxes in supersonic combustion, Figure 4.13 shows
the resulting oscillations in the temperature profile, which could cause unphysical
ignition to occur. Oscillations at both the contact discontinuity and shock wave
are visible, caused by oscillations in density at these locations.
81
Figure 4.13: Temperature profile for Sod’s shock tube problem, after 1500 itera-
tions
4.5 Viscous Fluxes
The calculation of the viscous flux in Equation 4.8 requires calculation of face
data for both the diffusion terms and gradient vectors. Simple averaging of the
neighbouring cell centred data is used to calculate the face values for the scalar
diffusion coefficients so the focus of this section is on the more complex calcula-
tions for face values of the gradient data.
4.5.1 Gradient Reconstruction
Cell-centred gradient data is required for calculation of the viscous fluxes and is
also used in the linear interpolation method for higher order upwind inviscid flux
discretisation. Green-Gauss and least-squares methods are commonly employed
82
for the gradient reconstruction. However, the Green-Gauss approach can provide
highly inaccurate results on mixed grids [119], particularly when a cell-centred
data structure is used [120]. The least-squares approach is capable of provid-
ing accurate results on arbitrary mesh types, independent of the data structure
employed, and is therefore the method of choice for gradient reconstruction in
PULSAR.
For highly anisotropic grids, such as those found in regions close to a viscous
wall or in a shear layer, the least-squares method can underestimate the gradients
by up to an order of magnitude [120]. It is therefore important to apply weights,
wij , to contributions from neighbouring cells and the inverse distance weighting
approach is employed:
wij =
1
lij
(4.45)
Figure 4.14 shows a typical region of stretched cells. Through use of the inverse
distance weighting method, where lij is the distance between cell centres, cells
A and C provide a greater contribution to the gradient calculation, significantly
improving results from the least-squares method on anisotropic meshes.
Figure 4.14: Stencil for least-squares gradient reconstruction in shaded cell.
Anisotropic grid displayed.
The weighted least-squares procedure for gradient reconstruction is quite in-
volved and is provided in the Appendix of [120].
4.5.2 Discretisation Technique
Face values for the diffusion coefficient, or any other scalar quantities required for
the viscous flux, is calculated using the centred approach of Equation 4.12. The
83
centred approach is also employed for the calculation of the face gradient values,
∇φf , where it is natural to take a simple average of the neighbouring data:
∇φ = ∇φ|i +∇φ|j
2
(4.46)
However, on regular hexahedral grids this simple average can lead to decoupling
of the solution [121] and subsequent generation of spurious oscillations. In order
to avoid this decoupling, modifications similar to those introduced by Rhie and
Chow [122] for pressure-velocity coupling in the Euler equations, can be applied
to Equation 4.46:
∇φf = ∇φ−
[
∇φ · ~tij − ∂φ
∂l
∣∣∣∣
ij
]
~tij (4.47)
where
∂φ
∂l
∣∣∣∣
ij
=
φj − φi
lij
(4.48)
and the distance between the centres of cells i and j, lij , and corresponding vector,
~rij , are used to define the unit vector, ~tij.
~tij =
~rij
lij
(4.49)
4.6 Turbulence Model
The Menter SST turbulence model is chosen for the calculation of both the eddy
viscosity in a RANS simulation and also the sub-grid viscosity for LES, using a
DDES hybrid RANS-LES approach.
This two equation model is capable of providing the turbulence length scales
required for use in combustion model source terms whilst combining the good
near wall and free-stream properties of the k − ω and k − ε models, respectively.
84
4.6.1 RANS
Although there is a transition from a k−ω to a k−ε turbulence model away from
the wall through use of a blending function, F1, only two transport equations for
k and ω are required due to the simple relationship with ε, given by Equation
2.47:
∂ρ¯k
∂t
+
∂ρ¯u˜ik
∂xi
= P˜ − β⋆ρ¯ωk + ∂
∂xi
[
(µ+ σkµt)
∂k
∂xi
]
(4.50)
∂ρ¯ω
∂t
+
∂ρ¯u˜iω
∂xi
=
ρ¯α
µt
P˜ − βρ¯ω2 + ∂
∂xi
[
(µ+ σωµt)
∂ω
∂xi
]
+ 2 (1− F1) ρ¯σω2
ω
∂k
∂xi
∂ω
∂xi
(4.51)
In the standard k − ω model [37], the dynamic eddy viscosity is calculated from:
µt,k−ω =
ρk
ω
(4.52)
However, for adverse pressure gradient regions of a boundary layer this formula-
tion leads to a severe over-prediction of the shear stress [123]. A better model for
the eddy viscosity in these regions is given by:
µt,adverse =
ρa1k
S
(4.53)
The eddy viscosity for the Menter SST turbulence model is therefore chosen as
the minimum of µt,k−ω and µt,adverse. However, in order to restrict the use of
the adverse pressure gradient modification to inside the boundary layer, a second
blending function, F2, is introduced:
µt =
ρ¯a1k
max (a1ω, SF2)
(4.54)
The blending function, F1, used to transition between the k − ω and k − ε tur-
bulence models, is given as:
F1 = tanh
(
arg41
)
(4.55)
85
where
arg1 = min
(
arg2,
4ρ¯σω2k
CDkωy2w
)
(4.56)
with
arg2 = max
( √
k
β⋆ωyw
,
500µl
y2wρ¯ω
)
(4.57)
and
CDkω = max
(
2ρ¯σω2
1
ω
∂k
∂xi
∂ω
∂xi
, 10−10
)
(4.58)
The second blending function, F2, is given by:
F2 = tanh
(
arg22
)
(4.59)
The production term used in the two transport equations is defined as:
P = ρ¯u˜
′
iu
′
j
∂u˜i
∂xj
(4.60)
where the turbulent stresses, ρ¯u˜
′
iu
′
j, are given by the Boussinesq approximation
of equation 2.27. In order to prevent the buildup of turbulence in stagnation
regions a limiter is applied to this production term, giving:
P˜ = min (P, 10β⋆ρ¯kω) (4.61)
All model constants are provided in Table 4.4. The constants appearing in
the k and ω transport equations are made up of a blend of the constants for the
k − ω and k − ε turbulence models. For example, the constant σk is defined as:
σk = σk1F1 + σk2(1− F1) (4.62)
86
α1
5
9
α2 0.44
β1 0.075 β2 0.0828
σk1 0.85 σk2 1
σω1 0.5 σω2 0.856
κ 0.41 a1 0.31
β⋆ 0.09
Table 4.4: Model constants for the Menter SST turbulence model.
4.6.2 DDES
The Menter SST turbulence model is also used as the sub-grid scale model under
a DDES framework. Transitioning from a RANS model near the wall to a sub-
grid scale model in the free-stream is enabled through a simple modification to
the turbulence length scale in the k equation destruction term:
∂ρ¯k
∂t
+
∂ρ¯u˜ik
∂xi
= P˜ − ρ¯k
3
2
l˜
+
∂
∂xi
[
(µ+ σkµt)
∂k
∂xi
]
(4.63)
Specific details of model length scales, l˜, and the blending employed were provided
in Section 2.4.4.
4.6.3 Wall Distance
The distance to the nearest wall, yw, required for the Menter SST turbulence
model is found using a simple search procedure. For each cell, every wall boundary
point is searched to find the smallest distance to a viscous boundary. As one might
expect, this procedure is computationally expensive and more efficient methods
are available [124]. However, these efficient methods are more complicated to
implement, so the speed-up of the searching procedure was attempted.
Computational costs are brought down to a reasonable level through fully
parallel operation, whereby only wall distance calculations for the cells on each
partition are required. In order for this calculation to be performed only once,
the results are output to a data file which can be quickly read in for subsequent
runs of a simulation.
As an example, the time required to calculate the wall distances for the LES
87
mesh of the wall bounded combustor flow to be presented in Chapter 7 is ap-
proximately 460 seconds. The mesh consists of 36.8M cells and is partitioned
onto 2640 processors, giving approximately 17,000 cells per partition. The calcu-
lations were performed on the HECToR cluster. This one-off computational time
is considered acceptable.
4.6.4 Compressibility Corrections
For highly compressible flows, corrections can be applied to the turbulence model
in order to reduce the eddy viscosity to more realistic levels. Scaling is applied
to the destruction source terms in both the k and ω equations, by modifying the
β⋆ and β coefficients to:
β⋆cc = β
⋆[1 + ξ∗F (Mt)] (4.64)
βcc = β − β⋆ξ∗F (Mt) (4.65)
where β⋆cc and βcc are the new scaled coefficients to be used in the turbulence model
transport equations. ξ∗ and F (Mt) are defined by the particular compressibility
correction employed, where Mt is the turbulence Mach number, defined as:
M2t =
2k
a2
(4.66)
with a being the speed of sound. The Sarkar, Wilcox and Zeman compressibility
corrections [125] are implemented in PULSAR:
Sarkar compressibility correction:
ξ∗ = 1, F (Mt) =M
2
t (4.67)
Wilcox compressibility correction:
ξ∗ = 2, F (Mt) = max(M
2
t −M2to, 0) (4.68)
where Mto is a cut-off Mach number equal to 0.25.
88
Zeman compressibility correction:
ξ∗ =
3
4
, F (Mt) =
[
1− e− 12 (γ+1)(Mt+Mto)2/Λ2
]
H(Mt −Mto) (4.69)
where Mto = 0.1
√
2/(γ + 1) and Λ = 0.6.
4.7 Boundary Conditions
Correct specification of boundary conditions on the flow in question is of sig-
nificant importance in order to accurately replicate the physics encountered in
reality. Inlet, outlet and surface boundary conditions are applied for simulations
throughout this research.
4.7.1 Inlet
The eigenvalues of the one-dimensional Euler equations are given as:
u− a
u
u+ a
where u is the flow velocity and a is the speed of sound, and define the directions
in which information can propagate. The methods employed for specification of
inlet boundary conditions therefore depend on the Mach number of the incoming
flow.
4.7.1.1 Supersonic
For inlet Mach numbers above unity, information can only propagate downstream
since all eigenvalues are positive. Therefore, all flow parameters must be specified
at the inlet plane. PULSAR requires specification of the stagnation pressure p0,in,
static pressure pin and stagnation temperature T0,in. The static temperature Tin,
velocity normal to the inlet Vin and density ρin, can then be calculated from:
89
Tin = T0,in
(
pin
p0,in
)γ−1
γ
(4.70)
Vin =
√
2cp(T0,in − Tin) (4.71)
ρin =
pin
RTin
(4.72)
where γ and R are dependent upon temperature and composition.
4.7.1.2 Subsonic
For an inlet Mach number below unity the inlet velocity is lower than the speed
of sound and the u−a eigenvalue is negative, meaning information can propagate
in both the up and downstream directions. For this reason, one of the boundary
variables must be interpolated from inside the domain. In PULSAR the outgoing
Riemann invariant,
R− = ui,c · ni − 2ac
γc − 1 (4.73)
is used to specify the speed of sound at the inlet [119], ain:
ain =
−R−(γc − 1)
(γc − 1)cos2θ + 2
{
1 + cosθ
√
[(γc − 1)cos2θ + 2]a20
(γc − 1)(R−)2 −
γc − 1
2
}
(4.74)
where the subscripts c and 0 correspond to data at the centre of the computational
cell inside the domain and stagnation values, respectively, and θ is the flow angle
relative to the boundary. The stagnation temperature and pressure are prescribed
for the inlet, and the static temperature is calculated from:
Tin = T0,in
(
a2in
a20
)
(4.75)
allowing the inlet pressure, velocity and density to be calculated from Equations
4.70, 4.71 and 4.72, respectively.
90
4.7.1.3 Turbulence
Boundary conditions for the k and ω variables used in the Menter SST turbulence
model also need to be specified at the inlet plane, with the following range of
variables recommended [126]:
10−5µlVin
ρinLk−ω
< kin <
0.1µlVin
ρinLk−ω
(4.76)
Vin
Lk−ω
< ωin <
10Vin
Lk−ω
(4.77)
where Lk−ω is a prescribed length scale.
Turbulent inlets are also possible, where coherent structures are convected into
the domain through specification of velocity fluctuations. The method employed
to generate this turbulent velocity data is described in Section 4.10.
4.7.2 Outlet
Similarly to the inlet boundary condition, the method used to handle outlet flows
depends on the outlet Mach number. In PULSAR, for supersonic flows, the
required boundary data is simply interpolated from inside the domain using a
first order, zero gradient approach.
For subsonic flow one variable needs to be prescribed at the outlet to deal with
the negative u − a eigenvalue. The static pressure, pout, is the variable applied
and all other boundary data is interpolated from inside the domain in the same
fashion as for supersonic conditions.
4.7.3 Wall
The presence of viscous surfaces can have a significant impact on the physics
of a flow, due to the no-slip condition at the wall. Since scramjet combustors
provide a wall bounded problem, the accurate modelling of the physics involved
is of paramount importance. Viscous surfaces apply a shear stress to the near
wall fluid and can also provide a heat flux through use of isothermal boundary
conditions. Correct modelling of turbulent boundary layers also requires the
correct boundary condition application for the k and ω variables.
91
4.7.3.1 Velocity
A flow tangential to a solid surface encounters a viscosity dominated region called
a boundary layer, where the velocity tends to zero at the wall, called the no-slip
condition. The structure of a high Reynolds number flat plate zero pressure
gradient boundary layer can be described in terms of non-dimensional numbers
for wall distance (y+) and velocity (u+), where
y+ =
ρywuτ
µ
(4.78)
u+ =
utang
uτ
(4.79)
with
uτ =
√
|τw|
ρ¯
(4.80)
yw is the wall normal distance from the viscous surface to the centre of the near
wall cell, uτ is the shear velocity, τw is the wall shear stress and utang is the
velocity in the near wall cell, tangential to the surface.
Figure 4.15 shows the profile from a flat plate zero pressure gradient boundary
layer simulation. It can be seen that close to the wall there is a linear region,
called the viscous sublayer, where y+ scales directly with u+. The linear region is
valid up to a y+ value of approximately 10, but above a y+ of about 6 the errors
in this linear assumption begin to grow.
For 30 < y+ < 1000 the boundary layer profile is well represented by a log-law
[127], where u+ is proportional to the log of y+, given as:
u+ =
1
κ
ln(Ey+) (4.81)
where κ is the von Ka´rma´n constant, taken as 0.41, and E is a constant taken as
9.5. However, in the range 10 < y+ < 30 there exists a buffer layer, which does
not obey either the linear or log laws. Above a y+ of approximately 1000 exists
the outer region of the boundary layer.
92
Figure 4.15: Dimensionless profile for a flat plate zero pressure gradient boundary
layer.
It is the wall shear stress, τw, which is required to be applied as a boundary
condition to the momentum equations. Through manipulation of Equations 4.79
and 4.80 this wall shear stress can be given as:
τw = ρ¯
(
u+
utang
)2
(4.82)
It can be seen from this equation that as long as the u+ value and flow properties
for the near wall cell are known, the shear stress can be found.
As can be seen in Figure 4.15, Spalding [128] developed a single equation
which is capable of continuously defining the viscous, buffer and log-law regions:
y+ = u+ + 0.1108
[
e0.41u
+ − 1− 0.41u+ − (0.41u
+)2
2!
− (0.41u
+)3
3!
]
(4.83)
93
An iterative Newton-Raphson method can be applied to find the y+ and u+ values
for the near wall cell. Once the y+ and u+ values have been obtained, uτ can
be found from Equation 4.78 or 4.79 and subsequently the wall shear stress from
Equation 4.82.
However, the application of a log-law collapses under more complex flow con-
ditions [54]. In the presence of physics such as pressure gradients or shock-wave
boundary layer interactions it is recommended that the near-wall cell has a y+
of less than 5 [61], so that the assumptions of linearity in the viscous sublayer
are used to calculate the wall shear stress, rather than the now invalid law of the
wall.
4.7.3.2 Temperature
Adiabatic and isothermal wall boundary conditions are implemented into PUL-
SAR. Using the isothermal boundary condition, constant temperature walls can
be used to calculate the levels of heat flux, q˙w, into the near wall cell. The heat
flux is defined as [129]:
q˙w =
ρ¯cp(Tw − T˜ )uτ
T+
(4.84)
where T+ is the dimensionless variable for temperature, Tw is the specified wall
temperature and T˜ is the temperature in the near wall cell. Similarly to the
velocity boundary conditions, a linear relationship between T+ and y+ is assumed
to exist near to the wall:
T+ = y+Prl (4.85)
and a log relationship away from the wall:
T+ =
1
κ˜
ln(E˜y+) (4.86)
Once T+ is known, the heat flux can be found from Equation 4.84. Since y+
is already know from the velocity boundary conditions, iterative methods are not
required here. Determination of T+ for near wall cells inside the viscous sublayer
is straightforward through use Equation 4.85. For y+ values in the log-law region,
94
rather than attempting to determine the variables κ˜ and E˜ in Equation 4.86, the
temperature and velocity log laws are combined to produce [129]:
T+ =
κ
κ˜
(u+ + P ) (4.87)
where P is the pee function:
P =
1
κ
ln
(
E˜
E
)
(4.88)
empirically given by [130]:
P = 9.24(σ˜
3
4 − 1)[1 + 0.28e−0.007eσ] (4.89)
with
σ˜ =
Prl
Prt
(4.90)
4.7.3.3 Turbulence
In order to correctly define the near wall distribution of eddy viscosity, boundary
conditions are required for k and ω. A simple zero flux boundary condition is
applied at the wall for the k equation, but solutions for ω in the near wall linear
and logarithmic regions are respectively given by [131]:
ωlin =
80µl
ρ¯y2w
(4.91)
and
ωlog =
1
0.3κ
uτ,blend
yw
(4.92)
where uτ,blend is a blending of the shear velocity values from the linear and loga-
rithmic regions:
uτ,blend =
4
√
u4τ,lin + u
4
τ,log (4.93)
with
95
uτ,lin =
utang
y+
(4.94)
and
uτ,log =
utangκ
ln(Ey+)
(4.95)
For small y+ this blending tends to the linear value and for large y+ to the
logarithmic value. Similarly, a smooth blending can then be used to define the
near wall cell value, ωw:
ωw =
√
ω2lin + ω
2
log (4.96)
4.8 Combustion
Implementation of reaction rate modelling is not the only concern when develop-
ing a code to be used for simulations of combustion. The underlying thermody-
namics is also of paramount importance and both will be discussed here.
4.8.1 Thermo-chemistry
The stagnation specific internal energy equation introduced in Chapter 2, re-
produced here for clarity:
e˜0 = h˜− p˜
ρ¯
+
1
2
u˜iu˜i + k
handles the thermodynamics of the flow. The influence of chemical composition
on the thermodynamics is clearer when the energy equation is re-written as:
e˜0 =
Ns∑
α=1
Y˜αhα − RmT˜ + 1
2
u˜iu˜i + k (4.97)
where the definition of static enthalpy from Equation 2.13 and the ideal gas
law have been employed, with Rm being the mixture value for the specific gas
constant, given by:
96
Rm =
Ns∑
α=1
Y˜αRα. (4.98)
and Rα being the specific gas constant for species α. The only remaining unknown
in Equation 4.97 is the specific enthalpy for each species, hα. Since a reacting flow
encounters large variations in temperature, the constant specific heat assumptions
of a perfect gas can not be used and so hα must be defined through use of the
integral form:
hα =
∫ T
Tref
cpαdT + h
ref
α (4.99)
where cpα is the mass-based specific heat capacity at constant pressure for species
α and hrefα is the value of hα at the reference temperature Tref . For a semi-perfect
gas the specific heat capacity can be described through use of a polynomial in
temperature [132]:
cpα =
Np∑
k=1
a
(l)
α,kT˜
k−1 (4.100)
where a
(l)
α,k are the coefficients of the polynomial with degree Np, for species α in
temperature interval l. Performing the integration over successive temperature
intervals, it can be shown [132] that:
hα =
Np∑
k=1
a
(l)
α,k
k
T˜ k + a
(l)
α,Np+1
(4.101)
Inserting this into Equation 4.97 and arranging terms into corresponding powers
of T , a polynomial for the temperature can be formed:
0 =
[(
1
2
u˜iu˜i + k − e0
)
+
Ns∑
α=1
Y˜αa
(l)
α,Np+1
]
+
[
Ns∑
α=1
Y˜α
(
a
(l)
α,1 − Rα
)]
T˜ +
Np∑
k=2
[
Ns∑
α=1
Y˜α
a
(l)
α,k
k
]
T˜ k (4.102)
97
Employing the same Newton-Raphson iterative method as used for Equation
4.83, the solution to this equation can be found, providing the temperature. The
corresponding pressure can be found through use of the ideal gas equation:
p¯ = ρ¯RmT˜ (4.103)
and the temperature dependent dynamic molecular viscosity is given by:
µl =
λ
cp
Prl (4.104)
where λ is the mixture thermal conductivity, which can be found using the fol-
lowing relationship [132]:
λ
cp
= Aλ
(
T˜
Tref
)r
(4.105)
with Aλ, r and Tref being constants.
The molar polynomial coefficients, a¯
(l)
α,k, are provided by the CHEMKIN database
and the mass based coefficients, a
(l)
α,k, are calculated from:
a
(l)
α,k =
R0
Wα
a¯
(l)
α,k (4.106)
where R0 is the universal gas constant.
4.8.2 Assumed PDF Combustion Model
As discussed in Chapter 3, the assumed PDF combustion model is used to find
the mean reaction rate through integration of the instantaneous reaction rate,
¯˙ω =
∫
ω˙
(
ρˆ, Tˆ , Yˆ1, . . . , YˆNs
)
P
(
ρˆ, Tˆ , Yˆ1, . . . , YˆNs
)
dρˆdTˆdYˆ1, . . . , dYˆNs
where statistical independence is assumed between temperature, composition and
density:
P
(
ρˆ, Tˆ , Yˆ1, . . . , YˆNs
)
= PT (Tˆ )PY (Yˆ1, . . . , YˆNs)δ(ρˆ− ρ¯)
98
With the instantaneous reaction rate given by Equation 3.17, this leads to
¯˙ωα = Wα
Nr∑
r=1
(ν ′′αr − ν ′αr)
kfrcνMrM Ns∏
γ=1
c
ν′γr
γ − kbrcνMrM
Ns∏
γ=1
c
ν′′γr
γ
 (4.107)
where
kr =
∫ ∞
0
kr(Tˆ )PT (Tˆ )dTˆ (4.108)
and
cνMrM
Ns∏
γ=1
c
νγr
γ =
∫
cˆνMrM
Ns∏
γ=1
cˆνγrγ PY (Yˆ1, . . . , YˆNs)dYˆ1, . . . , dYˆNs (4.109)
with
cˆνMrM
Ns∏
γ=1
cˆνγrγ =
Ns∑
γ=1
(
ηγ,M ρ¯Yˆγ
Wγ
)νMr Ns∏
γ=1
(
ρ¯Yˆγ
Wγ
)νγr
(4.110)
where νMr is unity if third body reactions are present in reaction r and zero
otherwise and ̂ corresponds to sample space variables.
4.8.2.1 Clipped Gaussian PDF
Numerical integration is required in order to evaluate the mean reaction rate con-
stant in Equation 4.108, since no analytical solution is available. The instanta-
neous reaction rate constant is integrated with a Gaussian PDF for temperature:
PT (Tˆ ) =
1√
2πσT
exp
[
−(Tˆ − T˜ )
2
2σT
]
(4.111)
where T˜ and σT are the mean and variance values.
However, the integration limits on Tˆ of 0 and ∞ cause problems in the eval-
uation of kr(Tˆ ), since the Arrhenius expression is not valid at very low or very
99
high temperatures [76]. Therefore, the integration is modified to
kr =
∫ Tˆmax
Tˆmin
kr(Tˆ )PT (Tˆ )dTˆ = T1 (4.112)
where a clipped Gaussian profile [133] is employed with Tˆmin and Tˆmax equal to
300K and 3000K, respectively:
PT (Tˆ ) =
1√
2πσT
exp
[
−(Tˆ − T˜ )
2
2σT
] [
H(Tˆ − Tmin)−H(Tˆ − Tmax)
]
(4.113)
+ A1δ(Tˆ − Tmin) + A2δ(Tˆ − Tmax)
A1 and A2 are the areas under the PDF below and above the Tmin and Tmax
limits, respectively. They are lumped at the two extremes through use of the
delta function, in order to stay consistent with Equation 4.111.
On-the-fly numerical integration is computationally expensive and so the look-
up table approach is preferred. At the start of a simulation matrices of mean
forward and backward reaction rate constants are evaluated over a user-defined
range of temperature and temperature variance values. The user also defines the
matrix density, by specifying the number of rows and columns (number of tem-
perature and variance values) to use. The composite Simpsons rule is employed
to integrate the reaction rate constant for each matrix entry. The sample space
from 300K to 3000K is split into 1000 intervals in order to capture PDFs of a
sharp nature.
During a simulation, linear interpolation between four tabulated values is used
to find the required mean reaction rate constant from simulated temperature and
temperature variance values.
4.8.2.2 Multivariate Beta PDF
The multivariate beta PDF of Girimaji [79; 80] is employed for the species mass
fraction PDF:
100
PY (Yˆ1, . . . , YˆNs) =
Γ(
∑Ns
γ=1 βγ)∏Ns
γ=1 Γ(βγ)
[
δ
(
1−
Ns∑
γ=1
Yˆγ
)
Ns∏
γ=1
Yˆ βγ−1γ
]
(4.114)
where δ and Γ are the delta and Gamma functions, respectively. An analytical
solution is available [76] for this beta PDF to evaluate the integration in Equation
4.109, defined as:
cνMrM
Ns∏
γ=1
c
νγr
γ = T2T3T4 (4.115)
with
T2 = ρ¯
νMr
Ns∏
γ=1
(
ρ¯
Wγ
)νγr
= ρ¯mr
Ns∏
γ=1
(
1
Wγ
)νγr
(4.116)
T3 =
[
Ns∑
γ=1
ηγ
Wγ
(βγ + νγr)
]νMr
(4.117)
T4 =
∏Ns
γ=1
∏νγr
k=1(βγ + νγr − k)∏mr
l=1(B +mr − l)
(4.118)
where
mfr = ν
Mr +
Ns∑
γ
ν
′
γr, mbr = ν
Mr +
Ns∑
γ
ν
′′
γr (4.119)
and the Beta function parameters are
βγ = Y˜γB, B =
∑Ns
γ=1 Y˜γ(1− Y˜γ)
σY
− 1 (4.120)
where Y˜ and σY are the mean and variance for the species mass fraction.
101
4.8.2.3 Mean Reaction Rate
Feeding Equations 4.112 and 4.115 into Equation 4.107, the mean reaction rate
can finally be evaluated as:
¯˙ωα = Wα
Nr∑
r=1
[(
ν
′′
αr − ν
′
αr
)
(T1fT2fT3fT4f − T1bT2bT3bT4b)
]
(4.121)
4.8.2.4 Backward Reaction Rate Constant
Evaluation of the forward rate constant for reaction r, kfr, is straightforward from
the definition of Equation 3.15, using data from the chosen chemical mechanism.
However, evaluation of the backward reaction rate constant, kbr, is a little more
involved.
The backward reaction rate constant is found from [132]:
lnkbr = lnkfr +
Ns∑
α=1
(ν
′′
αr − ν
′
αr)
( g¯α
R0T
+ ln
pref
R0
− lnT
)
(4.122)
where g¯α is the molar Gibbs function for species α and pref is the thermodynamic
reference pressure. The molar Gibbs function can be expressed using polynomial
coefficients:
g¯α
R0T
=
a¯
(L)
α,Np+1
T
− a¯α,1lnT + (a¯(L)α,1 − a¯(L)α,Np+2)−
Np∑
k=2
a¯
(L)
α,k
k(k − 1)T
k−1 (4.123)
where the polynomial coefficients are provided by the CHEMKIN database. The
thermodynamic reference pressure is a user supplied value but is taken as 1 bar
for all reacting simulations presented.
4.9 Temporal Discretisation
Significant effort has been applied to the implementation of an upwind method
for the discretisation of the inviscid fluxes, in order to ensure a monotonic solu-
tion for a discontinuous flow field. However, it is also possible [134] that spurious
102
oscillations can be introduced by the scheme employed for the temporal discreti-
sation. For this reason, a three-step third order TVD Runge-Kutta scheme [134]
is implemented in PULSAR, given by:
(ρφ)(1) = (ρφ)n +∆tL((ρφ)n)
(ρφ)(2) =
3
4
(ρφ)n +
1
4
(ρφ)(1) +
2
3
∆tL((ρφ)(1)) (4.124)
(ρφ)n+1 =
1
3
(ρφ)n +
2
3
(ρφ)(2) +
2
3
∆tL((ρφ)(2))
for a differential equation of the form
∂ρφ
∂t
= L(ρφ) (4.125)
where ∆tL((ρφ)i) is the change in conserved variable ρφ at step i, given by
equation 4.5.
This scheme has a maximum stable CFL number of 1. Although there are
Runge-Kutta schemes with maximum CFL numbers above 1, theoretically al-
lowing larger time steps to be used, due to the stiffness of the reacting species
transport equations a CFL number below unity is in practice required for stabil-
ity. Therefore, this scheme is used to make use of its TVD properties, third order
of accuracy and efficient three-step formulation.
4.10 Turbulence Initialisation
The generation of turbulent structures in LES, either for initialisation purposes or
turbulent inlet boundary conditions, is often desired. Fourier transform methods
are often used, where a specific turbulence spectrum can be prescribed. However,
due to the use of unstructured meshes by PULSAR, this approach is not possible.
Instead, the method of Kempf et al. [64] is implemented, which is capable of
generating turbulence for arbitrary geometries with a prescribed Reynolds stress
tensor and turbulence length scales.
As a starting point, a different Gaussian profile of random numbers is applied
to all three velocity components, ui. To account for the variations in cell size,
103
the prescribed velocities in each cell are multiplied by square root of the volume
of the cell. These velocities are then diffused to form turbulent structures of a
prescribed length scale, through use of the following diffusion equation:
∂ui
∂t
= D
∂2ui
∂x2j
(4.126)
This equation can be solved using a simple iterative method with prescribed time
step, ∆t, and number of steps, n, where the diffusion coefficient, D, can be found
from:
D =
L2t
2πn∆t
(4.127)
where Lt is the required turbulence length scale. This length scale does not need
to be constant throughout the domain and in particular it is useful to be able to
prescribe a variable length scale for wall bounded flows. Near wall eddies can be
scaled with distance from the wall, while a maximum free-stream length scale,
Lmax, (corresponding to the integral length scale) is prescribed. This results in
the following definition for the spatially varying length scale used to calculate a
spatially varying diffusion coefficient:
Lt = min(Lmax, αwyw) (4.128)
where αw is a constant used to scale the near wall eddies with wall distance.
Once the random numbers have been diffused to form turbulent structures
the required Reynolds stresses can be applied through use of a transformation by
Lund et al. [135], forming the final velocity field ui,L:
ux,L = a11ux
uy,L = a21ux + a22uy (4.129)
uz,L = a31ux + a32uy + a33uz
where the Reynolds stress tensor, Rij, is applied through the tensor aij :
104
aij =

√
R11 0 0
R21/a11
√
R22 − a221 0
R31/a11 (R32 − a21a31)/a22
√
R33 − a231 − a232
 (4.130)
When using such turbulence profiles as inlet conditions to large eddy simu-
lations, the prescribed Reynolds stresses and length scales can have a significant
impact on the resulting flow. Therefore, knowledge of experimental turbulence
properties is of paramount importance for accurate computational results. In
Chapter 6, the influence of prescribed Reynolds stresses and near wall scaling
parameter αw, on the results of a LES study of a supersonic coaxial jet, is inves-
tigated.
4.11 Parallel Capabilities
In order to run on multiple processors the computational mesh must be decom-
posed so that each processor has its own subset of the domain on which to solve
the governing fluid flow equations. The ParMETIS [136] code is used for domain
decomposition.
In order to allow the computation of fluid flow between partitions a single layer
of overlapping halo cells is used and parallel communication is enabled through
the implementation of the message passing interface (MPI). The results presented
in this thesis could not have been obtained without these parallel capabilities due
to the significant computational requirements. Two high performance clusters
were used for the simulations.
4.11.1 Stokes
Stokes is a 42 node cluster in the Department of Engineering at the University of
Cambridge. Each node has two Intel Xeon E5540 quad-core processors and 24Gb
of DDR3 memory, giving a total of 336 cores on the machine. An InfiniBand
network is used for inter-node communication. Each job submitted is limited to
36 hours of run time.
105
4.11.2 HECToR
The HECToR XT6 machine is the UK national supercomputing service which
consists of 464 compute blades housed in 20 cabinets. Each blade has four nodes
with each node having two 12-core AMD Opteron 2.1GHz Magny Cours proces-
sors, giving a total of 44,544 cores over the machine. Each 12-core processor has
access to 16Gb of shared memory. Each job submitted is limited to 12 hours of
run time.
The XT6 machine started life with each processor coupled to a Cray SeaStar2
routing and communications chip, but this has recently been upgraded to the
Cray Gemini interconnect.
4.11.3 Scaling
The scalability of PULSAR has been evaluated using the HECToR cluster. This
study corresponds to a strong scaling study, whereby the size of the computational
mesh is held constant and split between an increasing number of cores, starting
with 264 and 528 cores and then increasing the number of cores by 528 up to
a maximum of 4224. The computational mesh for the SCHOLAR large eddy
simulation to be presented in Chapter 7 is used, consisting of 36.8M hexahedral
elements. A reacting simulation using the assumed PDF combustion model with
second order upwind spatial and three step Runge-Kutta temporal discretisations
is conducted.
The normalised speed of a simulation is calculated as the time per iteration
for the 264 core calculation, over the time per iteration for the x core simulation
in question,
Normalised Speed =
titer,264
titer,x
(4.131)
where the iteration time is calculated as an averaged of the first 300 time steps.
Figure 4.16 shows the scaling performance of PULSAR on the HECToR XT6
machine before and after the Gemini interconnect upgrade. It can be seen that
with the upgraded communication between nodes, the scaling is improved. The
time per iteration was also improved, which reduced from 18.1s to 16.4s for the
264 core simulation.
106
Figure 4.16: Strong scaling of PULSAR on the HECToR XT6 cluster, before and
after interconnect upgrade
107
Chapter 5
Supersonic Non-Reacting Coaxial
Jet - RANS
After development of a new code, significant validation is required. Since PUL-
SAR is to be used for the simulation of combustion, validation of its mixing
prediction capabilities is of primary importance. This analysis is more straight-
forward for non-reacting flows, hence a non-reacting coaxial jet is simulated for
this purpose. Results are to be presented for the simulation of a supersonic coax-
ial helium jet, since this test case was developed with the validation of CFD
software in mind, particularly the validation of codes to be used in the simu-
lation of supersonic combustion. Therefore, as well as providing validation for
PULSAR, this test case can simultaneously be used to investigate the capabilities
of computational methods for the simulation of mixing in a scramjet combustor.
5.1 Description
The experiment in question [100] was conducted at NASA Langley Research
Center. As can be seen from Figure 5.1, the coaxial jet assembly consists of both
a central jet and a co-flow. Both jets have an exit Mach number of 1.8 (see Figure
5.2), but since the central jet consists mostly of helium (see Figure 5.3) its density
is much lower than that of the air co-flow, giving it a much higher speed of sound
and therefore a much higher exit velocity (see Figure 5.4).
108
18.26
1.59
60.47
10.50
10.00 15.87
Static pressure tap
76.20
25.40
19.84
15.88
41.91
152.27 Pref, CJ
Helium / 5% Oxygen
or Air Tt, CJ
48.01
246.39
Center jet 
pressure tap
Plenum
pressure tap
Pref, coflow
All dimensions in mm
Screens
Air
Tt, coflow
24.61
12.66
Figure 5.1: Coaxial jet assembly [100]
Figure 5.2: Mach number profile
This generates a compressible shear layer between the central and co-flow jets
[100], as would be found in a scramjet combustor due to fuel injection into a
supersonic flow. The use of helium for the central jet is to replicate the light gas
fuel hydrogen and to simulate the subsequent mixing process, avoiding the need
109
for the simulation of combustion when using this test case for code validation.
The remaining mass fraction of 0.2961 for the central jet is oxygen, which
corresponds to a mole fraction of just 0.05. This oxygen is added to the central
jet to allow the use of an oxygen flow-tagging technique for non-intrusive velocity
measurements.
Figure 5.3: Helium mass fraction profile
Figure 5.4: Velocity magnitude profile, m/s
5.1.1 Experimental Data
Both mean and rms values for the axial component of velocity were obtained
using the Raman excitation plus laser-induced electronic fluorescence (RELIEF)
technique. The flow was also probed to obtain pitot pressure, total temperature
and gas concentration measurements.
110
Since the concentration measurements were actually measurements for mole
fraction of the centre jet gas, they are effectively normalised measurements for he-
lium, which have been processed to provide both mole and mass fraction profiles.
For direct comparisons of experimental and computational data, the numerical
helium mass fractions, YHe,CFD, should be normalised by the centre jet helium
mass fraction:
YHe,norm =
YHe,CFD
0.7039
(5.1)
Pitot pressure, mole fraction and velocity measurements are available at 14
locations down the axis of the jet. However, the locations for velocity data do not
correspond to the locations at which pitot pressure and concentration measure-
ments were taken. The experimental profiles are in the radial direction across the
jet.
Since the flow is supersonic, some post-processing is required to convert the
static pressure data from numerical computations to pitot pressure measurements.
Figure 5.5 shows the resulting pitot probe bow shock, which can be assumed to
be a normal shock on its centreline.
Figure 5.5: Pitot tube shock schematic
The pitot pressure, ppitot, can therefore be calculated from the free-stream
static pressure, p, and Mach number, M :
ppitot = p0s = p
(
γ + 1
2
M2
) γ
γ−1
(
2γ
γ + 1
M2 − γ − 1
γ + 1
) 1
1−γ
(5.2)
where p0s is the post-shock stagnation pressure. However, in subsonic regions of
111
the flow the pitot pressure is equivalent to the free-stream stagnation pressure,
p0:
ppitot = p0 = p
(
1 +
γ − 1
2
M2
) γ
γ−1
(5.3)
5.1.2 Computational Mesh
Figure 5.6 shows the mesh used for the RANS simulations. ANSYS ICEM CFD
is used to generate a three-dimensional mesh using the multi-block technique,
employing O-grids to handle the coaxial nature of the problem. The mesh is
purely hexahedral and contains 3.8M cells.
Figure 5.6: Slice through RANS mesh at z = 0, near the jet exit
Due to the use of O-grids, boundary layer cells far from the centreline of
the jet can have very high aspect ratios. This problem is encountered for the
boundary layer cells in the co-flow jet and therefore the wall normal spacing was
relaxed in this region. For the co-flow surfaces the near wall spacing transitions
from 0.5mm at inlet to 2× 10−6m near the jet exit. However, for the central jet
a near wall spacing of 2× 10−6m is uniformly used. This means the y+ values in
the co-flow are mostly outside the viscous sublayer, as shown by Figure 5.7, with
Spalding’s equation used to obtain the wall shear stress. For the central jet the
112
y+ values are less than three everywhere, with the near wall cells lying inside the
linear region of the boundary layer.
A total of 24 cells are used across the 0.25mm central jet tip and the axial
spacing in the free-stream is equal to 2.5mm. The mesh downstream of the jet
extends 0.5m (50 central jet diameters) and 0.44m (44 central jet diameters) in
the axial and radial directions, respectively. The large radial extent ensures the
presence of the far-field boundary does not have an influence on the jet behaviour.
Figure 5.7: y+ values: Scaled to show distributions for the co-flow (top) and
central (bottom) jet surfaces
113
5.1.3 Boundary Conditions
Although the jets have an exit Mach number above unity, this is caused by flow
acceleration through a converging diverging nozzle. All three inlets to the domain
therefore have subsonic boundary conditions. The inlet stagnation properties and
flow composition data are presented in Table 5.1.
Centre Jet Co-flow Jet Ambient
Stagnation pressure, p0 (Pa) 628300 580000 101325
Stagnation temperature, T0 (K) 306.0 300.0 294.6
YHe 0.7039 0.0000 0.0000
YO2 0.2961 0.2300 0.2300
YN2 0.0000 0.7700 0.7700
Table 5.1: Coaxial jet inlet boundary conditions.
The inlet boundary conditions for the k and ω turbulence model variables are:
kin =
0.001µU∞
ρLref
(5.4)
ωin =
5U∞
Lref
(5.5)
where the reference length is set to 1mm, to be on the order of the tip thickness.
The outlet boundary analyses whether to apply subsonic or supersonic con-
ditions, since the outlet boundary in the outer shear layer between the co-flow
and ambient air will transition between these two regimes. For supersonic outlet
Mach numbers the boundary values are simply interpolated from inside the do-
main. For subsonic outlet Mach numbers an exit static pressure of 101325 Pa is
applied.
The no-slip condition is applied to all jet surfaces and an inviscid condition is
applied to the far-field boundary.
5.1.4 Computational Methods
The AUSM+UP scheme is used for the spatial discretisation, with second order
of accuracy achieved through use of linear interpolation with the Van Rosendale
114
gradient limiter. The three-step, third order TVD Runge Kutta method is used
for temporal integration, with a CFL number of 0.5. The Menter SST turbulence
model is employed with constant values for the turbulent Prandtl and Schmidt
numbers.
Since convergence acceleration methods, such as multi-grid, are not imple-
mented in PULSAR and since this shear layer jet is diffusion dominated, the
computational solution may still be changing between time steps even after resid-
uals appear to have leveled out. For this reason, monitoring points are used to
study the change in variables over time. Once variables such as the species mass
fraction, temperature, pressure and velocity have achieved a steady state, nu-
merical convergence can be declared. An example of this convergence monitoring
can be seen in Figure 5.8, which is the convergence history for the helium mass
fraction, down the centre-line of the jet at an axial location of 0.3m, for the fine
mesh to be presented in Section 5.2.5.
Figure 5.8: Convergence studied through data from monitor point
115
Simulations are carried out on 24 cores of the Stokes cluster. After initialisa-
tion with an inviscid solution, convergence is obtained in approximately 60 hours
after approximately 45,000 iterations.
5.2 Results
Due to the presence of an inner shear layer between the central and co-flow jets
and an outer shear layer between the co-flow and ambient air, this coaxial jet is
diffusion dominated. Since validation of the species mixing prediction capabilities
of PULSAR is a primary objective, the turbulent diffusion term in the species
transport equation (Equation 2.38) will play an important role in the simulations;
replicated here for clarity:
µt
Sct
∂Y˜α
∂xj
It is therefore sensible to first investigate the influence on computational results
of model parameters in this term.
5.2.1 Turbulent Schmidt Number
Since the species diffusion term is proportional to the inverse of the turbulent
Schmidt number, it is important to investigate how sensitive the computational
results are to changes in this model constant. Keeping the turbulent Prandtl
number fixed at 0.9 for now, but choosing two values for the turbulent Schmidt
number of 0.7 and 1.0, these sensitivities can be investigated.
It can be seen from Figure 5.9 that for both turbulent Schmidt number values
good agreement with experimental data for normalised helium mass fraction is
obtained near the jet exit, but at locations further downstream a gross over
prediction of the spreading rate of the central jet is obtained.
Increasing the turbulent Schmidt number to 1.5 provides computational re-
sults which are in better agreement with the experimental data, but it is generally
accepted that Schmidt numbers far above 1 are too high.
Figure 5.10 shows the influence of varying turbulent Schmidt number on the
116
Figure 5.9: Radial (r) profiles of normalised helium mass fraction at several axial
(x) locations, with varying Schmidt number and a Prandtl number of 0.9
117
Figure 5.10: Radial (r) profiles of normalised pitot pressure at several axial (x)
locations, with varying Schmidt number and a Prandtl number of 0.9
118
pitot pressure profiles, where the normalised pitot pressure is defined as:
ppitot,norm =
ppitot
580000
(5.6)
with 580000 Pa corresponding to the stagnation pressure at the co-flow inlet.
It can be seen that variations in the turbulent Schmidt number have a similar
influence on the pitot pressure profiles as for the species mass fraction. The results
can be explained by analysing the magnitude of the species diffusion coefficient,
where small turbulent Schmidt numbers correspond to high levels of diffusion
corresponding high losses in stagnation pressure.
However, none of the results presented are satisfactory, since it is a turbulent
Schmidt number of 1.0 which gives the best agreement with experimental data for
the pitot pressure but a corresponding poor agreement with experimental data
for the helium mass fraction.
5.2.2 Compressibility Correction
Looking back at the species turbulent diffusion term, it can be seen there is also
a dependence on the eddy viscosity, which can be scaled through application of
compressibility corrections to the turbulence model. Compressibility corrections
reduce the levels of eddy viscosity in regions of high turbulence Mach number, as
discussed in Section 4.6.4.
Figure 5.11 compares computational results using the Sarkar, Zeman and
Wilcox compressibility corrections, to experimental data. The turbulent Prandtl
and Schmidt numbers are held constant at 0.9 and 1.0, respectively.
It can be seen that all compressibility corrections improve the mixing predic-
tion when compared to results from the simulation without corrections applied,
but to varying degrees. The Wilcox compressibility correction provides the small-
est improvement due to the fact that it is only triggered in regions of the flow
which have a turbulence Mach number above 0.25. It can be seen from Figure
5.12 that, because of the high speed of sound due to the low density of helium,
only small regions of the central jet flow satisfy this criteria. Although excellent
agreement with experimental data is obtained at x/D = 12.1361, further down-
stream of this location the Wilcox compressibility correction is not activated for
119
Figure 5.11: Radial (r) profiles of normalised helium mass fraction at several axial
(x) locations, with different compressibility corrections and turbulent Schmidt
and Prandtl numbers of 1.0 and 0.9, respectively
120
the inner shear layer and hence no further improvements are visible. The Sarkar
compressibility correction gives the best agreement with experimental data, but
of course these results could all be scaled through alteration of the turbulence
Schmidt number.
Figure 5.12: Turbulence Mach number profile
Figure 5.13 shows that the introduction of a compressibility correction has a
significant impact on the pitot pressure predictions, having a detrimental effect
on the accuracy of the results obtained. Again this is due to the reduction in the
species diffusion coefficient causing a reduction in stagnation pressure loss but
also because the scaled eddy viscosity is also present in the turbulent diffusion
term for the energy equation:
µtcp
Prt
∂T˜
∂xj
with a reduction in the eddy viscosity also causing a reduction in the stagnation
energy, and hence pressure, loss.
5.2.3 Turbulent Prandtl Number
This turbulent energy diffusion term also introduces the influence of the turbu-
lent Prandtl number on the results obtained. Figure 5.13 also demonstrates the
significant influence of the variation of this model constant on the pitot pressure
profiles. Decreasing the turbulent Prandtl number from 0.9 to 0.5 provides a
121
Figure 5.13: Radial (r) profiles of normalised pitot pressure at several axial (x)
locations, showing the influence of the Sarkar compressibility correction and vari-
ation of the turbulent Prandtl number, with the turbulence Schmidt number held
constant at 1.0
122
better agreement with experimental data, when the turbulent Schmidt number
is held constant at 1.0 and the Sarkar compressibility correction is employed.
Since the turbulent Prandtl number does not arise in the species transport
equation, the influence of its alteration on the species mass fraction distribution
is minimal, as shown in Figure 5.14.
5.2.4 Velocity Distribution
The computational setup has been tuned to give good agreement with experimen-
tal data for composition and pitot pressure profiles. With the turbulent Schmidt
number and turbulent Prandtl number held constant at 1.0 and 0.5, respectively,
and the Sarkar compressibility correction employed to scale the eddy viscosity,
computational predictions for the mean and fluctuating velocity components can
be evaluated.
Experimental data is only available for the axial velocity component and it
can be seen in Figure 5.15 that reasonable agreement is obtained with computa-
tional results for the mean values, however there is some disagreement with the
magnitude of the central jet velocity at several locations down its axis.
The Boussinesq approximation in Equation 2.27 used to calculate the Reynolds
stress tensor can be employed to estimate the levels of fluctuating velocity:
√
u˜
′
iu
′
j =
√√√√ 23 ρ¯kδij − 2µt (S˜ij − 13 ∂u˜k∂xk δij)
ρ¯
(5.7)
as was done by Baurle and Edwards [15]. Another basic approximation can also
be obtained from the turbulence kinetic energy of Equation 2.28:
√
u˜
′
iu
′
i =
√
2k
3
(5.8)
where the 3 comes from summation over all three velocity directions.
The Boussinesq approximation is capable of providing direction dependent
rms velocities due to the presence of the strain tensor, S˜ij , whereas the rms
velocities calculated from the turbulence kinetic energy are isotropic due to its
scalar nature.
123
Figure 5.14: Radial (r) profiles of normalised helium mass fraction at several
axial (x) locations, with different turbulent Prandtl numbers and the turbulent
Schmidt equal to 1.0
124
Figure 5.15: Radial (r) profiles of mean axial velocity at several axial (x) locations.
125
Figure 5.16 compares the Boussinesq and turbulence kinetic energy approxi-
mations to the rms axial velocity with experimental data. It can be seen that both
approximations are capable of providing good predictions for the magnitude of
the peaks in the rms velocity data and the general profile is well captured. How-
ever, there is poor agreement with experimental data for the centre of the central
and co-flow jets. The agreement starts to improve for the central jet profile at an
axial location of x/D = 15.3 but remains poor in the co-flow.
In order to investigate why the Boussinesq and turbulence kinetic energy ap-
proaches give comparable predictions for the axial rms velocity, the magnitude of
the terms in the Boussinesq approximation can be compared, with the turbulence
kinetic energy, strain and divergence terms respectively defined as:
2
3
ρ¯k, 2µt
∂u˜
∂x
,
2
3
µt
(
∂u˜
∂x
+
∂v˜
∂y
+
∂w˜
∂z
)
(5.9)
Figure 5.17 clearly shows the turbulence kinetic energy term dominates the
Boussinesq approximation at all axial locations and explains why comparable
approximations to the rms axial velocity are found from the Boussinesq and
turbulence kinetic energy approaches. This confirms the importance of including
the turbulence kinetic energy term in both the Boussinesq approximation and
mean stagnation energy equation (2.29).
The reason for the poor agreement with experimental data in the central
and co-flow jet core regions can be explained by Figure 5.18 which shows the
absence of turbulence kinetic energy in these areas, since it is concentrated in
high velocity gradient shear layer regions. Once the central jet shear layers merge
at the centreline, better agreement with experimental data for the core region is
obtained. This is due to the Boussinesq approximation requiring the presence of
velocity gradients to become active.
In order to see if the directional strain term plays a more significant role in the
Boussinesq approximation for the transverse rms velocities, the corresponding val-
ues are compared to the turbulence kinetic energy approximation in Figure 5.19.
It can be seen that both approaches are again in excellent agreement, confirming
the turbulence kinetic energy term in Equation 5.7 dominates in all directions.
The Boussinesq approximation therefore produces isotropic Reynolds stresses for
126
Figure 5.16: Radial (r) profiles of the axial rms velocity, comparing the Boussinesq
and turbulence kinetic energy approximations to experimental data at several
axial (x) locations.
127
Figure 5.17: Radial (r) profiles of the terms involved in the Boussinesq approxi-
mation to the Reynolds stress.
128
Figure 5.18: Turbulence kinetic energy profile
the test case in question; unfortunately no experimental data is available for rms
velocities in the y and z directions to confirm whether this behaviour is correct.
5.2.5 Mesh Convergence
In order to check mesh convergence for the RANS results presented, a simulation
is conducted on a finer mesh, which is discussed in more detail in Chapter 6. The
fine mesh contains 22.2M hexahedral cells and the simulation was run on 128
cores of the Stokes cluster.
Figures 5.20 and 5.21 compare normalised helium mass fraction and pitot
pressure profiles, where ’coarse’ corresponds to the mesh presented in Section
5.1.2 and ’fine’ for the LES mesh of Chapter 6. The turbulent Prandtl and
Schmidt numbers are held constant at 0.9 and 1.0, respectively, and the Sarkar
compressibility correction is employed. It can be seen that minimal differences
are present between the two sets of data, confirming mesh convergence for the
RANS results obtained.
5.3 Summary
The sensitivities of RANS to choices in model constant values has been presented.
Although it appears these model constants can be tuned to give accurate results,
different model constants would be required for the simulation of different test
cases, since they scale the eddy viscosity which is used to model the problem
dependent large scales of the flow. It is therefore very difficult to obtain accurate
129
results using RANS methods in a research environment, where the experimental
data required to tune model constants may not be available. It is particularly
difficult to use these methods for combustion research due to sensitivities of the
turbulent diffusion of reacting species to the turbulent Schmidt number.
However, it should be stated that although computational accuracy appears
sensitive to the choice of values for the model constants, RANS methods are
capable of predicting the correct trends in the results obtained and can therefore
still have an important role to play in early stage design activities.
In order to avoid the problems presented with constant turbulent Schmidt
and Prandtl numbers, models attempting to vary these parameters in space and
time have been attempted [137]. However, these methods can be expensive to use
through the introduction of additional transport equations and still suffer from
modelling constraints. It is more desirable to employ a computational method
which is less sensitive to the choice of such parameters, through reducing the
modelling effort and attempting to directly capture a higher level of the physics
involved; for example LES.
130
Figure 5.19: Radial (r) profiles of the transverse rms velocities, comparing the
Boussinesq (red and blue) and turbulence kinetic energy approximations at sev-
eral axial (x) locations.
131
Figure 5.20: Helium mass fraction profiles confirming mesh convergence
132
Figure 5.21: Pitot pressure profiles confirming mesh convergence
133
Chapter 6
Supersonic Non-Reacting Coaxial
Jet - LES
Since LES directly captures the problem dependent large scales of the flow and
applies modelling solely to the more universal small scales, the influence of empir-
ical constants and their tuning requirements should be reduced, when compared
to RANS. Due to a smaller range of the turbulence spectrum being accounted for
through the modelling process, the levels of modelled viscosity are lower, mean-
ing the turbulent transport terms in the Favre filtered system of equations play
a less dominant role in computations. Computational results should therefore
prove less sensitive to variations in the turbulence Prandtl and Schmidt numbers,
improving over the deficiencies of the RANS approach. LES should also generate
more physically meaningful results and provide the capability of extracting mean
and fluctuating statistical data.
In order to investigate this hypothesis, large eddy simulations of the non-
reacting coaxial jet introduced in Chapter 5, are presented. Baurle and Edwards
[15] compared results from RANS and LES for the test case in question, but
did not investigate the influence of the turbulent Prandtl and Schmidt numbers
on the LES results. Poor agreement was obtained with experimental data for
the LES ensemble averaged data, possibly due to the low levels of SGS viscosity
employed.
134
6.1 Delayed Detached Eddy Simulation
Due to the high Reynolds numbers encountered in high speed flows, the delayed
detached eddy simulation (DDES) hybrid RANS-LES method is used to reduce
near wall computational costs. The Menter SST turbulence model is used in both
the near wall RANS region and as the sub-grid scale model in the free-stream.
6.1.1 Computational Mesh
In order to further minimise the computational cost, the extent of the compu-
tational domain in the axial direction is reduced from that used in the RANS
simulations of Chapter 5. The distance from the central jet tip to the domain
outlet is now 0.27m, which is large enough to contain all experimental data points.
The y+ distributions in Figure 5.7 can be used to design an efficient but
accurate near wall mesh. In order for these time accurate simulations to be
affordable the near wall spacing is slightly relaxed, since it is this dimension
which governs the time step to be used. The central jet near wall cell size is fixed
at 6.0 × 10−6m for its entire length. Again, in order to avoid the generation of
very high aspect ratio cells on the curved surfaces of the co-flow jet (due to the
o-grid blocking mesh employed) the near wall cell size transitions from the inlet
to the jet exit plane. The near wall spacing at the inlet is set to 0.5mm which
linearly transitions to 6.0×10−6m and 1.0×10−5m half way down the jet length,
for the inner and outer surfaces of the co-flow, respectively. They then linearly
transition again to 3.0× 10−6m and 6.0× 10−6m, respectively, before the jet exit
plane. It can be seen in Figure 6.1 that a y+ of approximately four exists over the
majority of the viscous surfaces, lying within the linear viscous sublayer. This
provides sufficient resolution whilst allowing a computationally efficient time step
to be used. A time step of 3.075 × 10−9s is required to achieve computational
stability.
The axial mesh spacing is set to 0.25mm in order to try and obtain isotropic
cells in the central and co-flow jet core regions, as shown in Figure 6.2. Since
turbulent structures are three dimensional in nature it is advantageous to use
isotropic cells and simplifies the implicit filtering operation and choice of filter
135
width calculation. However, due to the meshing method employed, high aspect
ratio cells can not be avoided in the shear layer regions and their impact on the
computational results will be investigated. A purely hexahedral o-grid multi-
block mesh is employed, resulting in 22.22M cells, with 12 cells used to resolved
the central jet tip.
Figure 6.1: y+ distribution for coaxial jet DDES
6.1.2 Boundary Conditions
The same steady state boundary conditions used in the RANS simulations of
the previous Chapter are employed here. The simulations are initialised with a
steady state solution, mapped from the coarser RANS mesh onto the fine mesh
described above.
6.1.3 Computational Methods
Again, the second order AUSM+UP scheme with the Van Rosendale gradient
limiter is used for the spatial discretisation and the third order three step TVD
Runge-Kutta scheme for temporal integration.
136
Figure 6.2: Fine DDES mesh, showing high aspect ratio cells in the shear layer
regions and isotropic cells elsewhere.
6.1.4 Results
Figure 6.3 shows an instantaneous image from a simulation employing the DDES
method. It can be seen that there is a significant delay before the inner shear
layer transitions to turbulent behaviour. Through comparison of this image with
the experimental schlieren in Figure 6.4 it can be seen that the jet is significantly
more turbulent in reality.
Figure 6.3: Instantaneous helium mass fraction distribution for coaxial jet DDES
137
Because the jet is lacking in transient behaviour, it is easy to see that the en-
semble averages would be in poor agreement with experimental data and therefore
need not be calculated. In fact, at the axial locations of x/D = 12.1361 and x/D
= 15.0825, due to the lack of turbulent motion, the instantaneous profiles can be
compared with experimental results. It can be seen from Figure 6.5 that a signif-
icant under-prediction of the spreading rate of the jet exists, suggesting enhanced
turbulent mixing is required closer to the nozzle exit.
Figure 6.4: Experimental schlieren image [100], with angled shroud removed
Figure 6.5: Radial (r) profiles of normalised helium mass fraction at two axial (x)
locations for the DDES
138
This behaviour was also encountered by Baurle and Edwards [15] who needed
to use a very low value for the Smagorinsky constant in their SGS model in order
to encourage jet-breakup. However, this resulted in an over-prediction of the
spreading rate of the jet and hence another solution to this problem is required.
It is possible the levels of numerical dissipation provided by the second order
upwind scheme contribute to this result. However, in order to try and limit the
levels of subgrid viscosity in the simulation, the filter width in Equation 2.57
employing the cube root of the cell volume, is used. Due to the presence of high
aspect ratio cells in the shear layer regions, the use of this filter width should
provide a significantly reduced subgrid viscosity compared to Equation 2.58.
Since the sub-grid model employed aims to mimic the purely dissipative
Smagorinsky SGS model, the absence of backscatter capabilities could play a
role in the lack of generation of shear layer instabilities. It is possible movement
of energy towards the large scales could provide the shear layer perturbations
required to initiate jet-breakup.
The high axial velocity of the central jet severely exacerbates this delayed
transition problem. Any perturbations to the shear layer near the jet exit are
rapidly convected downstream, travelling a significant distance before being am-
plified to cause the required shear layer instabilities.
Since the near wall regions are modelled using the RANS approach, there
is no resolved turbulent content in the boundary layers. Since these boundary
layers separate to form the shear layers of the flow, they are unable to provide
the perturbations required to initiate jet-breakup. This is not a new problem;
Keating and Piomelli [63] employed stochastic forcing at the boundary between
the RANS and LES regions in order to increase the levels of resolved stresses at
the interface. Alternatively, it has been suggested [15] that the introduction of
turbulence structures at the inlet may help the jet transition. The influence of the
addition of coherent structures to the computational domain will be investigated.
6.2 Turbulence Inflow
Turbulence data is often provided at domain inlets; however, this is not feasible
for the jet in question due to the wide range of velocities encountered. Although
139
high velocities exist at the nozzle exits due to the supersonic conditions there, the
velocities at the domain inlets are subsonic and below 100ms−1 (see Figures 5.2
and 5.4). For temporal accuracy a global time step is used, meaning it could take
hundreds of thousands of iterations just for the turbulence information provided
at the inlet to reach the jet exit plane and the coherent structures may even
have been significantly dissipated by this time. The magnitude of the velocity
fluctuations at the inlet may also be required to be on the order of 50% of mean
inlet velocities in order to obtain turbulent intensity values of a few percent at jet
exit, possibly leading to simulation stability issues. For these reasons, a method
which enables this turbulence information to be provided closer to the jet exits
is implemented in PULSAR and this approach is employed here.
The definition of the input turbulence requires knowledge of the experimental
turbulence intensity and length scales. Since this data is not available, estimates
must be made.
6.2.1 Computational Approach
The position of the turbulence inlet plane is somewhat arbitrarily chosen to be
1.15cm (1.15 central jet exit diameters) upstream of the central jet tip, as indi-
cated in Figure 6.6. Turbulent fluctuations are imposed by modifying numerical
fluxes through the corresponding cell faces at this location, where the fluctuating
data is generated prior to the simulation.
In order to generate this fluctuating data a slice is taken through the mesh at
the chosen location, as demonstrated in Figure 6.6. This slice is then extruded in
the axial direction (perpendicular to its plane), to generate two cylindrical shaped
domains. These domains are meshed with hexahedral cells, and the mesh used
for the plane parallel to the extracted slice must correspond to the mesh used for
the jet geometry at this location, but an evenly space distribution is used in the
axial direction, as shown by Figure 6.7.
Approximately 400,000 cells are used for the central jet domain and 1.6M for
the co-flow. 200 layers of 0.5mm cells are employed in the axial direction, giving
a total domain length of 0.1m. This corresponds to 10 central jet diameters and
is deemed sufficient to avoid problems with repetition of inlet fluctuating data.
140
Figure 6.6: Location of turbulence inlet plane and corresponding geometry cross-
section
Figure 6.7: Mesh used to generate turbulence inlet data. Half of the mesh pro-
vided to show internal characteristics.
141
However, this corresponds to only four co-flow length scales, with the length
scale given by the distance between the outer and inner co-flow surfaces. Since
the interaction of the central and co-flow velocity fluctuations at the inner shear
layer is not periodic this is deemed acceptable. The outer co-flow domain length
is also restricted by memory considerations imposed by the cell count.
The extruded domains are filled with coherent structures using the method
of Kempf et al. [64] described in Section 4.10. In order to generate at least
three large eddy structures across a jet width the maximum length scale, Lmax,
is chosen to be 3mm, which is less than a third of the central jet exit diameter.
A variable length scale formulation is employed to handle the presence of viscous
walls, with parameter αw in Equation 4.128 set to 0.4, as was used in [138]. Figure
6.8 shows the resulting turbulence profile, employing overall isotropic properties
with Rii set to 20m
2s−2.
Figure 6.8: Generated inlet turbulence velocity data, in m/s. Half of the domain
provided to show the internal profile.
142
Taylor’s hypothesis [139] is used to impose planes of turbulent fluctuating
data on the numerical fluxes at the mesh location described. From here on, any
reference to inlet turbulence corresponds to the turbulent fluctuations imposed
towards the jet exit.
6.2.2 Results
Figure 6.9 shows the turbulent structures convecting into the domain at the
chosen location and their interaction with both the inner and outer shear layer
can clearly be seen. Use of the z-velocity component (perpendicular to the page)
to visualise the turbulent structures is somewhat unconventional, but enables
their clear visualisation and shear layer interaction.
Figure 6.9: z-velocity component in m/s, showing injected coherent structures
and shear layer interactions.
Figure 6.10 is an instantaneous image showing the resulting Helium mass frac-
tion distribution. Significant jet breakup is achieved, removing the problematic
transition delay from the simulation.
143
Figure 6.11 shows the resulting turbulent nature of the jet using positive Q
iso-surfaces coloured by helium mass fraction, where Q is given by:
Q =
1
2
(ΩijΩij − SijSij) (6.1)
with
Ωij =
1
2
(
∂ui
∂xj
− ∂uj
∂xi
)
(6.2)
Positive values of Q help to identify the low-pressure tubes which are associated
with coherent vortices [140].
Figure 6.10: Instantaneous helium mass fraction profile for the coaxial jet DDES
simulation with imposed turbulent fluctuations.
It can clearly be seen that the central jet has a fine turbulent structure whilst
the outer shear layer is made up of larger coherent vortices.
In order to study the influence of the turbulent Prandtl and Schmidt numbers
on LES results, three simulations are conducted employing the combination of
parameters in Table 6.1. The isotropic turbulence profile in Figure 6.8 is used
for the inlet fluctuations. Since no information on the experimental levels of
turbulence is available, the value of 20m2s−2 is chosen for the applied Reynolds
stresses, Rii, to provide peak inlet fluctuations on the order of 1-2% of the central
jet exit velocity.
144
Figure 6.11: Positive Q iso-surfaces coloured by helium mass fraction.
Prt Sct
Simulation 1 0.9 1.0
Simulation 2 0.9 0.5
Simulation 3 0.5 1.0
Table 6.1: Turbulent Prandtl and Schmidt numbers for each simulation con-
ducted.
The centre-line velocity profile from the RANS simulation employing the
Sarkar compressibility correction with turbulent Prandtl and Schmidt numbers of
0.5 and 1.0, respectively, is used to define a flow-through time for the LES com-
putational domain from the central-jet tip to the outlet. One flow-through time
turns out to be 2.762 × 10−4s, which corresponds to 89821 iterations. This can
be used to estimate the number of iterations required to flush initial transients
out of the system and then the number of iterations required to collect statistical
data.
In order to compare computational results to experimental data the mean
and rms values must be calculated. Since the co-flow exit velocity is significantly
below that of the central jet, multiple flow through times are needed to gather
the required statistics. Data collection is initiated after 225,000 iterations, cor-
responding to approximately 2.5 flow-through times, when initial transients have
convected out of the domain. Ensemble averaged data are then collected over
765,000 iterations, corresponding to approximately 8.5 flow-through times, which
are then employed to calculate the rms data over the period of another 8.5 flow-
through times. Mean and rms values are therefore evaluated after 11 and 19.5
145
flow-through times, respectively. Statistical data are calculated for the velocity
vector components and the pitot pressure, stagnation temperature and species
mass fraction variables.
This simulation is conducted on 1440 cores of the HECToR cluster with 45,000
iterations achieved in just under 12 hours of run time. The total run time for
each case is therefore approximately 19 days in real time or over 650,000 CPU
hours. With this corresponding to over 74 years of run time for a serial code,
parallel capabilities are obviously essential for such research.
Similarly to the study of solution convergence for a RANS simulation, the
convergence of statistical data is observed through monitor points in the domain.
Figure 6.12 shows both the instantaneous profile and ensemble average conver-
gence history for the helium mass fraction, from 225,000 to 990,000 iterations on
the centreline of the jet at an axial location of 0.27m.
Figure 6.12: Helium mass fraction time history (left) and mean convergence
(right) on the centreline at x=0.27m.
The turbulence energy spectrum from Simulation 1 is shown in Figure 6.13,
evaluated using the time history of y-component of velocity over the iteration
range from 225,000 to 1,755,000 on the centreline of the jet at an axial location
of 0.27m. It can be seen that the mesh resolution is adequate, since evidence
of the inertial range of the turbulence spectrum is visible. It appears that the
inertial range is in agreement with Kolmogorov’s k−5/3 hypothesis, whilst the
compressibility corrected gradient of k−8/3 proposed by Ingenito and Bruno [19]
appears to only agree after initiation of fall-off in the spectrum.
146
Figure 6.13: Turbulence energy spectrum for coaxial jet DDES
6.2.2.1 Helium Mass Fraction
It can be seen from Figure 6.14 that the LES results are indeed insensitive to
the choice of values for the turbulent Prandtl and Schmidt numbers, significantly
improving over the RANS method in this respect.
Although computational results for the helium mass fraction distribution are
in good agreement with experimental data near to the jet exit, poor agreement
is obtained further downstream. This could be due to the assumptions made for
the intensity and length scales of the input turbulence fluctuations. It may also
be due to the numerical diffusion supplied by the second order upwind scheme,
adding to the spreading rate of the jet at downstream locations.
The over prediction obtained for the spreading rate can be further highlighted
by the comparison in Figure 6.15 of the helium mass fraction distributions be-
tween the ensemble averaged data from the LES and from the RANS simula-
tion employing the Sarkar compressibility correction with turbulent Prandtl and
Schmidt number of 1.0 and 0.5, respectively, which was in good agreement with
experimental data. The shorter core region and premature spreading of the jet
in the LES simulation is clearly visible.
147
Figure 6.14: Radial (r) profiles of mean normalised helium mass fraction at several
axial (x) locations, with varying Prandtl and Schmidt numbers
148
Figure 6.15: Comparison of ensemble averaged DDES (top) and RANS (bottom)
helium mass fraction profiles.
6.2.2.2 Pitot Pressure
Similar results are obtained for the ensemble averaged pitot pressure profiles,
where it can again be seen from Figure 6.16 that computational results are in-
dependent of the turbulent Prandtl and Schmidt numbers employed. However,
the numerical results are again in poor agreement with the experimental data at
downstream locations.
The poor agreement is particularly evident for the outer shear layer, with
larger than expected losses in stagnation pressure and a larger spreading rate
than found in the experimental data. This may be due to the large scale eddies
in the outer shear layer causing more widespread mixing than experienced in
practice. As can be seen in Figure 6.17, these large scale structures also have a
significant influence on the inner shear layer. The unsteady behaviour of the outer
shear layer is significant enough to generate shock waves which interact with the
helium jet. It can be seen that significant distortion of the central jet results from
the presence of large turbulent structures and shock wave interactions and hence
an increased spreading rate for the helium mass fraction profile is obtained.
149
Figure 6.16: Radial (r) profiles of mean normalised pitot pressure at several axial
(x) locations, with varying Prandtl and Schmidt numbers.
150
Figure 6.17: Numerical schlieren showing interactions between outer and inner
shear layers.
Such severe unsteady behaviour for the outer shear layer and subsequent shock
generation is not evident in the experimental schlieren of Figure 6.4 and hence
this may play a significant role in the discrepancy between experimental and
computational data. The outer shear layer breakdown is caused by the intense
fluctuations in the inlet turbulence profile at the outer surface of the co-flow,
as can be seen in Figure 6.8. These high intensities are caused by the Reynolds
stress and length scale parameters chosen but also the volume normalisation of the
turbulence initialisation method and gradient reconstruction approach employed.
This is investigated in more detail in Section 6.4.
Lower than expected values for the pitot pressure are also evident in the inner
shear layer for several diameters downstream of the exit plane. The shear layer is
formed by the boundary layers separating from central jet and co-flow surfaces,
which are solved using the RANS technique. Hence there is a lack of resolved
turbulent content in the shear layer close to the exit plane, which if present would
cause mixing with the high velocity fluid, increasing the stagnation pressure in
this region.
6.2.2.3 Velocity
Experimental data for the mean and rms axial velocity are available, where the
rms values are particularly valuable for the validation of the higher order statistics
151
available from LES.
Figures 6.18 and 6.19 confirm the independence of both mean and fluctuat-
ing axial velocity data from the values employed for the turbulent Prandtl and
Schmidt numbers.
The mean axial velocity profiles follow a similar pattern to the helium mass
fraction and pitot pressure, with poor agreement with experimental data obtained
at downstream locations. Again this may be due to the large scale unsteadiness
of the outer shear layer and the parameters used for the generation of inlet tur-
bulence. Similarly to the pitot pressure profile, a region of lower than expected
velocity is present in the inner shear layer for several diameters downstream of
the jet exit, due to the lack of mixing.
Although the computational axial rms velocity results prove insensitive to
model constant values, they over predict the turbulence intensity levels by ap-
proximately 50%. This disagreement with experimental data may again be due
to the assumptions made for the imposed turbulence profile and the large outer
shear layer eddies may impose significant velocity fluctuations on the central jet
flow. The initial low turbulence intensities confirm the lack of mixing just down-
stream of the central jet tip. Although the magnitude of the fluctuations is poorly
predicted, the correct profiles are obtained.
Figure 6.20 compares all three components of fluctuating velocity from Sim-
ulation 2, showing that the y and z components have a magnitude significantly
below the axial rms velocity values. This is contradictory to the RANS results
presented in Section 5.2.4, where the fluctuating velocity data in all three direc-
tions was found to be the same, since the Boussinesq approximation used in the
modelling is dominated by the scalar turbulence kinetic energy. This implies that
the Boussinesq approximation may not be suitable for modelling of the Reynolds
stresses for this test case, although it was shown to produce good agreement with
experimental data for the axial rms velocity in regions of high shear. This state-
ment can not be confirmed due to a lack of experimental data for the y and z rms
velocities, but is consistent with what was found by Baurle and Edwards [15].
Profiles of the inlet turbulence velocity components can be seen in Figure 6.21,
confirming that an isotropic field is used, demonstrating that the disparity in rms
velocity components is not caused by anisotropy in the inlet turbulence profile.
152
Figure 6.18: Radial (r) profiles of mean axial velocity at several axial (x) locations,
with varying Prandtl and Schmidt numbers.
153
Figure 6.19: Radial (r) profiles of rms axial velocity at several axial (x) locations,
with varying Prandtl and Schmidt.
154
Figure 6.20: Radial (r) profiles of rms velocity components at several axial (x)
locations, from Simulation 2.
155
Figure 6.21: x (left), y (middle) and z (right) velocity profiles on a sample slice
of the turbulence inlet data, in m/s.
6.2.2.4 Statistical Convergence
In order to further demonstrate convergence of the ensemble averaged data, com-
putational results after 11 and 19.5 flow-through times are compared. The en-
semble averaged values obtained at the end of the first cycle of 765,000 iterations
are frozen and used to calculate the rms data over the next 765,000 iterations,
but are also continually updated over this time and stored in a separate variable.
It can be seen from Figures 6.22 and 6.23 that there is little difference between
the two sets of ensemble averaged data for the species mass fraction and axial
velocity and so confirmation of statistical convergence for the values used in rms
calculations is confirmed.
156
Figure 6.22: Radial (r) profiles of mean normalised helium mass fraction at several
axial (x) locations, after 11 and 19.5 flow-through-times, for Simulation 1.
157
Figure 6.23: Radial (r) profiles of mean axial velocity at several axial (x) locations,
after 11 and 19.5 flow-through-times, for Simulation 1.
158
Figure 6.24: Radial (r) profiles of mean normalised pitot pressure at several axial
(x) locations, after 11 and 19.5 flow-through-times, for Simulation 1.
159
It is a slightly different story for the pitot pressure profiles shown in Figure
6.24, however. Although convergence can be declared for the mean pitot pressure
values in the central jet, there are distinct variations in the co-flow and outer
shear layer, where the agreement with experimental data deteriorates over time.
This is due to the large scale structures which cause an over prediction of the
outer shear layer spreading. Since the outer shear layer separates the co-flow
and ambient air, subsonic flow velocities are present and hence a larger number
of iterations are required to fully develop this region and a larger number of
iterations are required to obtain statistical convergence.
The downstream locations also confirm the severity of the outer and inner
shear layer interactions, where they appear to merge; contradictory to experi-
mental data.
Convergence of the axial rms velocity is displayed in Figure 6.25, which shows
the time history for this value on the centreline of the jet at an axial location of
0.27m.
Figure 6.25: Axial rms velocity convergence
6.3 Parameter Sensitivities
Since the poor agreement between experimental and computational results at
downstream locations may be due to the assumptions made for the length scales
and Reynolds stresses of the imposed turbulence fluctuations, it is important
160
to investigate the sensitivities to these parameters. The choice of filter width
calculation and mesh resolution may also have a significant influence on the com-
putational results obtained.
6.3.1 Computational Mesh
Such an investigation requires multiple simulations to be conducted, varying the
parameters in question between runs. In order to reduce the computational cost
of such a study the computational domain is reduced in the axial direction from
that presented in Section 6.1.1. The location to which the domain is truncated
is chosen to ensure that important data points still remain within the bounds
of the computational mesh. Recalling the results in Section 6.2, whatever com-
putational setup is employed, reasonable agreement between experimental and
computational data is obtained close to the nozzle exits. However, further down-
stream, significant differences arise and it is therefore important to evaluate the
ensemble averaged data at these locations. The computational mesh is therefore
terminated at an axial location of 0.158m, ensuring the data point at x/D =
15.0825 is still available.
An identical nozzle mesh is used to that described in Section 6.1.1, but the
axial resolution in the free-stream is relaxed from 0.25mm, to 0.5mm. The result-
ing multi-block mesh consists of 9.35M hexahedral cells, which is still split into
1440 partitions and run on the HECToR cluster, with 90,000 iterations achieved
in approximately 11 hours of run time. Since the near wall mesh resolution is
unchanged, the time step remains at 3.075× 10−9s. Because the computational
domain has been reduced in the axial direction, one flow-through time now turns
out to be 1.437×10−4s, which corresponds to 46731 iterations. In order to further
reduce the computational cost, the collection of ensemble averaged data is initi-
ated after 180,000 iterations, corresponding to approximately four flow-through
times, and continues for 450,000 iterations, corresponding to approximately ten
flow-through times. Due to the computational cost associated with additionally
evaluating the rms data, only the influence of parameters on the mean values is
to be studied in detail here.
161
6.3.2 Filter Width
The cube root of the cell volume filter width of Equation 2.57 is used in Sections
6.1 and 6.2 in order to reduce the level of subgrid viscosity in the simulation, in an
attempt to compensate for use of an upwind method for the spatial discretisation
and to encourage the growth of shear layer instabilities. However, it is possible
the maximum length filter width of Equation 2.58 could reduce the high levels
of velocity variance encountered in Section 6.2.2.3. It is therefore important
to investigate the influence of the filter width calculation on the computational
results obtained.
Two simulations employing the filter widths discussed are conducted and the
turbulence inlet Reynolds stress, Rii, is set to 50m
2s−2 and an Lmax of 3mm and
a near wall scaling coefficient, αw, of 0.4, are used, as before.
Due to the minimal influence of variations in computational setup on nu-
merical results at the first two axial locations previously presented, they will
be ignored from now on with a concentration on behaviour at locations further
downstream.
Figure 6.26 shows that the choice of filter width can have a significant impact
on the computational results obtained, with the helium mass fraction profiles
showing the maximum length filter width significantly reduces the spreading rate
of the jet, for the given turbulence inlet conditions. From Figure 6.27 it can be
seen that this is because of the higher levels of subgrid viscosity generated, due to
the high aspect ratios cells present in the shear layer regions. However, this results
in a very dissipative simulation, with the instantaneous helium mass fraction
profile shown in Figure 6.28. It can be seen the high levels of subgrid viscosity
kills the smallest resolved scales in the flow, when compared to a simulation
employing the cube root filter width. It is therefore correct to employ the cube
root of volume filter width, in order to obtain the jet behaviour expected in
practice. Simulations employing the maximum length filter width would most
likely be more sensitive to the values chosen for the turbulent Prandtl and Schmidt
numbers, since the increased levels of subgrid viscosity would increase the role
played by the turbulent diffusion terms in the filtered system of equations.
162
Figure 6.26: Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, comparing the cube root
of volume and cell maximum length filter width calculations.
163
Figure 6.27: Instantaneous SGS viscosity profiles after 630,000 iterations, com-
paring the cell maximum length (top) and cube root of volume (bottom) filter
width calculations.
It can also be seen from Figure 6.26 that the more dissipative filter width
provides a sharper resolution of the co-flow pitot pressure profile. This is due
to the high levels of subgrid viscosity in the outer shear layer killing all shear
layer instabilities, leading to a constant smooth profile as shown in the numerical
schlieren of Figure 6.29. However, the less dissipative filter width allows signif-
icant turbulent behaviour in the outer shear layer, which leads to a significant
spreading of the ensemble averaged pitot pressure profile.
164
Figure 6.28: Instantaneous helium mass fraction profiles after 630,000 iterations,
comparing the cell maximum length (top) and cube root of volume (bottom) filter
width calculations.
Although it has been presented that velocity variance values from a simulation
employing the cube root filter width are too high, it can be seen from Figure 6.4
that unsteadiness in the outer shear layer is present in practice and hence the
level of subgrid viscosity provided by the maximum length filter width is too high.
The high levels produced suggest the circumferential cell dimension dominates the
filter width in this region, due to the o-grid meshing method employed.
165
Figure 6.29: Instantaneous numerical schlieren images after 630,000 iterations,
comparing the cell maximum length (top) and cube root of volume (bottom) filter
width calculations.
Since use of the maximum length filter width is a strict requirement when
using the DES method, it is also important to ensure use of the cube root filter
width for DDES does not present any problems for the near wall modelling. If
the cube root filter width is used in DES, transition to the LES regime incorrectly
occurs deep inside the boundary layer, close to the wall. However, as can be seen
in Figure 6.30, since the DDES method uses details of the flow variables to handle
the RANS to LES transition there is no detrimental impact of employing the less
dissipative cube root filter width in these simulations.
166
Figure 6.30: RANS-LES blending function for the DDES method, with blue and
red corresponding to the RANS and LES regimes, respectively. Maximum length
(left) and cube root of volume (right) filter widths are shown at the tip of the
central jet nozzle.
In order to reduce the influence of the filter width calculation on the simula-
tion, the computational cells in the LES regime should be made as isotropic as
possible. Since the choice of filter width calculation alone can have a significant
impact on the computational results, no effort is made to tune the CDES constant
employed in the subgrid-scale model.
6.3.3 Reynolds Stresses
The diagonal terms of the Reynolds stress tensor, Rii, are set to 20m
2s−2 for
the input turbulence in the DDES simulation presented in Section 6.2, while the
off-diagonal Reynolds stresses are set to zero. It is important to investigate the
influence of the Reynolds stress magnitude on the computational results obtained.
Figure 6.31 displays the influence on the helium mass fraction and pitot
pressure distributions by comparing computational results using values for the
Reynolds stress magnitude of 20m2s−2, 50m2s−2 and 100m2s−2. It can be seen
that the higher the Reynolds stress values, and hence the higher the magnitude
of the velocity fluctuations and corresponding turbulence intensity, the more de-
structive the input turbulence, leading to a higher spreading rate for the central
jet profiles.
167
Figure 6.31: Radial (r) profiles of mean normalised helium mass fraction (left) and
pitot pressure (right) at several axial (x) locations, with varying inlet turbulence
Reynolds stress, in m2s−2.
168
However, the Reynolds stress magnitude appears to have a minimal impact on
the co-flow and outer shear layer pitot pressure profiles. This may be due to the
longer periods required to ensemble average this region of the flow, as discussed
in Section 6.2.2.4.
Figure 6.32: Instantaneous numerical schlieren images after 810,000 iterations,
comparing inlet turbulence Reynolds stresses of 20m2s−2 (top) and 100m2s−2
(bottom).
The schlieren images in Figure 6.32 show the variation in shear layer structure
with an increase in Reynolds stress. It can be seen that the severity of shear
layer turbulence increases with the inlet turbulence intensity, confirming this as
the reason for an increased spreading rate of the central jet profile. The schlieren
images also confirm an influence of the Reynolds stress magnitude on the outer
shear layer of the flow.
169
6.3.4 Near Wall Scales
The value employed for the near wall eddy scaling, αw, may also impact the
interaction of inlet turbulence with the jet shear layers. The region in which
this scaling takes effect is controlled by the maximum eddy length scale, Lmax,
which is fixed at 3mm. Through manipulation of Equation 4.128 it can be shown
that the near wall scaling defines the turbulence length scale in the region where
αwyw < 3mm. Since the diameter of the central jet inlet domain is approximately
9mm, the value of αw would need to be greater than 0.67 for the constant Lmax
length scale to take effect anywhere within it.
The influence of the turbulence length scale is studied by applying values of
0.2, 0.4 and 0.6 to αw for the generation of the imposed turbulence data. The
impact on the fluctuating velocities of varying this parameter is presented in
Figure 6.33, where it can be seen a reduced value leads to a distribution of finer
scales near the wall.
αw = 0.2 αw = 0.4 αw = 0.6
Figure 6.33: Comparing near wall scaling coefficients, αw, of 0.2 (left), 0.4 (mid-
dle) and 0.6 (right) on a sample slice of the turbulence inlet data. Velocity
magnitude shown, with scale 0 to 10 m/s.
170
Figure 6.34: Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, with varying near wall
scaling coefficient.
171
Figure 6.34 compares the resulting helium mass fraction and pitot pressure
profiles, where it can be seen that the smaller near wall length scales lead to
reduced spreading rate of the central jet. Since it is the near wall eddies which
will interact with the jet shear layers, and since it is the larger scales which are
the more energetic (as defined by the turbulence energy spectrum), the smaller
the eddies the smaller the resulting perturbation applied and hence the smaller
the resulting instability. Better agreement with experimental data is obtained
for the lower near wall scaling values. Again, little influence is apparent for the
outer shear layer pitot pressure profile.
Figure 6.35: Instantaneous numerical schlieren images after 810,000 iterations,
comparing inlet turbulence near wall scaling coefficients of 0.2 (top) and 0.6
(bottom).
172
This is supported by the numerical schlieren images in Figure 6.35 which show
the larger the near wall length scales the higher the intensity of the shear layer
fluctuations, increasing the spreading rate of the jet.
6.3.5 Mesh Resolution
Due to the implicit filtering applied by the computational grid, the mesh resolu-
tion can have an impact on the results obtained. The finer the mesh the higher
the level of physics directly captured, and the coarser the mesh the larger the
filter width and the larger the reliance on the SGS model. Here, results from
the Rii = 20m
2s−2 and αw = 0.4 simulation are compared to data from the finer
mesh calculations of Section 6.1.
It can be seen from Figure 6.36 that the coarser mesh predicts a reduced
spreading rate for the central jet profile. This is due to the larger filter width
providing higher levels of subgrid viscosity, as can be seen from Figure 6.37,
which dampens the intensity of the turbulent fluctuations and hence decreases
the severity of the jet breakup. However, the magnitude of this reduction is
limited by the filter width used; if the maximum length filter width was employed
an increase in the subgrid viscosity on the order of 100% would be incurred in
the central jet region, through a doubling of the axial mesh spacing. A finer
resolution of the turbulent structures is also evident with a higher mesh density.
It can also be seen from Figure 6.36, that for the turbulence inflow profile in
question, the coarser mesh gives results in better agreement with experimental
data for the central jet profiles. Better agreement is also obtained for the co-flow
pitot pressure profile and this is partly due to the higher filter width providing
higher levels of dissipation to control the large scale dynamics of the outer shear
layer. However, the ensemble averaged data for the finer mesh has been collected
over a significantly longer period and it was shown in Section 6.2.2.4 that due
to the large scale instabilities in the outer shear layer and presence of subsonic
velocities the statistics there take longer to converge, and hence this may not be
a fair comparison.
173
Figure 6.36: Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, on coarse and fine com-
putational meshes.
174
Figure 6.37: Instantaneous SGS viscosity profiles after 810,000 iterations on
coarse (top) and fine (bottom) computational meshes.
The turbulence energy spectrum in Figure 6.38 confirms the resolution of the
coarser computational mesh is high enough in all directions to provide a filter
width within the inertial range. Spectra are evaluated using all three velocity
components, with a time history from 270,000 to 810,000 iterations. Similarly
to the spectrum in Figure 6.13 the velocity data is obtained at the centreline
of the jet, but now at an axial location of x/D = 15.0825. It is difficult to tell
with spectra calculated from velocity data at a single point, but again it appears
the inertial range of the turbulence spectrum has a decay rate in agreement
175
with Kolmogorov’s k−5/3 law, rather than the steeper gradient of k−8/3 due to
compressibility effects proposed by Ingenito and Bruno [19].
Figure 6.38: Turbulence energy spectrum for coarse mesh using x-velocity (green),
y-velocity (blue) and z-velocity (red).
6.4 Modified Turbulence Initialisation
It has been suggested earlier that the poor agreement of the co-flow pitot pressure
profile with experimental data is due to the large scale dynamics of the outer shear
layer. It has also been seen from the numerical schlieren image in Figure 6.17
that this also leads to an interaction between the outer and inner shear layers,
possibly encouraging central jet breakup at downstream locations.
The severe turbulent nature of the outer shear layer is due in part to the high
levels of turbulence intensity for the imposed turbulent fluctuations at the outer
surface of the co-flow jet, as can be seen in Figure 6.8, which is a product of the
turbulence initialisation method employed.
176
As discussed in Section 4.10, the turbulence initialisation method [64] scales
the velocities pre-diffusion by the cell volume. Figure 6.39 shows the influence of
this cell volume scaling, where it can be seen the most uniform velocity magnitude
profile is obtained when no scaling is employed. Scaling by the inverse of the cell
volume significantly reduces the magnitude of velocity fluctuations in the majority
of the domain but high velocities exist in the very near wall cells due to division
by their low volume.
Figure 6.39: Showing influence of cell volume scaling on turbulence initialisation
with volume multiplication (left), no volume scaling (middle) and volume division
(right). Velocity magnitude shown, with scale 0 to 10 m/s.
It is possible that the weighted least squares method used for the gradient re-
construction in the turbulence initialisation calculation contributes to the higher
turbulence intensity at the outer co-flow surface. Such methods can provide poor
estimates of the gradient for highly stretched cells in regions of surface curva-
ture [120], such as exists in this region of the mesh. Strong gradients requiring
calculation may be initiated near the wall by the volume scaling process, due to
the presence of cell stretching in the wall normal direction for boundary layer
177
capture.
A simulation neglecting the volume scaling for the imposed turbulence fluctu-
ation calculation is conducted in an attempt to study the influence of the outer
co-flow surface turbulence intensity on the outer shear layer and of the outer
shear layer behaviour on computational results. Reynolds stresses Rii are set to
20m2s−2, with off diagonal terms fixed at zero, and the near wall length scaling
term, αw, is set to 0.4.
It can be seen from the schlieren image in Figure 6.40 that the unsteadiness
experienced in the outer shear layer is somewhat reduced, but significant tur-
bulent structures still remain and some interaction between the inner and outer
shear layers is still evident.
Figure 6.40: Instantaneous numerical schlieren image with no volume scaling
applied to the inlet turbulence initialisation.
It can be seen from the helium mass fraction profiles in Figure 6.41 that at
the x/D locations of 8.1102 and 12.1361 there is a mild reduction in the predicted
spreading rate of the jet. This is most likely to be caused by the corresponding
reduction in turbulence intensity of the central jet and inner co-flow surfaces
caused by the removal of the volume scaling from the turbulence initialisation
method (see Figure 6.39). However, at an x/D location of 15.0825 a significant
improvement over the results obtained with volume scaling employed, is found.
Since the interaction between inner and outer shear layers is more prominent at
178
downstream locations, the reduction in severity of the outer shear layer breakdown
may be the reason for these improved results.
Although there is an improved agreement with experimental data at the most
downstream location, it should be noted the predicted reduction in the spreading
rate experienced closer to the jet exit provides a worse agreement than previously
obtained.
The increased accuracy at downstream locations is reinforced by the pitot
pressure profiles in Figure 6.41. However, the intensity of the shear layer tur-
bulence is confirmed as too high by the discrepancy between computational and
experimental results in the outer shear layer region, at the x/D location of 15.0825.
The computational cost involved in evaluating fully converged rms velocities
is high. However, comparing the values after 90,000 and 180,000 iterations of
rms calculations in Figure 6.42 it can be seen that the approximate magnitude
of the mean fluctuating velocities can be evaluated, since there is no drastic
change in the shear layer rms values between these two evaluations. However,
the results show there is still a large disagreement with experimental data, despite
the improvement in central jet profiles for mean values.
The low rms values experienced at the first three axial locations corresponds
to the benign nature and corresponding delayed transition of the inner shear
layer and corresponding under prediction of the central jet spreading rate. This
reduced spreading rate leads to sharp peaks at a x/D location of 8.2, but the
unsteady shear layer behaviour at this location leads to rms values larger than
found in practice. The breakdown in the central jet leads to an agreement with
experimental data for the width of the profile at locations further downstream,
but the intensity of the velocity fluctuations still far exceeds what is required.
It is this transition from low levels of shear layer turbulence intensity to severe
jet breakup which causes problems. As well as the interaction of inner and outer
shear layers already discussed, the interaction of the turbulence structures in the
central and co-flow jets at the inner shear layer interface further complicates the
physics involved. To demonstrate the severity of this interaction, Figure 6.43
demonstrates the greatly reduced dynamics of the inner and outer shear layers
when the turbulence structures imposed in the co-flow are removed.
The differences between rms values presented in Figures 6.19 and 6.42 are
179
Figure 6.41: Radial (r) profiles of mean normalised helium mass fraction (left)
and pitot pressure (right) at several axial (x) locations, with original and no
volume scaling applied to the inlet turbulence initialisation.
180
partly due to mesh resolution effects and partly due to the reduced inlet turbu-
lence intensities through removal of the volume scaling.
Figure 6.42: Radial (r) profiles of rms axial velocity at several axial (x) locations,
with no volume scaling applied to the inlet turbulence initialisation, after 90,000
and 180,000 iterations of rms calculations.
It is possible that the downstream convection of inlet coherent structures
causes a continual interaction with the jet shear layers, further increasing the
181
Figure 6.43: Instantaneous numerical schlieren image with the co-flow inlet tur-
bulence removed.
severity of the jet breakup and resulting rms velocity magnitudes. Implementing
the method of Keating and Piomelli [63] whereby stochastic forcing is employed
at the RANS-LES interface, increasing the resolved stresses in this location, may
be more suitable. The turbulence intensity in the shear layers near the jet exit
might therefore be increased in line with the experimental data with no further
interactions downstream to exacerbate the shear layer instabilities.
The same length scales and intensity levels have been employed in both the
central and co-flow jets and this is another area of the computational setup which
could be altered. There is no-end to the modifications which can be made to the
simulation and imposed turbulence profiles in order to obtain good agreement
with the experimental data, but it has already become a costly exercise. Of course
the low levels of subgrid viscosity provided by the chosen filter width or chosen
mesh design could also have a detrimental impact on the results obtained. The
influence of the near wall scaling has been demonstrated but the maximum length
scale, Lmax, could also play a role in the dynamics of the simulation. However,
it is obvious that the imposed turbulent structures play a significant role in the
resulting jet behaviour and it is therefore very important for experimentalists to
try and provide turbulence length scale and intensity data in order for accurate
simulations to be possible.
It appears that, compared to RANS, LES is more computationally expensive
182
and still requires tuning of the simulation setup. However, LES is more physi-
cally meaningful and the parameters influencing the simulation are measurable
properties rather than constants used in the modelling process. The high phys-
ical content of a large eddy simulation can be used to investigate the complex
processes present in a supersonic combustor and therefore further improve the
understanding of the physics involved in supersonic combustion.
183
Chapter 7
SCHOLAR Test Case
In order to validate the combustion modelling aspects of PULSAR, the SCHOLAR
scramjet test case is to be simulated. Again, the capabilities of the RANS and
LES approaches to turbulence modelling are compared and an investigation into
the physics and regime of supersonic combustion is conducted.
7.1 Description
The SCHOLAR experiment [105] was designed with the validation of computa-
tional fluid dynamics software in mind, and in particular the validation of codes
to be used for the simulation of supersonic combustion. This test case has been
discussed in significant detail elsewhere [88; 105] and so only a basic overview is
presented here.
Figure 7.1 is a schematic of the SCHOLAR experiment, showing the simple
geometric configuration which makes up a scramjet combustor. Hot vitiated
gas is supplied to the combustor nozzle at subsonic velocities, from a heater
burning hydrogen with premixed oxygen and air. The enthalpy at the nozzle
inlet corresponds to a flight Mach number of 7 and the nozzle has an exit Mach
number of 2. Pure hydrogen is injected through the top wall of the combustor at
an angle of 30 degrees to the horizontal, as shown in Figure 7.2, and the injector
is designed to give an exit Mach number of 2.5.
184
Figure 7.1: Schematic of the SCHOLAR experiment [105] (a) Detail of nozzle
and combustor (b) Detail of combustor entrance and injector geometry
Figure 7.2: Hydrogen mass fraction distribution on the combustor centreline
The combustor is divided into three sections; a water cooled nozzle, a copper
section containing the isolator and injector, and a downstream carbon steel section
containing the remaining divergent section of the combustor. The sections are
connected with stainless steel flanges and carbon gaskets. The thickness of the
copper and steel sections allowed the combustor to run without cooling, however
185
this lack of cooling limited the possible run times of the experiment to just over 20
seconds, in order to prevent damage to the apparatus. After injection, the upper
wall of the duct diverges at a constant angle of three degrees to the horizontal.
7.1.1 Experimental Data
Figure 7.1(a) shows 5 locations down the length of the combustor (labelled 1, 3,
5, 6 and 7) at which measurements are taken using the CARS technique, provid-
ing profiles of static temperature along with oxygen and nitrogen mole fraction
distributions. The temperature and oxygen mole fraction data are presented in
Figure 7.3.
Figure 7.3: Planes of temperature and oxygen mole fraction experimental data
down the axis of the SCHOLAR combustor
Static pressure taps are located on upper, lower and side walls of the com-
bustor. Pressure measurements were taken at two second intervals during the
experiment, starting at 10 seconds and ending after 24 seconds of run-time. Fig-
ure 7.4 compares the first and last set of pressure measurements with the mean
186
profile for the top and bottom walls, showing little change in the data. The mean
profiles are to be compared to computational results.
Figure 7.4: Experimental pressure distributions for the upper (left) and lower
(right) walls of the combustor, after 10 and 24 seconds and the mean.
It should be noted that pressure taps in the copper section of the combustor
are located along the centreline of the bottom wall, but at a location of z =
-36.3mm for the top wall, where z is measured from the centreline [105]. For the
steel duct, pressure taps are located along the centreline for both upper and lower
walls. Thermocouples are also present on the top wall for surface temperature
measurements, which are discussed in more detail in the next section.
7.1.2 Boundary Conditions
The nozzle and injector inlet boundary conditions are summarised in Table 7.1.
The nozzle inlet species mass fractions are made up of H2O, O2 and N2 since it
was measured that these species make up 99% of the mixture molecular weight
from the heater outlet [88], however the unmeasured 1% of radical species is a
high concentration which could have a significant influence on ignition.
The wall temperature of the nozzle is set at 500 K, but the wall temperatures
of the copper and steel sections varied from about 320 K to 500 K and 890 K,
respectively. Due to the lack of cooling, the copper and steel sections did not
reach a steady state temperature and so the respective wall temperatures for the
simulations are set at 475 K and 700 K, as in [88]. However, it has been shown
[88] that computational results are not sensitive to the wall temperature values.
187
Parameter Nozzle Injector
P0 (MPa) 0.767 3.440
T0 (K) 1828 302
YH2 - 1.0
YO2 0.2321 -
YH2O 0.2041 -
YN2 0.5638 -
Table 7.1: Inlet boundary conditions for nozzle and injector, showing stagnation
pressure, stagnation temperature and species mass fraction values
7.2 RANS Simulations
As discussed in Section 3.4, it has been shown that the influence of turbulence-
chemistry interaction modelling has a negligible impact on the results obtained
when simulating a simple laboratory supersonic jet flame. Recent studies have
therefore neglected these interactions and used simple laminar chemistry to han-
dle the reaction rate calculations.
In order to see if turbulence chemistry interaction modelling plays a more
significant role in a practical scramjet combustor geometry, computational results
from simulations employing laminar chemistry and the assumed PDF combustion
model are to be compared. These simulations are also to be used to tune the
setup of an LES study by aiding the choice of combustion model and chemical
mechanism.
7.2.1 Computational Mesh
The computational mesh is based on that from a previous study [88] and is a pure
hexahedral multi-block mesh generated using ANSYS ICEM CFD. The mesh is
fully three dimensional and contains 12.18M cells. Details of the mesh in the
injector region can be seen in Figure 7.5, where an o-grid is employed to handle
the circular injector cross-section.
In order to reduce the computational cost, rather than simulating the full
length of the combustor the mesh is terminated at a length of 1.25m which is just
downstream of the final plane of experimental data. Due to the supersonic flow
188
Figure 7.5: RANS mesh for the SCHOLAR test case. Detail around the step and
injector shown.
conditions, the increased proximity of the domain outlet should not influence the
computational results upstream.
A near wall mesh spacing of 1 × 10−5m is uniformly applied to all surfaces,
providing the y+ distribution shown in Figure 7.6. It can be seen that the near
wall mesh resolution is sufficient to ensure y+ values of less than 5 over all com-
bustor surfaces. However, the y+ values on the surface of the injector are some-
what higher due to the low hydrogen density causing a higher speed of sound
and hence higher velocities for a given Mach number. As the velocity increases
through the injector throat the boundary layer thins and y+ values increase, but
are reduced again after the throat due to boundary layer growth, despite further
flow acceleration.
The axial mesh spacing transitions from approximately 1mm in the region
of injection, to 1.5mm in the steel section of the duct. A constant spacing of
approximately 1mm is employed in the z-direction (perpendicular to the page
as viewed in Figure 7.5), but due to the increasing height of the combustor the
y-direction mesh spacing (between upper and lower walls) transitions to approx-
imately 1.4mm at the outlet. A coarser mesh is employed in the nozzle section,
in the axial direction.
189
Simulations are run on 64 cores of the Stokes cluster and convergence is mea-
sured through the monitoring of primary variables, as can be seen in Figure 7.7.
Simulations are initialised with a non-reacting solution.
Figure 7.6: y+ distribution for the SCHOLAR test case RANS simulations
Figure 7.7: Convergence of the temperature and hydrogen mass fraction on the
combustor centreline at x = 0.777m
7.2.2 Computational Methods
Again, the second order AUSM+UP scheme with the Van Rosendale gradient
limiter is used for the spatial discretisation and the third order three step TVD
190
Runge-Kutta scheme for temporal integration. Since the assumed PDF combus-
tion model is to be employed, it is important to use the two-equation Menter SST
turbulence model in order to provide information about the turbulence length
scales, for use in source terms of the combustion model variance equations. The
k and ω variables are set in a way to enforce the turbulence intensity and ratio
of turbulent to molecular viscosity at the nozzle exit (x = 0.0m) to be equal to
5% and 200, respectively, roughly in line with values used in [88].
Tabulated data is used for the forward and backward reaction rate calculations
in the assumed PDF combustion model, which are calculated at the start of the
simulation. The range of temperature and temperature variance intensity values
used for the table are 300K to 3000K and 0.0001K to 0.8K, respectively, where
the temperature variance intensity, IσT , is defined as:
IσT =
√
σT
T˜
(7.1)
These ranges are each split into 300 equally spaced values, generating tables of
90,000 data points.
Since the purpose of these simulations is to investigate the influence of differ-
ent modelling approaches on the computational results obtained, model constants
will be fixed between simulations and will not be tuned to give excellent agree-
ment with experimental data, as was done in Chapter 5. The turbulent Prandtl
and Schmidt numbers are fixed at 0.9 and 0.7, respectively. No compressibility
correction is employed
Although no information is provided on the concentration of radical species
in the vitiated air at the inlet to the combustor, the high stagnation temperature
of the flow combined with subsonic velocities causes reactions to proceed in the
nozzle. These nozzle reactions are permitted in order to naturally generate highly
reactive radical species, which are convected into the combustor to aid the ignition
process.
7.2.3 Chemical Kinetics
As well as studying the influence of turbulence-chemistry interaction modelling it
is also important to investigate the effect of the chemical mechanism on computa-
191
tional results. Therefore, two chemical kinetics models are to be compared, which
are the full [91; 92] and reduced [94] Jachimowski mechanisms. As discussed in
Section 3.5, these mechanisms are two of the most widely used in supersonic
combustion research and have both been shown to give comparable results to the
more complex O’Conaire mechanism. The full Jachimowski mechanism uses 9
species and 19 reactions while the reduced mechanism is significantly more cost
effective, consisting of 7 species and 7 reactions. Details of each mechanism can
be found in the Appendix.
7.2.4 Results
In order to investigate the influences of turbulence-chemistry interaction mod-
elling and chemical kinetics, four simulations are conducted with the combination
of computational setups displayed in Table 7.2.
Reaction Rate Chemical Mechanism
Simulation 1 Laminar chemistry Reduced Jachimowski
Simulation 2 Assumed PDF model Reduced Jachimowski
Simulation 3 Laminar chemistry Full Jachimowski
Simulation 4 Assumed PDF model Full Jachimowski
Table 7.2: Setup of four RANS simulations of the SCHOLAR test case
Figure 7.8 compares the experimental temperature profiles obtained using the
CARS technique with computational data from the four simulations. The first
point to note is the significant disagreement at the second location (x=0.4266m),
where the high temperatures predicted in the simulations correspond to earlier
ignition than found in the experiment. As can be seen from Figure 7.9, early
ignition is caused by the presence of an oblique shock. The strong shock wave
boundary layer interaction causes an increase in downstream temperature which
is high enough to cause ignition to occur. It should be noted that the contours
of static pressure are overlaid onto a static temperature profile. Ignition through
this mechanism and flame anchoring at this location is evident in all simulations
conducted. The physics present in this ignition process is very complex and
presents a significant challenge to the computational methods employed. Due to
the high kinetic energy of the flow, if a slightly stronger shock wave boundary
192
layer interaction is predicted than occurred in the experiment the subsequent
deceleration would raise the local temperature sufficiently to cause ignition to
occur.
x Experimental (1) (2) (3) (4)
0.2742m
0.4266m
0.7770m
1.2342m
Figure 7.8: Comparison of mean temperature profiles from each RANS simulation
to experimental data
Figure 7.9: Shock induced ignition shown by contours of static pressure over a
static temperature profile
193
Comparing results from the two chemical mechanisms, it appears the full
mechanism in combination with the assumed PDF combustion model is not ca-
pable of predicting the high temperature levels found in the experimental data,
but the profiles obtained are consistent with those from the other three simula-
tions.
By comparing results from laminar chemistry and the assumed PDF com-
bustion model for each mechanism, it is evident that minimal differences occur
between temperature profiles, at all axial locations. Both laminar chemistry and
turbulence chemistry interaction modelling results give reasonable agreement with
experimental data, downstream of ignition. This suggests laminar chemistry does
a reasonable job at simulating supersonic combustion for this particular test case,
implying that interactions between turbulence and the reaction zone are minimal.
This conclusion can be reinforced through comparison of experimental and
computational O2 mole fraction profiles. It can again be seen from Figure 7.10
that when comparing results from laminar chemistry with those from the assumed
PDF combustion model, for the same mechanism, minimal differences occur.
There are however differences between results from the two chemical kinetics
models, with a faster destruction of oxygen occurring with the full Jachimowski
mechanism in the core region of the flow. However, with these simple visual
comparisons it is difficult to say with any certainty which mechanism is in better
agreement with experimental data.
Computational results for the wall pressure can also be compared to the mean
experimental data presented in Figure 7.4. Figure 7.11 shows this comparison for
the upper and lower combustor walls for all four simulations conducted, using
wall centreline computational data. It can be seen that computational results are
in reasonable agreement with experimental data, but the pressure peak in the
ignition region is poorly captured. Part of the discrepancy is due to the early
ignition predicted in the simulations, and hence the pressure profiles lie below
the experimental data downstream. Severe oscillations due to the presence of
shock waves are clearly visible in all computational profiles, but are absent in
the experimental data. It is possible the shock wave boundary layer interaction
points occur between the experimental pressure taps, but it is also possible the
severity of the interaction predicted by the computational methods is too high.
194
These oscillations can be seen to reduce in magnitude down the length of the
combustor due to natural and numerical diffusion effects.
x Experimental (1) (2) (3) (4)
0.2742m
0.4266m
0.7770m
1.2342m
Figure 7.10: Comparison of mean oxygen mole fraction profiles from each RANS
simulation to experimental data
Figure 7.11: Comparison of upper (left) and lower (right) wall pressure distribu-
tions to experimental data, for each RANS simulation
195
It can also be seen from Figure 7.11 that there is little difference in the pres-
sure profiles resulting from the four simulations, with Simulation 4 providing the
poorest agreement with experimental data, corresponding to the low tempera-
ture levels presented in Figure 7.8. In order for a more robust and quantitative
comparison of computational models to take place, more detailed experimental
data are required.
The computational pressure profiles presented in Figure 7.11 are obtained
along the centreline of the upper and lower combustor walls. However, the ex-
perimental data on the upper wall of the copper section is collected near to the
side wall. Figure 7.12 compares pressure profiles taken on both the centreline
and towards the side wall (at z = -36.3mm, measured from the centreline) for
Simulation 1. It can be seen that no significant differences are apparent, other
than slightly different shock wave boundary interaction locations, and so only
the centreline pressures will be compared to experimental data from now on, for
simplicity.
Figure 7.12: Comparison of upper (left) and lower (right) wall centreline and near
wall pressure distributions, for Simulation 1
It would be sensible to suggest that the minimal difference between results
from laminar chemistry and the assumed PDF combustion model may be down
to the assumptions made in the modelling effort, through assumed statistical
independence of temperature and composition, assumed shapes of PDFs and the
neglection of source terms in variance transport equations. However, Mo¨bus et al.
[75] have also compared the transported PDF approach to laminar chemistry for a
196
laboratory jet flame, again displaying only a small difference between the results
obtained, as discussed in Section 3.4. A deeper understanding of the physics
involved in supersonic combustion and compressible turbulence is required in
order to explain the behaviour observed.
7.2.5 Regime of Supersonic Combustion
Further insight into the regime of supersonic combustion can be gained by at-
tempting to place the physics on the diagram of Figure 3.2. The non-dimensional
Damko¨hler and turbulent Reynolds numbers, given by Equations 3.2 and 2.2, re-
spectively, can be extracted from the computational data and used to help define
the regime in which the reactions occur.
This analysis was conducted as a post processing exercise by Cocks et al.
[141], by applying 1-step chemistry to the temperature profile of Simulation 1 in
order to obtain the production rate of H2O for use in Equation 3.4, to define the
chemical time scale. Through this method, Damko¨hler numbers on the order of
50 to 200 are obtained, giving the combustion regime shown by the symbols in
Figure 7.13. The low Damko¨hler numbers at x = 0.2742 are for the pre-ignition
region where chemistry lags mixing, whilst the higher values obtained suggest
operation in the flamelet regime.
As discussed in Section 2.1 it has been suggested [19] that baroclinic and
dilatational effects lead to a steeper decay in the compressible turbulence energy
spectrum than is given by classic Kolmogorov scaling. This, coupled with the
high levels of molecular viscosity due to high temperatures damping the smallest
scales of the flow may lead to a truncated spectrum where the smallest turbulent
structures are larger than would be expected in the subsonic regime. This leads
to a picture of the smallest eddies being larger than the flame thickness and
hence only being able to provide wrinkling, rather than enter the flame and
cause distributed reactions to occur. If this flame distortion is a large length
scale phenomena, which can sufficiently be captured by a mesh of reasonable
resolution, the laminar chemistry approach would be capable of providing an
accurate description of the reaction rate terms.
It should be noted that 1-step chemistry was employed for this analysis in
197
Figure 7.13: Regime diagram showing data from 1-step chemistry post-processing,
at locations down the axis of the combustor
order for a direct comparison to be made to the work of Ingenito and Bruno
[19]. Ingenito and Bruno conducted an LES simulation of the SCHOLAR test
case using 1-step chemistry and subsequently proposed a regime for supersonic
combustion, with which the data in Figure 7.13 can be seen to be in excellent
agreement.
However, if the reaction rates for the production of H2O are directly extracted
from the complex chemistry RANS simulations, a very different picture is pro-
vided for the range of dimensionless Damko¨hler numbers. Figure 7.14 shows that
Damko¨hler numbers on the order of unity are found for all four simulations. This
suggests finite-rate effects play a more significant role than previously suggested,
with turbulence and chemical times scales of a comparable magnitude. Figure
7.14 also shows that comparable reaction rates are provided by laminar chemistry
and the assumed PDF combustion model.
198
x (1) (2) (3) (4)
0.2742m
0.4266m
0.7770m
1.2342m
Figure 7.14: Damko¨hler profiles down the axis of the SCHOLAR combustor for
each complex chemistry RANS simulation
The importance of finite rate chemistry can be reinforced by analysing the
eddy turnover times calculated from Equation 3.3 using information from the
two-equation turbulence model. It can be seen from Figure 7.15 that these are
of the order of 5× 10−4 seconds. Through analysis of the profile of axial velocity
down the length of the combustor, a flow-through time from the backwards facing
step to the domain outlet can be given as 7.5 × 10−4 seconds, showing that not
only are the reaction rates comparable to the turbulence time scale but also to
the residence time of reactants within the combustor.
The large discrepancy between the 1-step chemistry analysis and reaction
rate information extracted from the complex chemistry simulations is due to
the poor suitability of this 1-step mechanism to the simulation of supersonic
combustion, as discussed in Section 3.5. Through analysis of ignition delay and
induction times, both Gerlinger et al. [98] and Berglund et al. [68] observed
that the reaction rates provided by this mechanism are far too fast. It was
199
x = 0.2742m x = 0.4266m x = 0.7770m x = 1.2342m
Figure 7.15: Distribution of large eddy turnover time for Simulation 3
found that significant improvements in the reaction rate predictions are obtained
through the use of more complex chemical kinetics models. Therefore, the results
of Ingenito and Bruno obtained through the use of the 1-step mechanism may
be somewhat questionable, with the complex mechanism data presented here
being more reliable. This suggests that a distributed reaction regime may be
more likely, for this particular test case, with finite rate effects of paramount
importance. However, the increased likelihood of distributed reactions occurring
suggests the interaction between turbulence and chemistry is more significant
than previously discussed, making it harder to explain why laminar chemistry is
capable of providing results in reasonable agreement with experimental data.
It is possible the physics encountered in supersonic combustion is sufficiently
different to that studied under subsonic conditions that the simple extension of
current combustion modelling techniques is not capable of capturing the inter-
actions present. A deeper understanding of the physics involved in supersonic
combustion and the interaction of turbulence and chemistry is sorely required.
Methods such as direct numerical simulation (DNS) are ideal for these investi-
gations, but the question is whether such simulations of high Reynolds number
flows are affordable.
The turbulence encountered (and hence it’s interaction with chemistry) is also
very case dependent, with the backwards facing step and large diameter injector
likely to generate large coherent structures in the flow. Combustors employing
hydrocarbon fuels or multiple injectors may experience smaller scales at the low
200
wavenumber end of the spectrum. The turbulent Reynolds number used to define
a region on the regime diagram is also highly dependent on the characteristics
of the turbulence at the combustor inlet. Hence the regime in which supersonic
combustion occurs is likely to also be case dependent, as discussed in Section 3.1.
7.2.6 Inlet Turbulence Influence
It was shown by Rodriguez and Cutler [88] that the levels of turbulence at the
nozzle exit can have a significant impact on the wall pressure results obtained.
This is to be confirmed here, whilst also investigating the sole influence of implied
turbulence length scales on a RANS solution.
Through use of Equation 2.28, the turbulence intensity, I, can be defined as:
I =
u˜′
u˜
=
1
u˜
√
2k
3
(7.2)
and by combining Equations 2.45 and 2.47 the turbulence length scale is given
by:
Lturb =
k
1
2
β⋆ω
(7.3)
Therefore, keeping the turbulence intensity constant and increasing the ratio of
turbulent to molecular viscosity through a reduction in ω corresponds to an in-
crease in the turbulence length scale. Figure 7.16 shows the pressure distributions
resulting from modifications to the turbulence levels at the nozzle exit, for both
the laminar chemistry and assumed PDF combustion model simulations using
the reduced chemical mechanism. The low turbulence data corresponds to no
inlet turbulence, allowing it to be self-generated by the physics of the problem in
question. The high turbulence data corresponds to simulations where the turbu-
lence intensity is kept at 5%, but the ratio of turbulent to molecular viscosity is
increased to 500, through a reduction in ω at the nozzle exit.
It can be seen that the higher the levels of turbulence at the nozzle exit, the
higher the wall pressure values at both the upper and lower surfaces of the com-
bustor. This confirms the findings of Rodriguez and Cutler, whilst additionally
showing that both the turbulence intensity and implied turbulence length scales
201
Figure 7.16: Influence of inflow turbulence on the upper (left) and lower (right)
wall pressure distributions for Simulations 1 and 2.
can have a significant influence on the computational results obtained from a
RANS simulation. This reinforces the conclusions of Chapter 6, providing fur-
ther support to the need for experimentalists to measure turbulence parameters
and also displays it is not only LES that can suffer from a lack of this data.
It should also be noted that even at higher turbulence levels laminar chemistry
and the assumed PDF combustion model provide comparable results. In fact, the
turbulence intensity and levels of eddy viscosity have a larger impact on the results
obtained than the method chosen for the reaction rate calculations.
In order to demonstrate the dependence of the wall pressure measurements
on the temperature rise from combustion, the computational results from Sim-
ulation 1 are compared to those from a non-reacting simulation in Figure 7.17,
where the influence of combustion is clearly visible. The ’Cold’ simulation has
the same nozzle exit turbulence profile as Simulation 1, whilst the ’Cold low turb’
simulation allows self-generation of k and ω. It can be seen that the inlet tur-
bulence profile has no impact on the pressure distribution for the non-reacting
simulations, suggesting that the main influence of increased turbulence levels is
an increase in mixing and hence increased reaction rates and corresponding heat
release from combustion.
This can be confirmed through analysis of the temperature profiles resulting
from the low and high turbulence reacting simulations, where it can be seen from
Figure 7.18 that the high turbulence test case does indeed result in a much wider
distribution of elevated temperatures due to mixing.
202
Figure 7.17: Influence of heat release on the upper (left) and lower (right) wall
pressure distributions and influence of turbulence inflow on results from a non-
reacting simulation
x (a) (b) (c) (d)
0.2742m
0.4266m
0.7770m
1.2342m
Figure 7.18: Influence of turbulence inflow on temperature profiles down the
axis of the SCHOLAR combustor, showing reduced mechanism results with (a)
laminar chemistry and low turbulence, (b) assumed PDF combustion model and
low turbulence, (c) laminar chemistry and high turbulence and (d) assumed PDF
combustion model and high turbulence
203
Correspondingly, it can be seen in Figure 7.19 that the high turbulence test
case results in a more significant destruction of oxygen, both in the core of the
combustor and surrounding region.
x (a) (b) (c) (d)
0.2742m
0.4266m
0.7770m
1.2342m
Figure 7.19: Influence of turbulence inflow on oxygen mole fraction profiles down
the axis of the SCHOLAR combustor, showing reduced mechanism results with
(a) laminar chemistry and low turbulence, (b) assumed PDF combustion model
and low turbulence, (c) laminar chemistry and high turbulence and (d) assumed
PDF combustion model and high turbulence
7.2.7 Radicals Influence
Since the mass fractions of radical species entering the nozzle from the heater are
not measured, it is important to investigate their influence on the computational
results obtained. As discussed in Section 7.1.2, the radical species are allowed
to evolve in the high temperature nozzle region before the throat and are then
convected into the combustor to aid ignition. Figure 7.20 compares the top and
bottom wall pressure distributions from a simulation where the nozzle chemistry
204
is frozen, preventing radical formation, to the results from Simulation 1. Laminar
chemistry and the reduced chemical mechanism are employed.
Figure 7.20: Influence of inlet radical species concentration on upper (left) and
lower (right) wall pressure distributions for Simulation 1
From the pressure distributions it is apparent that the presence of radicals at
the inlet has a negligible impact on the computational results obtained. This is
because it is not the radicals which are influencing ignition, but the presence of a
shock wave. However, the temperature profiles in Figure 7.21 show that ignition
is actually slightly delayed in the no radicals case, even though the shock still
acts as the ignition mechanism. This is due to the increased time required for
radical formation.
Figure 7.21: Centreline static temperature profiles for Simulation 1 with nozzle
reactions (top) and without (bottom)
Slightly elevated temperatures also appear to be present in the case with
frozen nozzle chemistry, suggesting that the presence of radicals at the inlet can
have an influence on the results obtained, albeit a small one for this test case.
205
The interaction between the reaction zone and shock waves which reflect down
the length of the combustor can be seen by the flame distortion and patches of
higher temperature downstream of the discontinuities.
Of course, if ignition is not caused by shock waves in practice, the concentra-
tion of radical species at the inlet may play a more significant role in the ignition
process.
7.2.8 Mesh Convergence
It was hoped that mesh convergence could be proved by conducting a RANS
simulation on the fine LES mesh to be presented in Section 7.3.1, but due to
restraints on computational resources this is not possible. However, since the
mesh is of a comparable resolution to that used by Rodriguez and Cutler [88] and
since comparable computational results are obtained, a detailed mesh convergence
study is not deemed essential in order to demonstrate their validity. Through a
comparison of results to those obtained on a coarser computational grid Rodriguez
and Cutler found the influence of mesh resolution to be smaller than the influence
of parameters such as turbulence intensity at the combustor entrance.
7.2.9 Computational Cost
It is worth mentioning the additional computational costs incurred by both the
combustion modelling process and increase in complexity of the chemical kinetics.
Average times per iteration for the four baseline simulations are presented in Table
7.3.
Time per iteration (s)
Simulation 1 10.53
Simulation 2 11.92
Simulation 3 14.43
Simulation 4 15.46
Table 7.3: Time per iteration for each simulation, calculated over 100 time steps
Compared to the laminar chemistry simulations, the increase in computa-
tional cost through use of the assumed PDF combustion model is on the order
206
of 7% and 13% for the full and reduced Jachimowski mechanisms, respectively.
The percentage increase in computational cost for the full mechanism is below
that for the reduced mechanism since the percentage increase in additional trans-
port equations is lower. The percentage increase in computational cost incurred
through using the full mechanism, compared to employing the reduced mecha-
nism, is 30% and 37% for assumed PDF combustion model and laminar chemistry
simulations, respectively.
Of course, since the times per iteration are calculated over a period of 100 time
steps there is a degree of error in their comparisons, but the increased computa-
tional cost induced by combustion modelling and complex chemical mechanisms
is clear.
7.3 DDES Study
Since LES is computationally expensive in comparison to RANS, the results from
Section 7.2 can be used to tune the setup of a large eddy simulation of the
SCHOLAR scramjet test case in order to limit the number of runs required.
Again the DDES method is employed to bring the computational cost asso-
ciated with the LES of this high Reynolds number wall bounded flow down to
acceptable levels, since wall resolved LES of a supersonic combustor is beyond the
computational capabilities of today. The Menter SST two-equation turbulence
model is used to calculate the subgrid viscosity and fluid mechanical properties
of the flow.
The capabilities of LES for the study of supersonic reacting flows is to be
evaluated and an attempt to gain a deeper understanding into the physics involved
in the SCHOLAR test case is to be made.
7.3.1 Computational Mesh
Due to the high mesh resolution requirements, coupled with the significant length
of the nozzle-combustor assembly and number of iterations required to obtain con-
verged statistical data, the computational mesh is truncated at an axial location
of 0.81m in order to make the simulation affordable with available resources. This
207
effectively removes the final plane of experimental data from the simulation, with
the domain outlet now a small distance downstream of the 0.777m experimental
data location.
The RANS mesh described in Section 7.2.1 is refined in all directions in order
to generate a grid which can support the resolved turbulence content inherent in
LES. The chosen mesh density is comparable to that used by others [90]. The
majority of the domain contains cells of an isotropic nature, but high aspect ratio
elements are present in the upper half of the combustor in the injector region due
to both the continuation of the pre-step boundary layer mesh and concentration
of cells downstream of the step which expand with the combustor divergence, as
can be seen in Figure 7.5.
The combustor axial mesh spacing is set at 0.6mm around the region of in-
jection, stretching to 0.75mm for the remainder of the copper and steel sections.
Mesh spacing in the z-direction is set constant to 0.6mm whilst the outlet mesh
spacing is fixed at 0.6mm in the y-direction. It should be noted that these are
the maximum values in the core region of the domain, with near wall stretching
employed to capture the boundary layer.
This generates a pure hexahedral mesh with 36.8M elements, which is parti-
tioned to run on 2640 cores of the HECToR cluster. The same near wall mesh
spacing used in the RANS simulations is employed, which leads to a solution
time step of 4.03× 10−9 seconds. Using data from the centreline velocity profile
of a RANS simulation, the combustor flow-through time (from the step location
to domain outlet) is calculated as 4.6× 10−4 seconds, corresponding to just over
114,000 iterations.
7.3.2 Computational Methods
Since it is concluded from the RANS simulations that the neglect of turbulence-
chemistry interaction modelling through the use of simple laminar chemistry is
capable of providing a reasonable agreement with experimental data, the addi-
tional cost involved in modelling is not warranted and so need not be used here.
Since modelling requirements are lower in LES than RANS it is expected the in-
fluence of a combustion model on the computational results would be even lower
208
than already demonstrated.
The reduced Jachimowski mechanism was shown to provide results in better
agreement with experimental data than the full Jachimowski chemical kinetics
model, with significant computational cost savings, and so is employed here.
It has also been shown that the levels of turbulence entering the combustor can
have a significant impact on the computational results obtained. However, since
no experimental data is available for the turbulence intensity and length scales
it is difficult to know what turbulence profile to prescribe. Since fuel injection
into a supersonic cross-flow is expected to be a significantly more violent process
than the region of separation encountered around the thin central tip of the
coaxial jet simulated in Chapter 6, it is possible the combustor turbulence could
be sufficiently self generating. Peterson et al. [90] conducted a LES of a test
case similar to the SCHOLAR experiment, but with fuel injection normal to the
wall rather than at a significantly less intrusive 30 degrees to the horizontal, as
studied here. It was concluded that the normal jet is capable of breaking down
without the addition of synthetic turbulence in the boundary layer upstream of
injection. However, the computational results obtained in Section 7.2.6 when
allowing turbulence to self-generate in the combustor (’low turbulence’ test case)
suggest this method is not suitable here.
It possible turbulence emanating from both the nozzle and injector could have
a significant influence on the levels of mixing experienced. Since available com-
putational resources do not allow a large number of simulations to be conducted
in order to investigate the influence of various inflow parameters and simulation
configurations, it is decided that a single turbulence inflow is to be provided
just upstream of the combustor entrance, in an attempt to excite the shear layer
generated over the backwards facing step.
Experience from Chapter 6 is used to describe what is a reasonable turbu-
lence inflow profile, which is imposed (somewhat arbitrarily) only 4.1mm up-
stream of the combustor entrance, where the combustor entrance is defined as the
step location. It is worth pointing out that this location is further downstream
than the nozzle exit (x = 0.0m) where the levels of turbulence were imposed in
the RANS simulations conducted previously. Isotropic inflow turbulence with
Reynolds stresses, Rii, of 50m
2s−2, maximum length scale of 9mm and near wall
209
scaling factor, αw, of 0.4 is generated. The resulting turbulence field can be seen
in Figure 7.22, where positive Q isosurfaces are coloured by velocity magnitude.
As with the coaxial jet test case, the smallest structures and highest intensities
can be seen to exist in the near wall region.
Figure 7.22: Positive Q isosurfaces coloured by velocity magnitude in m/s, show-
ing the inflow turbulence profile
The inlet turbulence is generated on a domain which extends 0.2m in the
axial direction and consists of 200 layers of 1mm cells, totalling 4.86M hexahedral
elements.
7.3.3 Results
Ensemble averaged data are collected for the velocity, pressure, temperature and
species mole fractions, but the additional computational cost involved in evaluat-
ing the rms values is too high and so they are not calculated here. The simulation
is started from a RANS solution mapped onto the fine LES mesh and 180,000
iterations are used to flush initial transients out of the domain, which corresponds
to just over one and a half flow-through times. Due to the lack of subsonic regions
in the domain, it is not expected to take the flow any longer than this to develop.
The ensemble averaged data are then collected over an additional 990,000 itera-
tions, corresponding to over eight and a half flow-through times. The simulation
210
therefore runs for a total of 1.23M time steps, or 4.95× 10−3 seconds.
The ensemble averaged oxygen mole fraction profiles are compared to experi-
mental data in Figure 7.23. It can be seen that a reasonable agreement is obtained
just downstream of injection, at the x=0.2742m location. However, as moving
further downstream, a mole fraction profile emerges which appears somewhat
more elongated than presented in the experimental data. This continues to the
x=7770m plane, where it can also be seen that significant oxygen depletion is
present closer to the lower wall and further from the upper wall than found in
the experiment. The reasons for the results obtained are to be investigated later,
through a detailed analysis of the physics involved.
Experimental DDES
x = 0.2742m
x = 0.4266m
x = 0.7770m
Figure 7.23: Comparison of the ensemble averaged oxygen mole fraction to ex-
perimental data
Figure 7.24 compares the ensemble averaged temperature profiles. It can be
seen that the mean temperature values are significantly below those provided
in the experimental data and significant modification of the temperature scale
in column (b) is required in order to highlight the temperature distribution cal-
culated. This was also found by Peterson et al. [90] with their detached eddy
simulation of a similar test case with normal fuel injection. The low values are
due to the significant motion of the flame, which is due to the turbulent nature
211
of the simulation, with the high temperature regions not at a single location long
enough to contribute significantly to the ensemble averaged values. It should be
noted that the experimental data are not time averaged, but are polynomial fits
through data at a given location [105].
Experimental (a) (b)
x = 0.2742m
x = 0.4266m
x = 0.7770m
Figure 7.24: Comparison of the ensemble averaged temperature to experimental
data. Experimental data and (a) are scaled from 200K (blue) to 2200K (red),
whilst (b) is scaled from 200K (blue) to 1700K (red). (a) and (b) are the same
computational data.
The poor agreement with experimental data for the upper and lower wall
pressure distributions presented in Figure 7.25 is in line with the low values cal-
culated for the ensemble averaged temperatures. However, as will be shown, the
instantaneous flame temperatures are of the order of 2200K but the instantaneous
pressure profiles are in agreement with the ensemble averaged data. It is possible
the level of imposed turbulence may have an influence on the results obtained
since it has been shown in Figure 7.16 that the nature of the inflow turbulence
can have a significant impact on the wall pressure profiles when using RANS.
Reducing the turbulence intensity may lead to a more concentrated flame and
hence higher ensemble averaged values, but an increase in the intensity may lead
to higher mixing and a more widely distributed flame and hence higher heat re-
lease and wall pressure values. Also, even though the roles of the combustion
model and chemical mechanism were investigated in the RANS simulations pre-
212
Figure 7.25: Comparison of upper (left) and lower (right) wall pressure distribu-
tions to experimental data, showing instantaneous and ensemble averaged data
sented at the start of this Chapter, it is possible they have an impact on the results
obtained here. In order to investigate the influence of the inflow turbulence and
combustion model in the LES, two additional simulations are conducted. The
first simulation removes the turbulence inflow but keeps using laminar chemistry
for the reaction rate calculations, and the second simulation keeps the turbu-
lence inflow but uses the assumed PDF combustion model to handle subgrid
turbulence-chemistry interactions. Results from all three simulations are to be
compared.
Only a simple modification is made to the assumed PDF formulation to con-
vert it to a subgrid combustion model for handling the turbulence chemistry in-
teractions in LES. The subgrid viscosity and fluid mechanical properties provided
by the two-equation subgrid-scale model are now used in the variance transport
equations in place of the properties previously provided by RANS modelling. Due
to both the limitation in computational resources and additional cost involved in
evaluating the reaction rates using the assumed PDF combustion model, statis-
tical data is collected over only 600,000 iterations, compared to 990,000 for the
two laminar chemistry simulations.
An example of the time taken per iteration is presented in Table 7.4, where
the additional cost of using a combustion model compared to simple laminar
chemistry is clearly visible. The additional cost incurred by employing a turbu-
lence inflow appears to be below 1%, which is somewhat negligible, and this small
213
difference could even be attributed to other effects such as cluster load (e.g. I/O).
Simulation Time per iteration (s)
Laminar with turbulence 1.294
Laminar without turbulence 1.284
Assumed PDF with turbulence 1.481
Table 7.4: Time per iteration in seconds, averaged over 30,000 time steps.
It can be seen from Figure 7.26 that the presence of inflow turbulence has a
significant impact on the ensemble averaged oxygen mole fraction profiles, with a
wider region of oxygen depletion caused by increased mixing in case (a) compared
to case (b). However, the influence of including turbulence-chemistry interaction
modelling appears to have a minimal impact on the results obtained. Both of
these observations support the conclusions of the RANS results presented previ-
ously.
(a) (b) (c)
x = 0.2742m
x = 0.4266m
x = 0.7770m
Figure 7.26: Comparison of ensemble averaged oxygen mole fraction profiles for
simulations using (a) laminar chemistry and a turbulence inflow, (b) laminar
chemistry and no turbulence inflow and (c) the assumed PDF combustion model
and a turbulence inflow
214
Figure 7.27 compares the ensemble averaged temperature profiles for the three
simulations, which are scaled to a maximum temperature of 1700K in order to
be visible. Again it appears the influence of the turbulence inflow is far greater
than the influence of subgrid turbulence-chemistry interaction modelling. The
assumed PDF combustion model appears to predict slightly lower average tem-
peratures than the laminar chemistry simulation with a turbulence inflow, but
this may simply be due to the significantly fewer number of iterations over which
the assumed PDF statistics are calculated. The argument of significant flame
unsteadiness leading to lower mean temperatures is supported by the higher val-
ues provided by the laminar chemistry simulation without the turbulence inflow,
than with; particularly at the x = 0.4266m location. Further downstream, at x
= 0.7770m, the temperature profiles are significantly more concentrated when no
turbulence inflow is present.
(a) (b) (c)
x = 0.2742m
x = 0.4266m
x = 0.7770m
Figure 7.27: Comparison of ensemble averaged temperature profiles for simula-
tions using (a) laminar chemistry and a turbulence inflow, (b) laminar chemistry
and no turbulence inflow and (c) the assumed PDF combustion model and a
turbulence inflow
The influence of the turbulence inflow and combustion modelling on the com-
putational results can also be analysed through a comparison of the respective
upper and lower wall pressure distributions presented in Figure 7.28. As with the
215
RANS results, it is apparent that the inflow turbulence has a much more signif-
icant impact on the pressure profiles obtained than the addition of turbulence-
chemistry interaction modelling. The increase in pressure with the addition of
turbulence suggests that a significantly higher inlet turbulence intensity is re-
quired for the computational results to be in agreement with experimental data.
In order to demonstrate convergence of the statistical results obtained, the
time history of the ensemble averaged hydrogen mole fraction calculation is pre-
sented in Figure 7.29.
Figure 7.28: Comparison of upper (left) and lower (right) wall pressure distribu-
tions for the three DDES simulations
Figure 7.29: Convergence of ensemble averaged hydrogen mole fraction on the
centreline of the combustor at x=0.777m
216
Additionally, the ensemble averaged oxygen mole fraction and temperature
profiles after 780,000 and 1,230,000 iterations are presented in Figure 7.30. Whilst
the profiles after 1,230,000 iterations are significantly smoother, there is no dras-
tic change in the shape obtained. This confirms statistical convergence for the
laminar chemistry simulations, and also sufficient convergence for the assumed
PDF simulation for the visual comparisons conducted here.
x 780k 1230k 780k 1230k
0.2742m
0.4266m
0.7770m
Figure 7.30: Comparison of ensemble averaged oxygen mole fraction and tem-
perature profiles after 780,000 and 1,230,000 iterations for the laminar chemistry
simulation with a turbulence inflow
In order to demonstrate that the simulations have sufficiently evolved after
180,000 iterations for the commencement of statistics calculations, positive Q iso-
surfaces for all three simulations are presented in Figure 7.31, after both 180,000
and 780,000 iterations, and are coloured by hydrogen mass fraction. It can be
seen that even in the case where no turbulence inflow is present and the physics
is left to self-evolve, significant turbulent structures are present after 180,000
iterations, which are also comparable to those found after 780,000 time steps.
Although hairpin vortices from the boundary layer appear to contribute signif-
icantly to the mixing at downstream locations, it is clear that the turbulence
content for the simulation with no turbulence inflow is below that of the other
two.
In order to better understand the dynamics of the flow it is important to
further study the instantaneous data. Figure 7.32 shows the instantaneous tem-
perature profiles for the three simulations in question, at three locations down
217
Laminar chemistry with turbulence inflow
Laminar chemistry without turbulence inflow
Assumed PDF combustion model with turbulence inflow
Figure 7.31: Positive Q isosurfaces coloured by hydrogen mass fraction for the
three DDES simulations after 180,000 (top) and 780,000 (bottom) iterations, for
each case
218
the axis of the combustor. Three instances in time are also presented in order to
study the evolution of the flow profile over a period of 600,000 iterations. The
first thing to note is the high levels of symmetry present in case (b), the simula-
tion with no turbulence inflow. This corresponds to the lack of turbulent mixing,
which is three dimensional and asymmetric by nature. However, the flow does
appear to further develop over time, with more asymmetries introduced. This
suggests that although the initial transients from the RANS initialisation may
have disappeared after 180,000 iterations, the full development of the flow field
may take significantly longer, particularly when a turbulence inflow is not present
to aid this process.
In contrast, the dynamic nature of the flame in cases (a) and (d) is clearly
visible, corresponding to the wider distribution of temperature in the ensemble
averaged profiles.
Figures 7.33 and 7.34 show the instantaneous profiles of oxygen mole fraction
and hydrogen mass fraction, respectively. Again, the higher degree of asymmetry
and longer times required for flow evolution for case (b) are clearly visible. The
increased levels of turbulence in cases (a) and (c) corresponds to high mixing
which leads to a higher depletion of both oxygen and hydrogen. The lack of
mixing in case (b) causes a high concentration of hydrogen to still be present at
an axial location of 0.7770m.
The dominant flow feature which can clearly be seen in Figure 7.31, and are
also visible in the instantaneous images of Figures 7.32, 7.33 and 7.34, are the
two streamwise vortices emanating from the jet. Such vortices are classical of
the physics resulting from transverse injection into a moving fluid [142] and it is
likely their breakdown plays a significant role in the mixing process. However,
the vortices appear very persistent and a significant delay in their breakdown is
evident. This may be a large contributor to the discrepancies displayed between
computational and experimental data. It can be seen from Figure 7.31 that the
inflow turbulence is likely to have a significant impact on the vortex breakdown,
since the vortices appear to be more persistent when no turbulence inflow is
present.
219
(a) (b) (c)
x = 0.2742m
180k
480k
780k
x = 0.4266m
180k
480k
780k
x = 0.7770m
180k
480k
780k
Figure 7.32: Instantaneous temperature profiles for the (a) laminar chemistry
with a turbulence inflow, (b) laminar chemistry with no turbulence inflow and
(c) assumed PDF combustion model with a turbulence inflow, simulations, at
three axial locations and at three instances in time. Scaled from 200K (blue) to
2200K (red).
220
(a) (b) (c)
x = 0.2742m
180k
480k
780k
x = 0.4266m
180k
480k
780k
x = 0.7770m
180k
480k
780k
Figure 7.33: Instantaneous oxygen mole fraction profiles for the (a) laminar chem-
istry with a turbulence inflow, (b) laminar chemistry with no turbulence inflow
and (c) assumed PDF combustion model with a turbulence inflow, simulations,
at three axial locations and at three instances in time. Scaled from 0 (blue) to
0.24 (red).
221
(a) (b) (c)
x = 0.2742m
180k
480k
780k
x = 0.4266m
180k
480k
780k
x = 0.7770m
180k
480k
780k
Figure 7.34: Instantaneous hydrogen mass fraction profiles for the (a) laminar
chemistry with a turbulence inflow, (b) laminar chemistry with no turbulence
inflow and (c) assumed PDF combustion model with a turbulence inflow, sim-
ulations, at three axial locations and at three instances in time. Scaled from 0
(blue) to 0.7 (red).
222
However, the location at which vortex breakdown occurs appears to be a
transient phenomena, as can be seen by the images in Figure 7.35 which shows
positive Q isosurfaces for the laminar chemistry simulation with a turbulence in-
flow at 300,000 iteration intervals. Significant vortex breakdown is presented after
600,000 iterations, but after both 300,000 and 900,000 iterations their behaviour
appears significantly more benign.
Figure 7.35: Positive Q isosurfaces coloured by hydrogen mass fraction after
300,000 (top), 600,000 (middle) and 900,000 (bottom) time steps, for the laminar
chemistry simulation with a turbulence inflow.
Figure 7.36 presents slices down the axis of the combustor showing profiles
of vorticity magnitude, which can be used to highlight the behaviour of these
dominant streamwise vortices. It should be noted that these profiles are for the
laminar chemistry simulation with a turbulence inflow, and only results from this
simulation will presented from here onwards due to the poorer results from the
simulation without a turbulence inflow and negligible difference in results from
the assumed PDF combustion model simulation. Starting from the top left image
in Figure 7.36 and working along a row at a time, significant insight can be gained
into the development of the physics encountered in the combustor.
223
Figure 7.36: Profiles of vorticity magnitude down the axis of the combustor for
the laminar chemistry simulation with a turbulence inflow, after 1.2M iterations.
The images on the first five rows are in 0.01m steps, starting at x=0.16m and
ending at x=0.35m. The images on the last two rows are in 0.05m steps, from
x=0.40m to x=0.75m.
224
The unsteady boundary layer visible on the upper wall of the combustor pre-
injection is due to the presence of the backwards facing step further upstream.
As the fuel is injected a ring of high vorticity develops, which originates from the
injector boundary layer. Two counter-rotating vortices begin to form within this
ring of vorticity and develop further as they proceed downstream. Two corner
vortices become visible on the lower wall of the combustor (in the third row of
images), which are a product of a shock wave boundary layer interaction. On
the fourth row of images it is apparent that the two dominant vortices begin to
move away from the upper wall and break through the ring of vorticity, which
remains above the vortex pair but is drawn downwards by their entrainment.
The corner vortices grow in size and two small vortices become apparent on the
upper wall of the combustor, which become clearer further downstream (on the
fifth row of images). Rows six and seven take larger steps between planes and
show the significant interaction of the dominant counter-rotating vortices with
the lower combustor wall. This interaction has a significant influence on the
whole combustor and it can be seen in the final image that significant turbulent
structures are present on all walls as well as in the centre of the domain.
The flow physics encountered around the region of injection can be studied
in more detail by analysing the three-dimensional positive Q contours. Figure
7.37 shows the structures present, coloured by hydrogen mass fraction, with the
top wall of the combustor removed for clarity. It should also be noted that the
contours on each side wall of the combustor have be clipped away, in order to aid
visualisation.
The main flow features are labelled in the first image with (1) being the the
wake of the backwards facing step and resulting streaks in the boundary layer.
(2) corresponds to a horseshoe vortex wrapping around the jet, which is caused
by the adverse pressure gradient upstream of injection, leading to boundary layer
separation [142]. (3) corresponds to a region of shock wave boundary layer inter-
action induced separation and (4) are the dominant counter-rotating vortex pair.
(5) are the vortices on the upper surface, which are a pair of secondary vortices
arising due the flow field imposed on the upper wall by the counter-rotating vortex
pair and (6) appears to be a second horseshoe vortex which may be generated by
the presence of a bow shock, as will be discussed later. (7) and (8) are additional
225
Figure 7.37: Positive Q isosurfaces around the region of injection coloured by
hydrogen mass fraction, showing views from an angle (top), the side (middle)
and above (bottom).
226
regions of shock wave boundary layer interactions, caused by the reflection and
generation of shocks at (3). The importance of these interactions is also to be
highlighted.
x = 0.25m x = 0.30m x = 0.35m
Figure 7.38: Profiles of positive Q (top) and velocity vectors coloured by vorticity
magnitude (bottom) at three axial locations, showing the counter-rotating and
secondary vortices.
The generation of the secondary vortices on the upper wall of the combustor
can be analysed by studying the velocity vector fields in Figure 7.38, which are
coloured by vorticity magnitude. The top images in this figure are profiles of pos-
itive Q, which are used to highlight the location of the coherent structures. The
227
positive Q profile at x = 0.25m shows the presence of four dominant structures,
corresponding to the counter-rotating vortex pair emanating from the jet and
the induced secondary vortices on the upper surface of the combustor. It can be
seen that the left and right large vortices rotate in a clockwise and anti-clockwise
manner, respectively. Due to the close proximity of the upper wall to this velocity
field a spanwise boundary layer is generated, with respective velocities in the two
halves of the combustor converging at the centreline of the wall. This conver-
gence generates an adverse pressure gradient which causes the boundary layers
to separate and form the two secondary vortices seen. These vortices rotate in
the opposite sense to the larger vortices which generated them. It can also be
seen that the secondary vortices are still present after the larger vortices have
moved towards the bottom wall, but are separated by a larger distance.
One reason the secondary vortices may move apart can be seen in Figure 7.39,
which is a duplicate of the positive Q contours in Figure 7.37 but now coloured
by static pressure. It can be seen from the view above the combustor that a
significant shock-shock-boundary layer interaction causes a small region of high
pressure. It will be shown that it is this region of high pressure, and hence high
temperature, which causes ignition. The spot of high pressure and subsequent
expansion due to heat release from combustion may be the cause of the two
secondary vortices moving apart downstream of this location.
A high pressure region can also be seen on the lower wall of the combustor,
which corresponds to a significant shock wave boundary layer interaction and
subsequent separation and reattachment of the flow.
In order to further demonstrate the complex shock system arising in the com-
bustor, a three dimensional schlieren image is shown in Figure 7.40, which is
coloured by static pressure. The dominant flow features are numerically labelled,
with (1) corresponding to the expansion fan generated by flow acceleration around
the backwards facing step. When this flow meets the wall it must turn back on
itself, which is enabled through the oblique shock at (2). (3) is the bow shock
which is typical of injection into a supersonic cross-flow [101; 102] and a signifi-
cant interaction and combination of the oblique and bow shocks takes place. This
resulting shock causes a significant shock wave boundary layer interaction and
corresponding region of separation at (4). In order for the flow to climb over
228
Figure 7.39: Positive Q isosurfaces around the region of injection coloured by
static pressure, showing views from an angle (top), the side (middle) and above
(bottom).
229
this separation bubble it must turn into itself and hence a second oblique shock
is generated at (5). The strong shock causing the lower wall separation reflects
to generate the shock at (6). Two more shock wave boundary layer interactions
occur on the upper wall at (7) and (8). Such reflections continue down the length
of the combustor, generating a very complex flow field with interactions that
present a significant challenge to computational techniques. As well as shock
wave boundary layer interactions, shock-shock, shock-vortex, shock-turbulence
and shock-flame interactions are also present. An additional bow shock appears
to be present at (9), which could explain the additional horseshoe vortex at (6)
in Figure 7.37.
Figure 7.40: Three dimensional schlieren image coloured by static pressure
The shock wave causing the upper wall boundary layer interaction labelled
(8) in Figure 7.37 is obviously not flat, but combines with the flat oblique shock
generated by the region of separation on the lower wall, at the combustor centre-
line; causing the high pressure spot discussed. This is demonstrated further in
Figure 7.41, which shows pressure contours either side of the combustor centre-
line (top and bottom images) and one on the combustor centreline, coloured by
static pressure. From the off-centreline contour plots it is clear that two separate
shocks emanate from the separation bubble on the lower wall of the combustor.
However, at the centreline these shocks appear to merge causing both a signif-
icant interaction with the jet and upper wall boundary layer, leading to a high
pressure region downstream. It is likely the curved shock is a product of the jet
bow shock reflecting off the lower wall and it is also likely that the combination
230
of the two shocks may be sensitive to both the inflow boundary conditions and
geometrical setup, since any alteration in these parameters is likely to influence
shock angles and the location of the lower wall separation bubble and therefore
whether this shock combination and hence high pressure region on the upper wall
exists.
Figure 7.41: Pressure contours coloured by static pressure either side of the cen-
treline (top and bottom) and on the centreline (middle).
The contours of temperature and radical species mass fraction in Figure 7.42
show the location at which ignition occurs, which, from the two-dimensional
static pressure contours on the centreline, is clearly due to the strong shock wave
boundary layer interaction on the upper wall.
It appears from Figures 7.37 and 7.39 that the vortex pair emanating from
the jet follow the upper wall downstream of injection before tending to the lower
surface of the combustor. Due to the downwards momentum of the injected fuel
there is a tendency for the jet to move away from the upper wall, however the rate
at which this happens is significantly reduced by the entrained cross-flow fluid
231
Temperature
OH Mass Fraction
O Mass Fraction
H Mass Fraction
Figure 7.42: Three-dimensional contours of: Temperature = 1900K, YOH = 0.015,
YO = 0.008 and YH = 0.004, with a two-dimensional pressure contour on the
combustor centreline.
[142] and the vortices appear to stay close to the upper surface. The velocity
field generated may also cause a low pressure region above the vortices, further
reducing their downward motion. However, as can be seen even more clearly in
Figure 7.43 the motion of the vortices down the length of the combustor does
not appear monotonic and there is a distinctive change of direction towards the
lower wall, along with subsequent disturbances to the flow path. It is most likely
that this behaviour is the result of shock-vortex interactions, with a turning effect
imposed on the streamwise structures.
However, it should be noted that no such influence is imposed on the structures
breaking off in the upper half of the domain. It is possible the coherent vortices
would also be less susceptible to shock manipulation after breakdown has occurred
and hence a delay in vortex bursting may play a significant role in the resulting
elongated profiles exhibited in the ensemble averaged data.
232
Figure 7.43: Side view of positive Q contours coloured by hydrogen mass fraction
As well as increasing the intensity of the turbulence inflow upstream of the
combustion chamber, the introduction of coherent structures in the injector may
also have an influence on the vortex dynamics. It can be seen from the instan-
taneous image in Figure 7.44 that no shear layer instabilities are present imme-
diately after injection, but are often experienced with transverse injection into a
cross-flow [142].
Figure 7.44: Instantaneous hydrogen mass fraction profile on the centreline after
1.2M iterations.
In contrast, Figure 7.45 shows the distribution of species mass fraction vari-
ance from Simulation 2 of the RANS study, which employed the assumed PDF
combustion model for the reaction rate calculation. The high mass fraction vari-
ance displayed in the jet shear layer immediately after injection corresponds to
significant fluctuations in composition in that region, which are not exhibited in
the LES study and hence the addition of injector turbulence may be required.
Finally, in order to demonstrate that the mesh resolution is high enough to
support the resolved content inherent in LES a turbulence energy spectrum is
presented in Figure 7.46, where it can be seen an inertial range is indeed present.
The spectrum is calculated from a time history of the y-velocity over a period of
233
Figure 7.45: Centreline σY profile from Simulation 2 of the RANS study.
765,000 iterations of the laminar chemistry simulation with a turbulence inflow,
at centre of the combustor and an axial location of 0.777m. Again, it is difficult
to confirm or disprove the presence of a steeper decay than found in low Mach
number turbulence, with both the standard Kolmogorov scaling of k−5/3 and
steeper decay of k−8/3 appearing to be in better agreement with different parts
of the spectrum.
Figure 7.46: Turbulence energy spectrum using data for the y-velocity from
225,000 to 990,000 iterations on the centreline of the combustor at x=0.777m.
The extremely complex physics encountered in a scramjet combustor has been
demonstrated through use of the LES technique. The importance of vortex dy-
namics, shock-shock, shock boundary layer, shock-combustion and shock-vortex
234
interactions along with turbulence and its associated interactions have been high-
lighted. The influence of turbulence chemistry interaction modelling has again
been shown to have a minimal influence on the results obtained, with other aspects
of the computational setup having a more significant impact. The domination of
inflow turbulence has again been demonstrated, highlighting the importance for
experimentalists to provide this information in order for accurate simulations to
be conducted.
235
Chapter 8
Conclusions
8.1 Conclusions
8.1.1 Overview
This work had two aims: First, to compare the capabilities of RANS and LES
approaches as a tool for the simulation of supersonic combustion; second, that
insight would be gained into the physics involved in supersonic combustion, in
order to aid the development of turbulence and combustion modelling in this
regime.
A new code, PULSAR, has been successfully developed for both Reynolds av-
eraged and large eddy simulations of supersonic reacting flows. Significant atten-
tion has been paid to the discretisation of convective fluxes, ensuring a monotonic
solution is achieved with limited numerical dissipation. In order to avoid the re-
strictive reduction to first order of accuracy in discontinuous regions through use
of flux limiters, a gradient limiter based on that by Van Rosendale [117] has been
developed, allowing second order of accuracy under all flow conditions.
PULSAR is fully parallel through the implementation of MPI, with good scal-
ing performance demonstrated over thousands of processor cores. Such parallel
capabilities have been shown to be of paramount importance for high fidelity
simulations due to the significant computational costs involved.
The DDES method in combination with the Menter SST turbulence model
236
is implemented in order to conduct LES of high Reynolds number wall bounded
flows, with near wall modelling handled by the RANS technique. Due to the lack
of resolved content in the RANS-LES blending region the turbulence generation
method of Kempf et al. [64] is employed, which can be used to provide turbulence
inflow data for a simulation.
Finite rate combustion calculations are enabled through the implementation
of complex chemistry reactions, using multiple transport equations for the reac-
tive scalars. PULSAR is capable of neglecting turbulence-chemistry interactions
through the use of the laminar chemistry approach, or including them through
use of an assumed PDF combustion model.
8.1.2 Non-Reacting Flow
The non-reacting elements of PULSAR were validated through the simulation of
a supersonic coaxial jet, using helium for the central jet gas and air for the co-flow
in order to simulate compressible mixing of a light fuel. It was found that RANS
methods are capable of providing results for mean properties of the flow which
are in good agreement with experimental data, but only after significant tuning
of model constants. The high levels of eddy viscosity encountered in RANS due
to modelling of the whole turbulence spectrum causes the turbulent flux terms in
the system of governing equations to play a dominant role. The turbulent Prandtl
and Schmidt numbers were shown to have a significant influence on the results
obtained and compressibility corrections for the Menter SST turbulence model
were required to reduce the levels of eddy viscosity generated. However, results
were also observed to be dependent on the compressibility correction employed.
Such dependence on model parameters makes the accuracy of RANS results
unreliable, especially if no experimental data is available for tuning. Even if
tuning is possible, it can be a computationally expensive exercise due to the
number of simulations required and still the accuracy of extrapolation to flight
conditions is uncertain.
A tuned RANS simulation was capable of providing results for the rms axial
velocity in reasonable agreement with experimental data. However, due to the use
of the Boussinesq approximation, poor agreement is obtained in the core region
237
of the jet because of the lack of mean velocity gradients there. Also, due to the
use of the Boussinesq approximation for the calculation of the Reynolds stresses
and the dominance of the turbulence kinetic energy scalar in high speed flows,
the rms velocities in all three directions were found to be the same. The accuracy
of this result could not be confirmed due to the lack of experimental rms velocity
data in the two transverse directions.
Through simulation of the same non-reacting coaxial jet it was found that
LES results are insensitive to variations in the turbulence Prandtl and Schmidt
numbers, improving on this significant deficiency of the RANS technique. Since
the large scales of the flow are directly captured in LES and only the small
scales modelled, the levels of eddy viscosity are much lower than in RANS. The
turbulent flux terms in the governing system of equations, which are scaled by
these model constants, therefore play a significantly less dominant role. However,
their level of influence was found to be sensitive to the filter width used, with the
cube root of the volume providing the least dissipative results.
If the physics in the LES is left to develop by itself it was found that a
significant delay in the turbulent transition of the jet existed. This could be due
to the RANS-LES method employed for the simulation since no resolved content
is present in the near wall region and hence no turbulence information is present
to perturb the jet shear layer upon separation from the central jet tip. It is
also possible the levels of resolved turbulence in the nozzle LES region are not
in agreement with that present in the experiment, which is unknown since no
measurements were made. These effects are magnified by the high axial velocity
of the central jet, since any perturbation to the inner shear layer is convected
far downstream before it has time to cause turbulence transition to occur. The
purely dissipative nature of the subgrid scale model may also play a role, since
the presence of backscatter is not allowed for, which would provide energy to the
large scale disturbances in the flow.
In order to encourage jet breakup and to help solve the problems due to lack
of resolved content in both the near wall regions and free stream, the method of
Kempf et al. [64] was used to generate turbulence inflow data for the simulation.
However, this data was imposed towards the jet exit rather than at the domain
inlet due to the time scales of the flow making the latter computationally ex-
238
pensive. Turbulence length scales in both the near wall region and free stream,
along with Reynolds stresses, can be prescribed using this turbulence generation
technique. It was found that the ensemble averaged data are highly sensitive to
the nature of the inflow turbulence, through the prescription of both of these
parameters. Again, tuning of the simulation was required in order to get results
in reasonable agreement with the mean experimental data.
It was found that the breakdown of the shear layer between the co-flow and
ambient air can cause significant interactions with the shear layer between the
co-flow and core helium jet, having a major impact on the results obtained. This
interaction was avoided by Baurle and Edwards [15] by using a RANS method for
the outer region and applying LES to only the inner jet, but this approach neglects
an important aspect of the complex physics present. In order to reduce the
severity of the outer shear layer breakdown the turbulence initialisation method
was modified, through removal of the volume scaling, in order to reduce the
turbulence intensity generated on the outer wall of the co-flow. This was found to
reduce the inner and outer shear layer interactions although significant breakdown
of the outer shear layer still occurred.
Despite the reasonable agreement of the mean flow parameters, the magni-
tudes of the rms velocities calculated were in poor agreement with experimental
data, although the correct profiles were obtained. However, the two transverse
components of rms velocity data were shown to be significantly smaller than the
axial component, contradictory to the results obtained using the RANS technique.
Although experimental data for the transverse components are not available, this
result is in agreement with the LES results of Baurle et al. [15].
Tuning of LES is significantly more problematic than the tuning of RANS due
to the high computational costs involved. Due to the mesh resolution require-
ments and number of iterations needed to achieve converged statistical data,
massive parallel computing is required. However, tuning in LES concerns phys-
ical data, rather than model parameters as in RANS, meaning the tuning effort
can be significantly reduced if experimental data for the turbulence intensity and
length scales are available.
239
8.1.3 Reacting Flow
In order to validate the reacting aspects of PULSAR and to also conduct an inves-
tigation into the regimes and physics of supersonic combustion, the SCHOLAR
scramjet test case was simulated. The RANS and LES approaches were again
compared, with results from the RANS simulations used to tune the LES setup.
Although RANS has been shown to be sensitive to the choice of model con-
stants, it can be used to investigate trends in computational results brought
about through modification of the modelling process.
Using RANS, the laminar chemistry and assumed PDF combustion model
approaches to the simulation of reacting flows were compared in an attempt to
evaluate the influence of turbulence-chemistry interactions in supersonic combus-
tion. It was found that only minimal differences in results arise between the
two techniques, suggesting the influence of turbulence-chemistry interactions is
minimal. The current trend in supersonic combustion modelling appears to be
a movement to more complex and computationally expensive methods, such as
the transported PDF and LEM models. However, since the influence of the
turbulence-chemistry interactions appears to be small, it is possible the physics
involved could be captured by a simpler mathematical description.
The influence of chemistry was evaluated through the use of two complex
chemical mechanisms. Although the differences between results from the reduced
and full mechanisms was found to be small, they were more significant than the
differences between combustion modelling approaches. Slower oxygen depletion
in the core region of the combustor was evident when the reduced Jachimowski
mechanism is employed, and the full Jachimowski mechanism in combination
with the assumed PDF combustion model was unable to predict the temperature
levels evident in the experimental data.
The level of turbulence at the combustor entrance was found to have a signif-
icant impact on the results obtained, evaluated through comparisons of compu-
tational wall pressure distributions and both planes of temperature and oxygen
mole fraction to experimental data. Increased levels of turbulence were found
to enhance mixing in the combustion chamber, leading to increased heat release
and subsequent increased pressure levels on the walls of the combustor. Results
240
were independently influenced by turbulence intensity and implied turbulence
length scales, demonstrating that turbulence inflow data can have a significant
impact on reacting RANS simulations, as well as the LES dependencies found
with the coaxial jet. This reinforces the need for experimentalists to measure the
turbulence intensities and length scales present.
Whatever mechanism or combustion modelling approach was employed, early
ignition was found in the computational results when compared to experimental
data. This is due to a strong shock wave boundary layer interaction, causing
sufficient deceleration of the flow and subsequent rise in temperature to cause
ignition to occur. Influence of the radical species concentration in the facility
nozzle on this ignition process was investigated. With chemistry frozen in the
nozzle region, preventing the generation of radical species to be convected into
the combustion chamber, a small delay in ignition was found when compared to
a simulation with radicals present. However, the ignition mechanism is still the
shock wave boundary layer interaction, with the small delay due to the longer
time required for radical production. It is expected that if ignition is not caused
by such a violent process the radical species concentration at the combustor inlet
would play a much more significant role in this process and hence it important
for experimentalists to also attempt to provide such measurements.
The RANS results were used to conduct an investigation into the regime of
supersonic combustion, where both Damko¨hler and turbulent Reynolds numbers
were evaluated in an attempt to better understand the physics involved. As a
post-processing exercise 1-step chemistry was applied to the temperature profile
of a complex chemistry simulation in order to evaluate the production rate of
H2O, for use as the chemical time scale in the Damko¨hler number calculations.
This was done in order for a direct comparison to be made to the LES results
of Ingenito and Bruno [19], who employed 1-step chemistry for the simulation in
question. It was found that the results from this exercise were in excellent agree-
ment with the combustion regime proposed by Ingenito and Bruno, suggesting
a flamelet like behaviour. However, significantly lower Damko¨hler numbers were
found when the production rate of H2O was directly taken from the complex
chemistry simulations, presenting the importance of finite rate reactions and sug-
gesting operation between the flamelet and distributed reaction regimes. The
241
slower reaction rates provided by the complex chemical mechanisms are in agree-
ment with the work of Gerlinger et al. [98] and Berglund et al. [68], where
1-step chemistry was found to produce chemical time scales which are much too
short, with complex chemistry in much better agreement with available data.
This suggests the regime predicted by the complex chemical mechanisms is more
reliable.
A LES study was also carried out on the SCHOLAR test case, through use of
the DDES method to make such a simulation of this wall bounded flow affordable.
The results of the RANS simulations were used to specify use of laminar chemistry
and the reduced Jachimowski mechanism, along with a single turbulence inflow
upstream of the combustor entrance. The additional computational cost involved
in using the full Jachimowski mechanism or assumed PDF combustion model
make their use unwarranted. Poor agreement was found between the ensemble
averaged temperature profiles and experimental data, due to the lower mean
temperatures calculated from the LES. It was shown that this is not due to the
use of laminar chemistry, with very comparable results obtained when including
modelling for the turbulence-chemistry interactions, confirming the results from
the RANS investigation. However, it was shown that the ensemble averaged
results are significantly influenced by the level of mixing in the combustor through
use of a laminar chemistry simulation without a turbulence inflow.
The influence of inflow turbulence on the wall pressure profiles and poor agree-
ment of computational results with experimental data even when a turbulence
inflow is employed suggests a significantly higher level of mixing is required. The
lack of unsteadiness in the shear layer between fuel and oxidizer immediately after
injection suggests a turbulence inflow in the injector may also be needed. This is
supported through analysis of the species mass fraction variance variable from a
RANS simulation employing the assumed PDF combustion model, which shows
a high variance intensity in this region.
8.1.4 Revealed Physics
Since LES directly captures the large scales of the flow, greater insight can be
gained into the physical processes present. Despite the non-reacting coaxial jet
242
appearing to be a simple test case, the physics involved are very complex. The
presence of both inner and outer shear layers significantly complicates the dynam-
ics due to the significant interactions which can arise. It was found that shocks
could be formed between the two shear layers, severely disrupting their turbulent
breakdown. The turbulence length scales of the outer shear layer were found to
be much larger than those in the region between the central jet and co-flow and
the magnitude of the velocity fluctuations in the axial direction were found to be
higher than for the two transverse components.
Through simulation of the SCHOLAR test case, it was found that the domi-
nant feature of oblique wall injection is a pair of counter-rotating vortices, whose
breakdown plays an important role in the mixing process. This breakdown is
in turn influenced by the presence of turbulence at the combustor entrance, but
appears to be a dynamic process. The degree of mixing present was found to
have a significant impact on the wall pressure distributions due to strong mixing,
leading to higher heat release. However, due to the shallow injection angle and
hence smaller disturbance to the flow than would be encountered with wall normal
injection, the vortices appear persistent, limiting their mixing capabilities.
Although mixing was found to play a dominant role in the combustion pro-
cess, both RANS and LES results displayed that interactions between turbulence
and chemistry are minimal. The low Damko¨hler numbers provided by complex
chemistry calculations suggest some of the smaller turbulent scales may be able
to enter and disrupt the reaction zone, possibly causing extinction, whilst the
largest eddies wrinkle the flame. Since combustion modelling had a minimal in-
fluence on the results obtained, it is suggested that the effects on compressibility
on the turbulence spectrum, causing the smallest eddies to be larger than would
be expected under incompressible conditions, may limit the disruptive nature of
the smaller scales of the flow.
The behaviour of the dominant vortex pair is also influenced by the complex
shock system present, through alterations in their flow direction. It was found
that the ignition mechanism is due to the combination of two shock waves on the
combustor centreline and their subsequent strong interaction with the boundary
layer on the upper wall. It is therefore suggested that this ignition mechanism
could be influenced through alterations in geometry or boundary conditions. This
243
may explain why such ignition was not experienced in the experiment, due to
possible differences in these parameters.
Despite the streamwise vorticity present in the coherent vortex pair playing
the dominant role in fuel-air mixing, the injection of vorticity from the boundary
layer through the presence of hairpin vortices also plays a significant role in mixing
at downstream locations.
The velocity field generated by the vortices was found to induce a pair of sec-
ondary vortices on the upper wall of the combustor. A bow shock and horseshoe
vortex are also found to exist, which are common occurrences with wall injection
into a supersonic crossflow. The horseshoe vortex is a result of the adverse pres-
sure gradient generated upstream of injection, distorting the incoming vorticity
field.
The turbulence energy spectra obtained were not able to prove or disprove
the suggestion of a steeper inertial range due to compressibility effects, since the
difference in k−5/3 and k−8/3 slopes is relatively small. Further work is required
in order to better understand the nature of turbulence in a highly compressible
flow.
8.2 Future Work
Several areas of future work are evident from the research presented, from exper-
imental to computational aspects and from turbulence to combustion modelling.
8.2.1 Experimental Data
From an experimental point of view, there is a distinct need for both detailed
measurements of a supersonic reacting flow and for measurements of turbulence
intensity and length scales.
Despite the SCHOLAR test case and similar experiments providing a realistic
replication of combustion in a scramjet engine, it is very difficult to make detailed
measurements in such wall bounded flows. Data sets comparable to that available
for the non-reacting coaxial jet presented in this thesis are required, where both
244
mean and fluctuating data are available. As well as fluctuating data for the
velocity, for which transverse component measurements are required in order to
further evaluate the deficiencies of the Boussinesq approximation employed in
RANS turbulence modelling, fluctuating data for temperature and species mass
fractions are vital for the evaluation of combustion model accuracy and chemical
mechanism suitability.
Using LES to conduct investigations into mildly separated flows is currently a
challenge due to the significant sensitivities of computational results to the nature
of the turbulence inflow and lack of experimental measurements for this data.
Both the non-reacting and reacting test cases simulated in this thesis present a
significant challenge to the computational techniques employed due to the limited
existence of massively separated regions. Although the SCHOLAR test case was
designed to simplify modelling, through a reduction in large regions of separation
and re-circulation, this in fact introduces problems in LES due to the limited
generation of large scale instabilities. Hybrid RANS-LES studies of normal wall
or strut injection may be more successful due to the inherent turbulent nature of
the flow and hence reduced sensitivities to inflow turbulence [90].
Since the physics involved in the fuel injection process has a dominant influ-
ence on the resulting combustor flow, it may be sensible to study both the physics
and behaviour of computational techniques on smaller wall injection test cases,
such as [101; 103], before simulating the computationally expensive combustor as
attempted here.
8.2.2 Turbulence Modelling
Since the Boussinesq approximation leads to identical predictions for the axial
and transverse Reynolds stresses, which are not realistic according to results from
LES, the Reynolds stress model may provide a more accurate representation of the
physics involved, although such a turbulence model is computationally expensive
due to the large number of additional transport equations required.
It has been discussed that the purely dissipative nature of the Smagorinsky
SGS model employed in the DES method may contribute to the lack of turbulence
generation in the LES of mildly separated flows. A dynamic SGS model should
245
be incorporated into the method in order to modify the levels of subgrid viscosity
in both space and time, but to also introduce the modelling of backscatter into
the simulation. Both of these influences may aid the transition of shear layers
to the turbulence regime. The sole influence of backscatter can be evaluated
by allowing the generation of negative model constants in one simulation, but
restricting them to positive values in another.
8.2.3 Turbulence Inflow
The current method of choice for encouraging shear layer transition when using
LES is the application of a turbulence inflow, where slices of turbulence data from
a domain of previously calculated fluctuations are imposed at a given location.
This method applies coherent structures across the whole inflow, rather than just
in the boundary layer region as is done with some techniques.
It is possible this approach is not suitable for the non-reacting coaxial jet stud-
ied, although this is hard to evaluate in the absence of experimental turbulence
data. The computational rms results suggest too little interaction near the nozzle
exit, but too large an interaction further downstream. This could be caused by
the downstream convection of coherent structures in the core region of the flow.
In practice there may be limited turbulence in the core region of the jet, with
the shear layer breakdown mostly caused by the resolved content in the sepa-
rating boundary layer. Other methods, such as the near wall stochastic forcing
approach developed by Keating and Piomelli [63] may be more suitable, which
increases the levels of resolved stress at the RANS-LES interface. This removes
the presence of free stream turbulence from the simulation whilst providing the
shear layer disturbances required.
The downstream convection of free stream turbulence may however be impor-
tant for the SCHOLAR test case, with this statement supported by the improve-
ment of the RANS results when a significant turbulence inflow is employed. It is
important to study the influence of turbulence in the injector on the shear layer
immediately downstream of injection and subsequent influence on the dominant
vortices and impact on mixing and ensemble averaged results obtained.
246
8.2.4 Supersonic Combustion
In order for the turbulence-chemistry interactions in supersonic combustion to
be correctly modelled, even if they are small, a deeper understanding of these
interactions and the physics involved is required. This knowledge can then be used
to develop new methods for incorporating this physics into the RANS and LES
frameworks, rather than moving to more complex and more expensive combustion
models as currently appears to be the trend. DNS is the obvious choice for such
investigations, but further work is required to evaluate whether the simulation
of the high Reynolds number flow in question is affordable. A certain aspect of
the physics would need to be chosen for the investigation, such as the reacting
shear layer between fuel and oxidiser, since DNS of a whole jet or combustor is
not possible with today’s computational resources.
247
Appendix A: Jachimowski
Reaction Mechanisms
Reaction No. Reaction A n E
1 H2 +O2 ⇔ H2O+H 1.00× 1014 0.00 56000
2 H + O2 ⇔ OH+O 2.60× 1014 0.00 16800
3 O + H2 ⇔ OH+H 1.80× 1010 1.00 8900
4 OH + H2 ⇔ H2O+H 2.20× 1013 0.00 5150
5 OH +OH⇔ H2O+O 6.30× 1012 0.00 1090
6 H + OH+M⇔ H2O+M 2.20× 1022 -2.00 0
7 H + H +M⇔ H2 +M 6.40× 1017 -1.00 0
8 H + O+M⇔ OH+M 6.00× 1016 -0.60 0
9 H + O2 +M⇔ HO2 +M 2.10× 1015 0.00 -1000
10 HO2 +H⇔ OH+OH 1.40× 1014 0.00 1080
11 HO2 +H⇔ H2O+O 1.00× 1013 0.00 1080
12 HO2 +O⇔ O2 +OH 1.50× 1013 0.00 950
13 HO2 +OH⇔ H2O+O2 8.00× 1012 0.00 0
14 HO2 +HO2 ⇔ H2O2 +O2 2.00× 1012 0.00 0
15 H + H2O2 ⇔ H2 +HO2 1.40× 1012 0.00 3600
16 O + H2O2 ⇔ OH+HO2 1.40× 1013 0.00 6400
17 OH + H2O2 ⇔ H2O+HO2 6.10× 1012 0.00 1430
18 H2O2 +M⇔ OH +OH+M 1.20× 1017 0.00 45500
19 O + O+M⇔ O2 +M 6.00× 1013 0.00 -1800
Table 1: Full Jachimowski mechanism [91; 92]. Units are seconds, moles, cubic
centimeters, calories and Kelvin. Third-body efficiencies relative to N2 = 1 are
as follows: For reaction 6, H2O = 6.0; for reaction 7, H2 = 2 and H2O = 6.0; for
reaction 8, H2O = 5.0; for reaction 9, H2 = 2 and H2O = 16.0; and for reaction
18, H2O = 15.0.
248
Reaction No. Reaction A n E
1 H2 +O2 ⇔ OH+OH 1.70× 1013 0.00 48000
2 H + O2 ⇔ OH+O 1.20× 1017 -0.91 16500
3 OH + H2 ⇔ H2O+H 2.20× 1013 0.00 5150
4 O + H2 ⇔ OH+H 5.06× 1004 2.67 6290
5 OH +OH⇔ H2O+O 6.30× 1012 0.00 1090
6 H + OH+M⇔ H2O+M 2.21× 1022 -2.00 0
7 H + H+M⇔ H2 +M 7.30× 1017 -1.00 0
Table 2: Reduced Jachimowski mechanism [94]. Units are seconds, moles, cubic
centimeters, calories and Kelvin. Third-body efficiencies relative to N2 = 1 are
H2 = 2.5 and H2O = 16.0.
249
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