Technische Universitat¤ Munchen¤
Fachgebiet Stromungsmechanik¤
Direct and large-eddy simulation
of inert and reacting
compressible turbulent shear layers
Inga Mahle
Vollstandiger¤ Abdruck der von der Fakultat¤ fur¤ Maschinenwesen der Technischen Universitat¤
Munchen¤ zur Erlangung des akademischen Grades eines
Doktor-Ingenieurs
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr.-Ing. habil. N. A. Adams
Prufer¤ der Dissertation:
1. Univ.-Prof. Dr.-Ing., Dr.-Ing. habil. R. Friedrich i.R.
2. Univ.-Prof. W. H. Polifke Ph.D. (CCNY)
Die Dissertation wurde am 16.4.2007 bei der Technischen Universitat¤ Munchen¤ eingereicht und
durch die Fakultat¤ fur¤ Maschinenwesen am 10.7.2007 angenommen.Abstract
In the rst part of the thesis, Direct Numerical Simulations (DNS) of temporally evolving, tur-
bulent, compressible shear layers are discussed. at three different convective Mach
numbers (0:15, 0:7 and 1:1) were performed for both, inert and in nitely fast reacting gases. All
simulations were continued beyond the onset of a self-similar state in order to guarantee statistics
of general value. Self-similarity manifested itself by a collapse of suitably normalized pro les of
ow variables and a constant momentum thickness growth rate. During this state, the Reynolds
number based on the vorticity thickness of the shear layers was between 10000 and 40000 and
therefore in a fully turbulent regime. The relevance of the achieved results and parameter ranges
for practical applications can be seen from the fact that shear (or mixing) layers develop when
injecting fuel into the combustion chamber of an engine. Here, a good mixing of fuel and oxidizer
is of great interest for an ef cient combustion process.
The focus of the DNS data analyses was on the effects of compressibility and heat release due to
combustion on turbulence and scalar mixing. Both phenomena, and heat release,
were studied separately as well as in combination. Increasing compressibility, i.e. increasing
convective Mach number, resulted in a stabilization of the mixing layers: Instantaneous elds of
ow quantities became smoother, there were less turbulent uctuations and the growth rate of the
mixing layers reduced. The latter effect was related to a reduction in the production rate of the
streamwise Reynolds stress and a reduction in the pressure-strain correlations caused by changes
in the uctuating pressure eld. When heat release was present, the effects of compressibility
were similar as for the inert mixing layers, but they were less distinct, e.g. the reduction of the
growth rate with increasing Mach number was comparatively smaller. At rst sight, heat release
alone had similar consequences as compressibility: A stabilization of the shear layers, ow elds
with lower levels of uctuations and smaller spreading rates. However, when studied in more
detail, it could be seen that the consequences of heat release, were mainly ’mean density effects’,
i.e. a result of the reduction of the mean density by the high temperatures in the vicinity of the
ame sheets. This was not the case for the compressibility effects. Therefore, it is important to
distinguish between compressibility and heat release effects, even though they share the property
to be both detrimental for the turbulent mixing process.
In the second part of the thesis, Large Eddy Simulations (LES) of shear layers at a convective
Mach number of 0:15 were performed. By a coarsening of the grid, large reductions of computa-
tional time were achieved. A deconvolution approach in the form of a single explicit ltering step
was validated successfully for inert and reacting mixing layers by comparison with DNS data.
For the LES with chemical reactions, two differently detailed chemistry models were used for the
ltered chemical source term: one model taking into account the same in nitely fast, irreversible,
global reaction as in the DNS and one amelet model. The particular formulation of the amelet
equations allowed not only to take into account multistep Arrhenius chemistry, but also detailed
diffusion mechanisms. The evaluation of the results obtained with two different descriptions of
these mechanisms - one with Soret and Dufour effects as well as multicomponent diffusion andone without - showed differences for both, laminar amelets and turbulent mixing layers, in
quantities related to the ame dynamics and in the extinction behaviour.Acknowledgements
First of all, I would like to thank my supervisor Prof. Dr.-Ing. habil. Rainer Friedrich for the
opportunity to perform this work at the ’Fachgebiet Stromungsmechanik’.¤ He gave me constant
guidance and support and was always ready to answer questions or to discuss various aspects.
I am also thankful to Prof. Dr. Joseph Mathew for giving me the opportunity to spend four months
in his lab at the Indian Institute of Science in Bangalore. I appreciate his gracious hospitality and
the fruitful discussions that we had. The nal outcome of our joint work has been very rewarding.
I would also like to thank Prof. W. Polifke Ph.D. (CCNY) for taking over the role of the second
examiner and Prof. Dr.-Ing. habil. N. Adams for leading the board of examiners.
My thanks go also to the High Performance Computing Group of the ’Leibniz Rechenzentrum’
(LRZ) for providing help and support at nearly all times of the day, all days of the week. The
computations of this work were performed on the Hitachi-SR8000 and the Altix 4700 of the LRZ.
Financial contributions came from the Federal Ministry of Education and Research (BMBF) un-
der grant number 03FRA1AC and from ’Bayerischer Forschungsverbund fur¤ Turbulente Verbren-
nung’ (FORTVER).
Without my colleagues and the mutual support and encouragement within our group, this work
would not have been imaginable. In this context, I would like to mention especially Dr.-Ing.
Holger Foysi who gave me a lot of help and advice concerning the numerical codes and evaluation
programs. I would also like to thank Prof. Alexandre Ern from CERMICS, ENPC (France) for
providing the code EGlib.
Last but not least, I am deeply grateful to my family, in particular to my parents and grandparents,
for supporting me throughout the process of this work and throughout all my life.
Garching, March 2007
Inga MahleContents
1 Introduction 1
2 DNS of inert compressible turbulent shear layers 4
2.1 Introduction and literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The DNS code for inert gas mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Navier-Stokes equations for a gas mixture . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 The numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 The structure of the compressible shear layers . . . . . . . . . . . . . . . . . . . . 10
2.4.1.1 Inert shear layer atM = 0:15 . . . . . . . . . . . . . . . . . . . . . . 11c
2.4.1.2 Inert shear layer atM = 0:7 . . . . . . . . . . . . . . . . . . . . . . . 12c
2.4.1.3 Inert shear layer atM = 1:1 . . . . . . . . . . . . . . . . . . . . . . . 14c
2.4.2 The self-similar state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.3 Check of resolution, domain sizes and ltering . . . . . . . . . . . . . . . . . . . 23
2.4.4 The effect of compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.4.1 Turbulence characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 25
Mean ow variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Reynolds stresses, turbulent kinetic energy and anisotropies . . . . . . . . 26
Reynolds stress transport equations . . . . . . . . . . . . . . . . . . . . . 28
Analysis of the reduced growth rate . . . . . . . . . . . . . . . . . . . . . 32
Pressure-strain terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
TKE transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Thermodynamic uctuations . . . . . . . . . . . . . . . . . . . . . . . . . 38
Correlations of thermodynamic uctuations . . . . . . . . . . . . . . . . . 40
Behaviour of the pressure-strain correlations . . . . . . . . . . . . . . . . 41
Turbulent and gradient Mach numbers . . . . . . . . . . . . . . . . . . . . 46
Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48II CONTENTS
2.4.4.2 Scalar mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Mean pro le and variance . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Scalar pdfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Mixing ef ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Scalar variance transport equation . . . . . . . . . . . . . . . . . . . . . . 53
Scalar uxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Transport equations of scalar uxes . . . . . . . . . . . . . . . . . . . . . 55
Behaviour of the pressure-scrambling terms . . . . . . . . . . . . . . . . . 58
Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.4.3 Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Measurement of volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Visual thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Measurement of densities . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Particle statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Fractal nature of the mixing layer interface . . . . . . . . . . . . . . . . . 68
2.4.4.4 Shocklets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3 DNS of in nitely fast reacting compressible turbulent shear layers 78
3.1 Introduction and literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2 The DNS code with an in nitely fast chemical reaction . . . . . . . . . . . . . . . . . . . 81
3.2.1 In nitely fast chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2.2 Transport equations for in nitely fast reacting ows . . . . . . . . . . . . . . . . 83
3.2.3 The numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.1 The structure of the in nitely fast reacting shear layers . . . . . . . . . . . . . . . 86
3.4.1.1 In nitely fast reacting shear layer atM = 0:15 . . . . . . . . . . . . . 86c
3.4.1.2 In nitely fast reacting shear layers atM = 0:7 andM = 1:1 . . . . . 87c c
3.4.2 The self-similar state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4.3 Check of resolution and domain sizes . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4.4 Effects of compressibility and heat release . . . . . . . . . . . . . . . . . . . . . . 91
3.4.4.1 Mean heat release term . . . . . . . . . . . . . . . . . . . . . . . . . . 91CONTENTS III
3.4.4.2 Turbulence characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 91
Mean ow variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Reynolds stresses, turbulent kinetic energy and anisotropies . . . . . . . . 93
Reynolds stress transport equations . . . . . . . . . . . . . . . . . . . . . 96
Analysis of the reduced growth rate . . . . . . . . . . . . . . . . . . . . . 100
Pressure-strain terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
TKE transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Thermodynamic uctuations . . . . . . . . . . . . . . . . . . . . . . . . . 105
Correlations of thermodynamic uctuations . . . . . . . . . . . . . . . . . 106
Behaviour of the pressure-strain correlations . . . . . . . . . . . . . . . . 107
Turbulent and gradient Mach numbers . . . . . . . . . . . . . . . . . . . . 112
Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.4.4.3 Scalar mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Mean pro le and variance . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Scalar pdfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Mixing ef ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Scalar variance transport equation . . . . . . . . . . . . . . . . . . . . . . 119
Scalar uxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Transport equations of scalar uxes . . . . . . . . . . . . . . . . . . . . . 121
Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.4.4.4 Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Measurement of volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Visual thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Measurement of densities . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Particle statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Fractal nature of the mixing layer interface . . . . . . . . . . . . . . . . . 129
3.4.5 Shocklets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131IV CONTENTS
4 LES of inert and in nitely fast reacting mixing layers 134
4.1 Introduction and literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2 Description of the LES method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2.1 Implicit Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2.2 Applied lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.2.3 Filtered equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.2.4 Modeling of the ltered heat release term . . . . . . . . . . . . . . . . . . . . . . 142
4.2.4.1 The ltered density function . . . . . . . . . . . . . . . . . . . . . . . . 144
4.2.4.2 The conditionally ltered scalar dissipation rate . . . . . . . . . . . . . 145
4.2.4.3 The ltered scalar dissipation rate . . . . . . . . . . . . . . . . . . . . . 146
4.3 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.4.1 Inert mixing layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.4.1.1 Instantaneous elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.4.1.2 Pro les of averaged ow variables . . . . . . . . . . . . . . . . . . . . 148
4.4.1.3 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.4.1.4 Effect of ltering on dissipation rates . . . . . . . . . . . . . . . . . . . 153
4.4.1.5 Re nement of the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.4.2 In nitely fast reacting mixing layers . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.4.2.1 Instantaneous elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.4.2.2 Pro les of averaged ow variables . . . . . . . . . . . . . . . . . . . . 156
4.4.2.3 The ltered heat release term . . . . . . . . . . . . . . . . . . . . . . . 158
4.4.2.4 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.4.2.5 Effect of ltering on dissipation rates . . . . . . . . . . . . . . . . . . . 163
4.4.2.6 Re nement of the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164