von der
Fakult¨at Energie-, Verfahrens- und Biotechnik der Universit¨at Stuttgart
zur Erlangung der W¨urde eines Doktors der Ingenieurwissenschaften
(Dr.-Ing.) genehmigte Abhandlung
vorgelegt von Bosen Wang aus Shandong, P. R. China
Hauptberichter: Prof. Dr. Andreas Kronenburg Mitberichter: Prof. Dr.-Ing. Christian Hasse Tag der m¨undlichen Pr¨ufung: 20. 09. 2018
Institut f¨ur Technische Verbrennung der Universit¨at Stuttgart
Hiermit versichere ich:
1. dass ich meine Arbeit selbst¨andig verfasst habe,
2. dass ich keine anderen als die angegebenen Quellen benutzt und alle w¨ortlich oder sinngem¨ass aus anderen Werken ¨ubernommenen Aussagen als solche gekennzeichnet habe,
3. dass die eingereichte Arbeit weder vollst¨andig noch in wesentlichen Teilen Gegen-stand eines anderen Pr¨ufungsverfahrens gewesen ist,
4. dass ich die Arbeit nicht vollst¨andig ver¨offentlicht habe und,
5. dass das elektronische Exemplar mit den anderen Exemplaren ¨ubereinstimmt.
Stuttgart, September 2018
Parts of this thesis have been presented at conferences and published in the archival literature. The background and theoretical developments given in Chapters 2, 3 and 4 have been modified with respect to the originally published texts and been significantly extended to ensure completeness, coherence and consistency of the present manuscript.
The relevant papers are:
1. B. Wang, A. Kronenburg, D. Dietzel, and O. T. Stein. Assessment of scaling laws for mixing fields in inter-droplet space. Proc. Combust. Inst., 36(2):2451-2458, 2017.
• Data and results discussed in this paper are presented in Chapter5, Sec. 5.2. • Author’s contribution: Programming (100%), data generation (100%),
scien-tific originality (60%)
2. B. Wang, H. Chu, A. Kronenburg, and O. T. Stein. A Resolved Simulation Study on the Interactions Between Droplets and Turbulent Flames Using OpenFOAM. In Nagel, WE and Kroner, DH and Resch, MM, editor, High Performance Computing in Science and Engineering 17, pages 205-220, 2018.
• This paper has been published in the conference proceedings. Data and results discussed therein are presented in Chapter 5, Sec. 5.1.
• Author’s contribution: Programming (70%), data generation (70%), scientific originality (60%)
3. B. Wang, A. Kronenburg, G. L. Tufano, and O. T. Stein. Fully resolved DNS of droplet array combustion in turbulent convective flows and modelling for mixing fields in inter-droplet space. Combust. Flame, 189:347-366, 2018.
• Data and results discussed in this paper are presented in Chapter5, Sec. 5.3. • Author’s contribution: Programming (100%), data generation (100%),
scien-tific originality (70%)
IV
4. B. Wang, A. Kronenburg, and O. T. Stein. Modelling sub-grid passive scalar statis-tics in moderately dense evaporating sprays. 37th International Symposium on Combustion, accepted for presentation 2018.
• This paper has been accepted for presentation at the symposium. Data and results discussed therein are presented in Chapter6.
• Author’s contribution: Programming (100%), data generation (100%), scien-tific originality (80%)
5. B. Wang, A. Kronenburg, and O. T. Stein. A new perspective on modelling passive scalar conditional mixing statistics in turbulent spray flames. In preparation.
• This paper is now in preparation. Data and results discussed therein are presented in Chapter7.
• Author’s contribution: Programming (100%), data generation (100%), scien-tific originality (80%)
Firstly, I would like to express my sincere gratitude to my supervisor Prof. Dr. Andreas Kronenburg for his trust, patience and immense knowledge. He created an individual space for me to do research, gave me a freedom to find my own path and offered me guidance when I was trapped. His words and deeds encouraged me to grow as an independent researcher in the future.
I would also like to thank Dr. Oliver Stein for his straightforward criticism combined with heart-warming support on my research. His sense of responsibility as well as kindness will have an impact on me for a long time.
I would appreciate Prof. Dr. Ing. Christian Hasse being the examiner for my thesis. I am grateful to all of my friends who gave me help and shared pleasure with me in Stuttgart. I would especially thank Gregor Neuber, Daniel Loureiro, Marvin Sontheimer, Giovanni Luigi Tufano, Dirk Dietzel, Son Vo, Gregor Olenik, Milena Smiljanic, Carmen Straub, Jonas Kirchmann, Benedikt Heine, Hongchao Chu and Yawei Gao. I would also appreciate Ms. Ricarda Schubert giving me extraordinary help.
I would especially express my deep and sincere gratitude to my wife, my parents and my parents-in-law for their continuous love, encouragement and support.
Finally, I would acknowledge the financial support by CSC/Chinese Scholarship Coun-cil and the computational resources provided by HLRS (University of Stuttgart).
The objective of the present study is to provide suitable and accurate sub-grid scale clo-sures such as the mixture fraction probability density function (PDF) and its conditional scalar dissipation for large eddy simulation (LES) of spray combustion. These are needed quantities for closures in mixture fraction based combustion models such as flamelet and conditional moment closure (CMC). Most existing LES-flamelet and LES-CMC computa-tions of spray combustion have been conducted for dilute sprays at atmospheric pressure where the assumption of single droplet combustion can be invoked, and the local flame around the droplet in one LES cell is similar to a counter flow diffusion flame. Therefore, sub-grid mixture fraction conditional scalar dissipation and its PDF are typically modelled by closures that have been derived for single phase non-premixed combustion where the interactions between turbulence and scalar mixing are modelled. However, single droplet combustion is seldomly observed in practical applications such as gas turbines and IC engines, because higher pressure leads to denser sprays for the same droplet equivalence ratio and the group combustion is a predominant combustion mode. There, the evapora-tive fluxes from the different droplets interact and their effect on scalar mixing cannot be neglected. More advanced sub-grid closures that characterize the joint effects of turbulence and evaporation on the scalar dissipation and PDF are needed.
The strategy suggested by Klimenko and Bilger [67] is employed to facilitate the mod-elling of the characteristic mixing quantities in turbulent spray combustion. Following their ideas, inter-droplet space is split into two zones according to their different mixing characteristics: the near droplet zone (NDZ) and the Kolmogorov scale zone (KSZ) where mixing is dominated by diffusion and turbulence, respectively, and evaporation is involved to affect the mixing in both zones. The near droplet zone is limited to a thin wake in the immediate neighbourhood of the droplets and does not dominate the entire statistics of mixture fraction space.
This study complements the derivations of the scaling laws for scalar dissipation and PDF in the Kolmogorov scale zone based on the suggestions by Klimenko and Bilger [67] and uses direct numerical simulation (DNS) to assess the scaling laws. Fully resolved DNS of droplet array evaporation and combustion in turbulent convective flows is performed where the liquid-vapour interface between droplets and the surrounding air is resolved with
Abstract VIII
realistic boundary conditions for mass conservation, heat conduction and species diffusion. This forms a basis for the analysis of all the mixing characteristics ranging from the droplet surface to the space of the order of inter-droplet distance. The transition location between the near droplet zone and the Kolmogorov scale zone is located by comparison of the spatial mixture fraction and the corresponding scaling laws for each zone. The statistics from the Kolmogorov scale zone are extracted to evaluate the scaling laws for scalar dissipation and PDF for different inflow Reynolds numbers, turbulence intensities and integral length scales, droplet diameters, inter-droplet distances, droplet combustion regimes and various instants of the transient evaporation process. Suitable modelling constants inherent in the scaling laws are extracted from the DNS and their fitting functions are suggested.
The fully resolved DNS is limited in two respects: 1) the random character of the droplet position is neglected, 2) the large turbulent scales are “screened” by the fixed droplet spacing. Carrier-phase DNS (CP-DNS) is conducted to overcome these limitations where the flow and mixing fields are resolved but the droplets are treated as point particles. Only the Kolmogorov scale zone is focused on as the near droplet zone is not fully resolved in CP-DNS. The common approximation of the droplets as point particles and the lack of resolution of the phase interface do not impact on the DNS cells that contain droplets only: Many more DNS cells could be affected as the insufficiently resolved quasi-laminar wake would cover a much wider range and would modify mixing characteristics therein. The scaling laws for scalar dissipation and PDF that are calibrated by the fully resolved DNS are evaluated in both non-reacting and reacting flows, for different droplet number density, droplet Stokes number, droplet pre-evaporation rate, spray combustion regimes, droplet size and large eddy scales.
The work presented in this thesis provides a more physical perspective to understand the mixing in spray combustion and the scaling laws for turbulent micro-mixing may have better predictive capability than the existing sub-models for single phase combustion for flamelet and conditional moment closure (CMC) approaches.
Ziel der vorliegenden Studie ist es, geeignete und genaue Kleinskalen-Schließungsmodelle wie die Mischungsbruchwahrscheinlichkeitsdichtefunktion (PDF) und ihre bedingte skalare Dissipation f¨ur die Large Eddy Simulation (LES) der Sprayverbrennung bereitzustellen. Dies sind die ben¨otigten Gr¨oßen f¨ur mischungsbruch basierte Verbrennungsmodelle, wie beispielsweise das Flamelet-Modell oder die Conditional Moment Closure (CMC). Die meisten LES-Flamelet- und LES-CMC-basierten Simulationen von Sprayverbrennung wur-den f¨ur verd¨unnte Sprays bei Atmosph¨arendruck durchgef¨uhrt, bei denen die Annahme einer Einzeltropfenverbrennung herangezogen werden kann, und die lokale Flamme um den Tropfen in einer LES-Zelle ¨ahnlich einer Gegenstrom-Diffusionsflamme ist. Daher werden die konditionierte skalare Dissipation des Mischungsbruches und deren PDF typischerweise durch Schließungen modelliert, die f¨ur eine einphasige, nicht vorgemischte Verbrennung hergeleitet wurden, und bei der die Wechselwirkungen zwischen Turbulenz und skalarer Mischung modelliert werden. In der praktischen Anwendung hingegen, wie z.B. bei Gastur-binen oder Verbrennungsmotoren, wird selten eine Einzeltropfenverbrennung beobachtet, da ein h¨oherer Druck zu dichteren Sprays bei gleichem Tropfen¨aquivalenzverh¨altnis f¨uhrt und die Gruppenverbrennung vorherrschend ist. Die Verdampfungsfl¨usse der verschiede-nen Tropfen wechselwirken miteinander und ihre Wirkung auf die skalare Durchmischung ist nicht zu vernachl¨assigen. Fortschrittlichere Kleinskalen-Schließungsmodelle sind daher erforderlich, welche die gemeinsamen Auswirkungen von Turbulenz und Verdampfung auf die skalare Dissipation und PDF charakterisieren.
Die von Klimenko und Bilger [67] vorgeschlagene Strategie wird verwendet, um die Modellierung der charakteristischen Mischgr¨oßen bei turbulenter Sprayverbrennung zu erm¨oglichen. Ihre Ideen werden verfolgt und der Tropfenzwischenraum wird entsprechend der unterschiedlichen Mischcharakteristik in zwei Zonen aufgeteilt: die Nahtropfenzone (near droplet zone, NDZ) und die Kolmogorov-Skalenzone (Kolmogorov scale zone, KSZ), wo die Vermischung durch Diffusion bzw. Turbulenz dominiert wird und die Verdampfung die Vermischung in beiden Zonen beeinflusst. Die Nahtropfenzone ist auf eine d¨unne Schicht in unmittelbarer N¨ahe der Tropfen beschr¨ankt und dominiert nicht die gesamte Statistik des Mischungsbruchraumes.
Diese Studie erg¨anzt die Herleitungen der Skalierungsgesetze f¨ur skalare
Kurzfassung X
tion und PDF in der Kolmogorov-Skalenzone auf Basis der Vorschl¨age von [67] und verwendet direkte numerische Simulation (DNS), um die Skalierungsgesetze zu be-werten. Eine vollst¨andig aufgel¨oste DNS der Verdampfung und Verbrennung von Tropfenarrays in turbulenten konvektiven Str¨omungen wird durchgef¨uhrt, wobei die Fl¨ussigkeit-Dampf-Grenzfl¨ache zwischen den Tropfen und der Umgebungsluft mit re-alistischen Randbedingungen f¨ur Massenerhaltung, W¨armeleitung und Massendiffusion aufgel¨ost wird. Dies bildet die Grundlage f¨ur die Analyse aller Mischungseigenschaften von der Tropfenoberfl¨ache bis hin zu Abst¨anden im Bereich des Tropfenzwischenraum. Die ¨Ubergangsstelle zwischen der nahen Tropfenzone und der Kolmogorov-Skalenzone wird durch den Vergleich der r¨aumlichen Verteilung des Mischungsbruches und der entsprechenden Skalierungsgesetze f¨ur jede Zone ermittelt. Die Statistiken aus der Kolmogorov-Skalenzone werden extrahiert, um die Skalierungsgesetze f¨ur skalare Dissi-pation und PDF f¨ur verschiedene Reynoldszahlen, Turbulenzintensit¨aten und integrale L¨angenskalen, Tropfendurchmesser, Tropfenabst¨ande, Tropfenverbrennungsregimes und verschiedene Zeitpunkte des transienten Verdampfungsprozesses auszuwerten. Geeignete Modellierungskonstanten, die den Skalierungsgesetzen zugrunde liegen, werden aus der DNS extrahiert, und es werden Anpassungsfunktionen vorgeschlagen.
Die vollst¨andig aufgel¨oste DNS ist in zweierlei Hinsicht begrenzt: 1) der zuf¨allige Charakter der Tropfenposition wird vernachl¨assigt, und 2) die großen turbulenten Skalen werden durch den festen Tropfenabstand ”abgeschirmt”. Eine Carrier-Phase DNS (CP-DNS) wird durchgef¨uhrt, um diese Einschr¨ankungen zu ¨uberwinden, wobei die Str¨ omungs-und Mischungsfelder aufgel¨ost werden, die Tropfen aber als Punktpartikel behandelt wer-den. Nur die Kolmogorov-Skalenzone wird betrachtet, da die Nahtropfenzone in CP-DNS nicht vollst¨andig aufgel¨ost ist. Die ¨ubliche Approximation der Tropfen als Punktpar-tikel und die fehlende Aufl¨osung der Phasengrenzfl¨ache haben keinen Einfluss auf die DNS-Zellen, die nur Tropfen enthalten: Viele weitere DNS-Zellen k¨onnten betroffen sein, da der ungen¨ugend aufgel¨oste, quasi-laminare Nachlauf einen viel gr¨oßeren Bereich ab-deckt und die Mischungseigenschaften darin ver¨andert werden. Die Skalierungsgesetze f¨ur skalare Dissipation und PDF, die durch die vollst¨andig aufgel¨oste DNS kalibriert wer-den, werden sowohl in nicht-reagierenden als auch in reagierenden Str¨omungen f¨ur un-terschiedliche Tropfenzahldichten, Stokeszahlen, Tropfenvorverdampfungsraten, Sprayver-brennungsregimes, Tropfengr¨oßen und Großwirbelskalen ausgewertet.
Die in dieser Arbeit vorgestellte Arbeit liefert eine bessere physikalische Perspektive, um die Vermischung in der Sprayverbrennung zu verstehen, und die Skalierungsgesetze f¨ur turbulentes, kleinskaliges Mischen k¨onnen eine bessere Vorhersagef¨ahigkeit haben als die bestehenden Modelle f¨ur einphasige Verbrennung f¨ur Flamelet- und CMC-Ans¨atze.
Abstract VII
Kurzfassung VIII
Table of Contents XI
List of Figures XV
List of Tables XXI
Nomenclature XXIII
1 Introduction 1
1.1 Background and Motivation . . . 1
1.2 State of the Art. . . 2
1.3 Objectives . . . 6
1.4 Thesis Outline . . . 7
2 Scalar Mixing in Turbulent Sprays 11 2.1 Turbulent Flows . . . 11
2.2 Droplet Evaporation . . . 13
2.3 Characteristic Mixing Quantities . . . 16
2.4 Scaling Laws for Mixing in Inter-droplet Space . . . 18
3 Derivation of the Scaling Laws for the Kolmogorov Scale Zone 23 3.1 Mixture Fraction Conditional Scalar Dissipation . . . 23
3.2 The PDF of Mixture Fraction . . . 26
3.3 Summary . . . 28
4 Computational Approach 31 4.1 Fully Resolved DNS . . . 31
4.1.1 Governing Equations for Gas Phase . . . 32
4.1.2 Governing Equations for Liquid Phase . . . 33
CONTENTS XII
4.1.3 Boundary Conditions . . . 34
4.2 Carrier-phase DNS . . . 35
4.2.1 Modelling of Lagrangian Particle Source Terms . . . 36
4.3 Turbulence Generation . . . 37
4.4 Numerical Schemes . . . 39
5 Assessment of the Scaling Laws by Fully Resolved DNS of droplet arrays 41 5.1 Definition of the Domain Size and Mesh Resolution. . . 42
5.1.1 Computational Configurations and Numerical Setups. . . 42
5.1.2 Analysis of Single Droplet Combustion . . . 46
5.1.3 Analysis of Droplet Array Combustion . . . 49
5.1.4 Computational Performance . . . 53
5.1.5 Summary . . . 54
5.2 Assessment of the Scaling Laws in Non-reacting Flows . . . 55
5.2.1 Computational Setups . . . 56
5.2.2 Modelling Results . . . 57
5.2.3 Summary . . . 61
5.3 Assessment of the Scaling Laws in Reacting Flows . . . 62
5.3.1 Setups for Investigated Parameters . . . 63
5.3.2 Effects of Inflow Reynolds Number and Turbulence Intensity . . . . 66
5.3.3 Effect of Inter-droplet Distance . . . 71
5.3.4 Modelling for One Transient Evaporation Process. . . 75
5.3.5 Calibrations of Modelling Parameters in the Scaling Laws . . . 78
5.3.6 Summary . . . 82
6 Assessment of the Scaling Laws in Turbulent Evaporating Sprays by CP-DNS 85 6.1 Interpretation of the Scaling Laws . . . 86
6.2 Computational Configurations and CP-DNS setups . . . 87
6.3 Extraction of DNS Statistics . . . 90
6.4 The Validity of CP-DNS . . . 92
6.5 Modelling Mixing . . . 94
6.6 Summary . . . 98
7 Assessment of the Scaling Laws in Turbulent Spray Flames by CP-DNS 99 7.1 Interpretation of the Scaling Laws . . . 100
7.2 DNS Setups . . . 101
7.3 Extraction of DNS Statistics . . . 103
7.4 Effect of DNS Sampling Space. . . 107
7.6 Effect of Droplet Stokes Number . . . 115
7.7 Quantitative Analysis for the Deviation in the Flame Region . . . 117
7.8 Summary . . . 120
8 Conclusion & Outlook 121
8.1 Conclusion . . . 121 8.2 Outlook . . . 123
Bibliography 125
Appendix 139
A Derivation of the Scalar dissipation in the Near Droplet Zone 139
B Independence of Modelling Results on Integral Length Scale and Droplet
Size for Chapter 5 141
C Independence of Modelling Results on the Boundary Condtions at the
Droplet Center for Chapter 5 149
D Independence of Modelling Results on Computational Setups for
Chap-ter 7 153
2.1 Schematic diagram of the near droplet zone (NDZ) and the Kolmogorov
scale zone (KSZ) in droplets arrays. The circles with arrows represent
eddies of the Kolmogorov length scale, η. . . 19
5.1 Schematic diagram of the computational domain containing a regular
droplet array . . . 43
5.2 Comparisons of the flamelet solutions for the 7-species [62] and Luche’s
134-species [80] mechanisms for two different dissipation rates, Nf,0 = 10
and Nf,0= 100. . . 44
5.3 Three dimensional grid and schematic diagram of one single droplet domain 45
5.4 Instantaneous mixture fraction and temperature fields in a plane containing the droplet for mesh 2 . . . 46 5.5 Evaporation rate profiles for the three mesh resolutions . . . 47 5.6 The axial and radial variations of mixture fraction mean . . . 48 5.7 The mixture fraction conditional scalar dissipation (Nf) mean and the PDF
at x/d = 3 and x/d = 10 . . . 48 5.8 Instantaneous mixture fraction and temperature fields in a plane containing
the droplets for the two cases . . . 50 5.9 The axial and radial variation of mixture fraction in inter-droplet spaces 1
and 2 . . . 51 5.10 Conditional scalar dissipation in inter-droplet spaces 1 and 2 . . . 52
5.11 The turbulent energy dissipation rate (ε) mean at x/rc = 0.5 and x/rc =
0.75 in inter-droplet spaces 1 and 2 . . . 52
5.12 The mixture fraction PDF in inter-droplet spaces 1 and 2 . . . 53
5.13 The computational performance for the 12-droplet and 192-droplet domain 54
5.14 Instantaneous mixture fraction fields on the plane across one layer of droplets for case 1 (left) and case 2 (right). . . 57 5.15 Comparison of the mean value of mixture fraction along the transversal
distance for different planes perpendicular to the mean flow with the near droplet zone analytical model for case 2. . . 58
LIST OF FIGURES XVI
5.16 Comparison of the mean value of mixture fraction along transversal distance on different planes perpendicular to the mean flow with the near droplet zone analytical model for case 1. . . 59 5.17 Comparison of the mean value of mixture fraction along transversal distance
on different planes perpendicular to the mean flow with the near droplet zone analytical model for case 3. . . 59 5.18 Comparison of average mixture fraction variation along longitudinal
dis-tance with analytical model in the Kolmogorov scale zone for three cases. . 60
5.19 Comparison of scalar dissipation with the analytical model in the Kol-mogorov scale zone for three cases. . . 61 5.20 Comparison of PDF with the analytical model in the Kolmogorov scale zone
for three cases. . . 61 5.21 Instantaneous mixture fraction and temperature fields in a plane containing
the droplets for cases 1, 2 and 3. The white curves represent the iso-contour lines of the stoichiometric mixture fraction (f = 0.0645). . . 66 5.22 Comparisons of mixture fraction distributions in axial and radial directions
in inter-droplet space 1 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and
fmax,KSZ (Eqs. (2.37-2.40)), for cases 1, 2 and 3 . . . 67
5.23 Comparisons of mixture fraction distributions in axial and radial directions in inter-droplet space 2 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and
fmax,KSZ (Eqs. (2.37-2.40)), for cases 1, 2 and 3 . . . 69
5.24 Comparisons of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone of inter-droplet space 1 with the scaling laws, Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases
1, 2 and 3 . . . 70 5.25 Comparisons of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone of inter-droplet space 2 with the scaling laws, Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases
1, 2 and 3 . . . 71 5.26 Instantaneous mixture fraction and temperature fields in a plane containing
the droplets for cases 4, 5 and 6. The white curves represent the iso-contour lines of the stoichiometric mixture fraction (f = 0.0645). . . 72 5.27 Comparisons of mixture fraction distributions in axial and radial directions
in inter-droplet space 1 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and
fmax,KSZ (Eqs. (2.37-2.40)), for cases 4, 5 and 6 . . . 73
5.28 Comparisons of mixture fraction distributions in axial and radial directions in inter-droplet space 2 with the scaling laws, fN DZ (Eq. (2.37)), fmax,N DZ
5.29 Comparisons of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone of inter-droplet space 1 with the scaling laws, Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases
4 and 6 . . . 75 5.30 Instantaneous mixture fraction and temperature fields in a plane containing
the droplets for cases 7, 8 and 9. The white curves represent the iso-contour lines of the stoichiometric mixture fraction (f = 0.0645). . . 76 5.31 Comparisons of mixture fraction distributions in axial and radial directions
in inter-droplet space 1 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and
fmax,KSZ (Eqs. (2.37-2.40)), for cases 7, 8 and 9 . . . 77
5.32 Comparisons of mixture fraction distributions in axial and radial directions in inter-droplet space 2 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and
fmax,KSZ (Eqs. (2.37-2.40)), for cases 7, 8 and 9 . . . 78
5.33 Comparisons of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone of inter-droplet space 1 with the scaling laws, Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases
7, 8 and 9 . . . 79 5.34 Comparisons of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone of inter-droplet space 2 with the scaling laws, Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases
7, 8 and 9 . . . 79 5.35 The results of calibrations of the modelling parameters for scaling laws . . . 80 5.36 The deviations of the modelling for axial and radial variations of mixture
fraction in inter-droplet space 1 of case 2 induced by different values of modelling constants . . . 81 5.37 The deviations of the modelling for mixture fraction conditional scalar
dis-sipation and PDF in the Kolmogorov scale zone of inter-droplet space 1 of case 2 induced by different values of modelling constants . . . 81 6.1 Schematic of the two different zones in sprays: the near droplet zone (NDZ)
and the Kolmogorov scale zone (KSZ). The Kolmogorov eddies of length scale η and time scale τη are indicated by the blue circles. . . 86
6.2 Illustrations of the computational domain. Solid points: Droplets. . . 88
6.3 Instantaneous temperature and mixture fraction fields at the central slice
parallel to the mean flow direction (from left to right) for case C1 after 4 flow through times. A zoom of the region surrounding a droplet is depicted on the right part of the mixture fraction field. . . 90
LIST OF FIGURES XVIII
6.4 The variation of average droplet diameters Dd (left), average evaporation
rate of droplets Jm (middle) and mean relative velocity between droplets
and the carrier-phase Ud (right) in the flow direction for cases C1 to C6. . . 91
6.5 Instantaneous mixture fraction fields for fixed droplet positions in fully
resolved DNS and carrier-phase DNS. The flow direction is from left to right.. . . 93 6.6 Comparisons of the modelling for mixture fraction conditional scalar
dissi-pation and PDF for fully resolved DNS and carrier-phase DNS. . . 94
6.7 DNS statistics of conditional scalar dissipation and PDF in the KSZ and
modelling with different analytical models for the cases C1 and C2 of dif-ferent mesh resolutions. . . 95
6.8 DNS statistics of conditional scalar dissipation and PDF in the KSZ and
modelling with different analytical models for the cases C3 and C4 of dif-ferent Stokes numbers. . . 96
6.9 DNS statistics of conditional scalar dissipation and PDF in the KSZ and
modelling with different analytical models for the cases C5 and C6 of dif-ferent droplet number densities. . . 97 6.10 DNS statistics of conditional scalar dissipation and PDF in the KSZ and
modelling with different analytical models for the case C6a with the same droplet loading as C6 and a reduced evaporation rate. . . 97
7.1 Illustrations of the computational domain. Solid points: Droplets. . . 101
7.2 Temporal variation of the inflow mean velocity, Ut and the flame front
position, Xt, where TX = 1000K for case C1. XL is the length of the
domain along the mean flow direction. . . 103 7.3 Instantaneous temperature, T , heat release rate, dQ, fuel vapour mass
frac-tion, YF, CO mass fraction, CO, mixture fraction, f , and reaction progress
variable, C, fields for case C1. . . 104 7.4 The reaction progress variable conditional fuel vapour mass fraction, YF,
and oxygen mass fraction, YO, and heat release rate, dQ for case C1. . . 105
7.5 Comparisions of mixture fraction conditional statistics in the entire inter-droplet space with that in the Kolmogorov scale zone for case C1. The DNS statistics are taken within 0.02 < C < 0.99. The zoom plot for scalar dissipation within 0 < f < 0.1 is also presented. . . 106
7.6 Modelling of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone within 0.02 < C < 0.99 for case C1. The
DNS statistics are taken from three large eddy turn over time, 3τt. The
7.7 The mixture fraction fields at a plane across the droplets in the flow direc-tion (a) and at a plane crossing the flow direcdirec-tion in the Kolmogorov scale zone (b) (which is indicated by the white line in (a)) of a fully resolved droplet array with the inter-droplet distance of 10 droplet diameters (see Sec. 5.3.3). . . 109
7.8 Modelling of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.5), the homogeneous reaction zone (0.5 < C < 0.83) and the droplet burning zone (0.83 < C < 0.99) for case C1.. . . 110 7.9 Instantaneous heat release rate, dQ, and temperature, T , fields for cases C2
(the first row), C3 (the second row) and C4 (the third row). . . 111 7.10 The reaction progress variable conditional fuel vapour mass fraction, YF,
and oxygen mass fraction, YO, and heat release rate, dQ, for cases C2 (the
first row), C3 (the second row), C4 (the third row). . . 112 7.11 Modelling of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.5), the homogeneous reaction zone (0.5 < C < 0.86) and the droplet burning zone (0.86 < C < 0.99) for case C2.. . . 113 7.12 Modelling of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.5), the homogeneous reaction zone (0.5 < C < 0.89) and the droplet burning zone (0.89 < C < 0.99) for case C3.. . . 114 7.13 Modelling of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone within the pre-heat zone (0.02 < C < 0.3), the homogeneous reaction zone (0.3 < C < 0.8) and the droplet burning zone (0.8 < C < 0.99) for case C4. . . 115 7.14 Modelling of mixture fraction conditional scalar dissipation and PDF in the
Kolmogorov scale zone within 0.02 < C < 0.99 for cases C5 (the first row), C6 (the second row) and C7 (the third row). . . 116 7.15 Modelling of mixture fraction conditional scalar dissipation and PDF in the
reaction zone (0.5 < C < 0.99) for case C1. . . 118 7.16 The variations of the key parameters against mixture fraction in the reaction
zone (0.5 < C < 0.99) for case C1. . . 119 B.1 Instantaneous mixture fraction and temperature fields in a plane containing
the droplets for cases 1, 1a and 1b. The white curves represent the iso-contour lines of the stoichiometric mixture fraction (f = 0.0645). . . 144
LIST OF FIGURES XX
B.2 Comparisons of mixture fraction distributions in axial and radial directions in inter-droplet space 1 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and
fmax,KSZ (Eqs. (2.37-2.40)), for cases 1, 1a and 1b . . . 145
B.3 Comparisons of mixture fraction distributions in axial and radial directions in inter-droplet space 2 with the scaling laws, fN DZ, fmax,N DZ, fKSZ and
fmax,KSZ (Eqs. (2.37-2.40)), for cases 1, 1a and 1b . . . 146
B.4 Comparisons of mixture fraction conditional scalar dissipation and PDF in the near droplet zone of inter-droplet space 1 with the scaling laws, Nf,N DZ
(Eq. (2.41)) and Pf,N DZ (Eq. (2.43)), for cases 1, 1a and 1b. . . 147
B.5 Comparisons of mixture fraction conditional scalar dissipation and PDF in the near droplet zone of inter-droplet space 2 with the scaling laws, Nf,N DZ
(Eq. (2.41)) and Pf,N DZ (Eq. (2.43)), for cases 1, 1a and 1b. . . 147
B.6 Comparisons of mixture fraction conditional scalar dissipation and PDF in the Kolmogorov scale zone of inter-droplet space 1 with the scaling laws, Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases
1, 1a and 1b. . . 148 B.7 Comparisons of mixture fraction conditional scalar dissipation and PDF in
the Kolmogorov scale zone of inter-droplet space 2 with the scaling laws, Nf,KSZ (Eq. (2.42)), Pf,KSZ (Eq. (2.44)) and Pf,G (Eq. (2.45)), for cases
1, 1a and 1b. . . 148 C.1 The instantaneous temperature profile inside a droplet of different layers
for case 1 with zero gradient (upper) and fixed temperature 300K (lower) at the droplet center at quasi-stationary state. The hole in the centre is used to replenish the fuel. . . 150 C.2 Assessment of the scaling laws with the fitting functions for case 1 with
fixed temperature 300K at the droplet center. . . 151 D.1 Instantaneous heat release rate, dQ, and temperature, T , fields for cases
CD.1 (the first row) and CD.2 (the second row). . . 154 D.2 The reaction progress variable conditional mixture fraction, f , for cases
CD.1 and CD.2.. . . 155 D.3 Modelling of mixture fraction conditional scalar dissipation and PDF in the
Kolmogorov scale zone within 0.02 < C < 0.99 for case CD.1 (the first row) and within 0.02 < C < 0.6 for case CD.2 (the second row).. . . 156
3.1 The scaling laws for scalar mixing in inter-droplet space . . . 29 5.1 Characteristic quantities of three different mesh resolutions . . . 46 5.2 Characteristic quantities of two investigated cases. . . 49 5.3 Characteristic quantities of the three test cases. Retis the inflow turbulent
Reynolds number. η is the inflow Kolmogorov length scale. τη is the inflow
Kolmogorov time scale. . . 56
5.4 Characteristic quantities of six investigated cases with different inflow
Reynolds numbers, turbulence intensities and inter-droplet distances . . . . 65
5.5 Characteristic quantities of three investigated cases representing different phases of the evaporation process. The initial droplet diameter is d0 = 100µm. 65
6.1 The setup of the six cases . . . 89 7.1 DNS cases . . . 102 B.1 Characteristic quantities of three investigated cases . . . 141 B.2 Computational details on the three different domain sizes. ncells is the total
number of cells, ncores is the total number of cores, CPUh is the number of CPU hours.. . . 142 D.1 DNS cases for validation . . . 153
Roman Symbols A surface area [m2] a modelling constant A1 modelling constant Ae scaling constant AE scaling constant Ai pre-exponential factor Aτ scaling constant bi temperature exponent B limiter scalar
Bf Spalding number based on mixture fraction [-]
BY Spalding number based on mass transfer [-]
c droplet number density [m−3]
C reaction progress variable [-]
ck species mole concentration [kmol/m3]
CD drag force coefficient [-]
CN modelling constant
Cp mixture specific heat capacity [J/kg/K]
Cp,d droplet specific heat capacity [J/kg/K]
Cp,k single species specific heat capacity [J/kg/K]
Cp,s heat capacity at the droplet surface [J/kg/K]
Nomenclature XXIV
d droplet diameter [µm]
d0 initial droplet diameter [µm]
dQ heat release rate [J/s]
dt time step [s]
D molecular diffusivity [m2/s]
e
D Favre averaged molecular diffusivity [m2/s]
Dd droplet diameter [m]
Dd,0 initial droplet diameter [m]
Dt turbulent diffusivity [m2/s]
Dφ diffusivity of a scalar [m2/s]
E turbulent energy spectrum [m3/s2]
Ei activation energy [J/kmol]
f mixture fraction [-]
e
f Favre averaged mixture fraction [-]
f0 mixture fraction fluctuation [-]
f
f02 mixture fraction variance [-]
f1 minimum mixture fraction in KSZ [-]
f1o normalized collective evaporation term in KSZ [-]
f2 minimum mixture fraction in NDZ [-]
f2o normalized collective evaporation term in NDZ [-]
fd mixture fraction inside the droplet [-]
mixture fraction of the CP-DNS cells with droplets [-]
fg average mixture fraction in inter-droplet space [-]
fg,s mixture fraction at the droplet surface on the gas side [-]
fk normalized volumetric flow rate [-]
fKSZ mixture fraction in KSZ [-]
e
fKSZ Favre averaged mixture fraction in KSZ [-]
fmax,KSZ axial variation of maximum mixture fraction in KSZ [-] fmax,N DZ axial variation of maximum mixture fraction in NDZ [-]
fs mixture fraction at the droplet surface [-]
fst stoichiometric mixture fraction [-]
Fm mass flow rate [kg/s]
g0 modelling constant
hf g evaporation enthalpy [J/kg]
hk absolute enthalpy [J/kg]
hs sensible enthalpy [J/kg]
J volumetric flow rate [m3/s]
Jm droplet evaporation rate [kg/s]
Jm average evaporation rate [kg/s]
k turbulent kinetic energy [m2/s2]
kf,i forward reaction rate coefficients [1/s]
kr,i reverse reaction rate coefficients [1/s]
K droplet evaporation constant [m2/s]
Kc,i equilibrium constant of reaction
lt integral length scale [m]
lN DZ length of near droplet zone [m]
L domain size [m]
Lii integral length scale [m]
md droplet mass [kg]
˙
m droplet evaporation rate [kg/s]
˙
mRE Richardson extrapolated evaporation rate [kg/s]
˙
m00 mass flux of evaporation [kg/m2/s]
˙
m00l,s mass flux of evaporation on the liquid side [kg/m2/s]
M million
MF fuel molecular mass [kg/mol]
Mmix average molecular mass [kg/mol]
n normal distance [m]
Nomenclature XXVI
nj normal vector (n0i,n0j,n0k) [m]
N number of species [-]
number of droplets [-]
Nf mixture fraction conditional scalar dissipation [1/s]
Nf,KSZ scalar dissipation in KSZ [1/s]
Nf,N DZ scalar dissipation in NDZ [1/s]
Nr number of elementary reactions [-]
Nu Nusselt number [-]
e
Nf filtered scalar dissipation [1/s]
p pressure [P a] discretization order p∞ inflow pressure [P a] Pf mixture fraction PDF [-] Pf,KSZ PDF in KSZ [-] Pf,KSZ∗ PDF in KSZ [-] Pf,N DZ PDF in NDZ [-] Pr Prandtl number [-]
Prl Prandtl number of liquid [-]
Qi conditionally averaged mass fraction of species [-]
˙
Q00j heat flux [J/m2/s]
r radial distance [m]
rc inter-droplet distance [m]
rc average inter-droplet distance [m]
rii distance of two different locations [m]
rs droplet radius [m]
Re Reynolds number [-]
Red droplet Reynolds number [-]
Ri reaction rate for the ith elementary reaction [mol/s/m3]
Ru universal gas constant [J/mol/K]
Rc axial length of NDZ [m]
s stoichiometric mass ratio [-]
S average cross-sectional area of the wake [m2]
Sc Schmidt number [-]
Se source term of energy [J/kg/s/m3]
Sh Sherwood number [-]
Sm source term of mass [kg/s/m3]
SL laminar flame speed [m/s]
St Stokes number [-]
Su source term of momentum [kg/s2/m2]
T temperature K
T∞ inflow temperature K
Td droplet temperature K
Tg gas temperature K
Tg,s surface temperature on the gas side K
Tl liquid temperature K
Tl,s surface temperature on the liquid side K
Ts droplet surface temperature K
TX average temperature at position X K
t time [s]
τt large eddy time scale [s]
u velocity [m/s]
ˆ
u velocity mode amplitude [m/s]
u0 fluctuating velocity vector [m/s]
u0i fluctuating velocity vector (u0i,u0j,u0k) [m/s]
ui velocity vector (ui,uj,uk) [m/s]
ul liquid velocity [m/s]
U0 root-mean-square velocity [m/s]
U00 root-mean-square velocity [m/s]
Nomenclature XXVIII
U0 mean velocity [m/s]
Ud relative velocity [m/s]
Ud relative velocity mean [m/s]
Ut temporal mean velocity [m/s]
U∞ inflow mean velocity [m/s]
vki0 forward stoichiometric coefficient
vki00 reverse stoichiometric coefficient
V volume of one CFD cell [m3]
Vd droplet velocity vector [m/s]
Vg gas velocity vector [m/s]
W collective evaporation term [1/s]
x axial distance [m]
x cartesian coordinate vector [m]
ˆ
x normalized distance [-]
x1 axial distance [m]
x1 average axial distance [m]
xi cartesian coordinate vector (xi,xj,xk) [m]
xu characteristic undulation length of the smallest eddy [m]
Xd droplet position vector [m]
X distance in x direction [m]
XF,s mole fraction of fuel vapour at the droplet surface [-]
XL domain length [m]
Xt temporal position in x direction [m]
YF fuel vapour mass fraction [-]
YF,g fuel vapour mass fraction [-]
YF l liquid fuel mass fraction of liquid fuel [-]
YF,s fuel vapour mass fraction at the droplet surface [-]
YF g,s fuel vapour mass fraction at the droplet surface [-]
YF,∞ fuel vapour mass fraction in infinity [-]
Yig,s non-evaporating species mass fraction at the droplet surface [-]
YO mass fraction of oxidizer [-]
YO,0 mass fraction of oxidizer in the unmixed oxidant [-]
Yp mass fraction of products [-]
Z a conserved variable [-] Greek Symbols α ratio β β PDF function γ weighting coefficient δ boundary thickness [m] δij Kronecker delta
δM droplet film thickness [m]
∆t time step [s]
∆x characteristic mesh size [m]
∆τ time interval [s]
∆κ wave number interval [1/m]
ε turbulent energy dissipation rate [m2/s3]
e
ε filtered energy dissipation rate [m2/s3]
η Kolmogorov length scale [m]
κ wave number vector [1/m]
κ wave number [1/m]
κe most energetic wave number [1/m]
κη Kolmogorov wave number [1/m]
λ heat conductivity [W/m/K]
λl liquid heat conductivity [W/m/K]
Nomenclature XXX
µg dynamic viscosity of gas [P a-s]
µl dynamic viscosity of liquid [P a-s]
µs dynamic viscosity at the droplet surface [P a-s]
ν kinematic viscosity [m2/s] ρ density [kg/m3] ρ average density [kg/m3] ρd droplet density [kg/m3] ρg gas density [kg/m3] ρl liquid density [kg/m3]
σ Fourier mode direction
τ characteristic time scale [s]
τij viscous stress tensor [N/m2]
τp particle relaxation time [s]
τt large eddy turn over time [s]
τv evaporation time scale [s]
τη Kolmogorov time scale [s]
φ scalar
φ1 modelling constant
φ2 modelling constant
φc modelling constant
φd droplet equivalence ratio
φk normalized mixture fraction in KSZ [-]
φn modelling constant
ψ Fourier mode phase
ωF reaction rate of fuel [kg/s/m3]
ωi reaction rate of species (ωi, ωj, ωk) [kg/s/m3]
ωO reaction rate of oxidizer [kg/s/m3]
ωp reaction rate of products [kg/s/m3]
ωQ heat release rate [J/kg/m3]
Superscripts k kth droplet n nth Fourier mode Subscripts 0 initial state g gas phase i1 inter-droplet space 1 i2 inter-droplet space 1 l liquid phase
max maximum value
min minimum value
n nth time step
s droplet surface
∞ inflow or infinity
Shorthands
CFD computational fluid dynamics
CFL Courant-Friedrichs-Lewy
CDS central differencing scheme
CMC conditional moment closure
CP-DNS Carrier phase direct numerical simulation
DNS direct numerical simulation
Nomenclature XXXII
FVM finite volume method
HLRS High-Performance Computing Center Stuttgart
IC internal combustion
KSZ Kolmogorov scale zone
LEM linear eddy model
LES large eddy simulation
NDZ near droplet zone
PCAH poly-cyclic aromatic hydrocarbons
PDF probability density function
PISO Pressure-Implicit with Splitting of Operators
RANS Reynolds averaged Navier Stokes/simulation
RMS root mean square
SSF stochastic separated flow
ST synthetic turbulence
TFM thickened flame model
TVD total variation diminishing
Introduction
1.1
Background and Motivation
Turbulent spray combustion is one of the most widely used technical realizations for power production in devices such as IC engines, gas turbines and rocket propulsion chambers. According to the international energy outlook in 2017 [3], petroleum and other liquids have been the largest source of energy for decades and will re-main so until 2040. The report also pointed out the world consumption of liquid fuels is projected to continue to rise by more than 1% annually. Further optimiza-tion of spray combusoptimiza-tion devices is related to industrial development and our life which includes improving combustion efficiency and reducing pollution emissions [83, 103]. Modern spray combustors aim at ensuring a relatively fine and disinte-grated spray after the so-called secondary break-up of a liquid jet which is injected from an atomizing nozzle [77, 119]. The subsequent mixing of the evaporated fuel vapour with the surrounding gas is a typical multi-scale problem [13, 44, 110] and it is challenging for experimental and numerical investigations. The large turbu-lent flow scales imposed by the geometry of the combustion chamber tend to be much larger than the inter-droplet distances and lead to droplet number density fluctuations. In contrast, small scale mixing at length scales ranging from scales of the order of the droplet diameter and Kolmogorov scales to scales proportional to inter-droplet spacing determines the combustible mixture preparation [67, 132]. Computational fluid dynamics (CFD) is one of the most advanced tools for improve-ments of combustion devices. In recent decades, it is widely recognized that large eddy simulation (LES) holds the largest potential to predict turbulent combustion efficiently and accurately [59, 97, 102, 135]. The sub-grid closures for the mixing of fuel vapour and oxidizer severely affects the accuracy of LES of turbulent spray
1.2. STATE OF THE ART 2
combustion. Most of the relevant studies are limited to the dilute sprays where the sub-grid models for mixing are based on so-called single-droplet models [2,105,121] and existing strategies for diffusion single-phase combustion [14]. For more dense sprays, the combustion and evaporation of different droplets interact and are addi-tionally affected by turbulence. These interactions can strongly influence the mixing characteristics in inter-droplet space and are highly likely not to be negligible. More advanced sub-grid closures for mixing are needed in less dilute sprays which may affect the modelling of chemical reactions, such as the formation of soot and other pollutants, e.g. poly-cyclic aromatic hydrocarbons (PCAH) and NOx [67]. Exper-imental measurements in turbulent spray flames do not resolve the details of the interactions between droplet evaporation, combustion and turbulence that occur at the micro scale in the immediate vicinity of the droplets [78, 84]. This study uses direct numerical simulation (DNS) to generate the database which then aids the development of sub-grid closures for the small scale mixing in LES. The DNS will allow a quantification of different regions of different mixing characteristics for the first time and the -to date- unknown effects of the determining parameters on the characteristic mixing quantities can be evaluated.
1.2
State of the Art
An Euler-Lagrange method is widely implemented in commercial CFD and open source codes to simulate the turbulent droplet laden flow. The gas phase is solved efficiently and accurately on an Eulerian mesh in the large eddy simulation (LES) contexts while the droplets are modelled as Lagrangian source point particles [16, 37, 109]. The so-called single droplet evaporation model is usually employed for modelling droplet evaporation where the inter-phase heat and mass transfer can be computed with the aid of the Spalding transfer number [2,121] and the Nusselt and Sherwood numbers [42]. The modelling of droplet-turbulence dynamic interactions such as droplet dispersion and the modifications of the turbulence properties by the liquid phase can be described by stochastic separated flow model (SSF) [4, 32, 43]. For reacting flows, combustion sub-models are required. LES coupled with the linear eddy model (LEM) [65, 85], eddy break-up model (EBU) [70, 122] or thick-ened flame model (TFM) [18,28] were applied to turbulent spray combustion in early studies. Since an accurate prediction of the oxidation of a complex fuel tends to involve hundreds even thousands of species and elementary reactions, it is extremely costly to solve all the transport equations in LES of turbulent combustion [76, 95].
Mixture fraction based combustion models such as flamelet [94,98] and conditional momentum closure (CMC) [12, 67, 89] were derived and further applied to spray combustion [47, 87, 92]. Mixture fraction is a conserved (non-reacting) scalar that describes the mixing of the fuel and oxidizer stream and the reactive scalars are strongly correlated with mixture fraction [8,10]. Flamelet and CMC models require the sub-grid closures for characteristic mixing quantities such as mixture fraction conditional scalar dissipation and its probability density function (PDF). It is im-portant to note here that spray combustion typically occurs in the non-premixed mode, i.e. the reaction is largely a mixing controlled process between fuel and oxi-dizer and the accurate modelling of scalar dissipation and PDF is a prerequisite for predictions of the combustion process. For instance, the basic CMC equation for the mixture fraction (f ) conditional species (Qi = hYi | f i) is given by [12, 66]
∂Qi ∂t = −huj | f i ∂Qi ∂xj + hNf | f i ∂2Q i ∂f2 − 1 ρPf ∂ ∂xj (hu00iYi00 | f iρPf)) + hωi | f i, (1.1)
where huj | f i, hu00iYi00 | f i and hωi | f i are the conditional velocity, conditional
turbulent flux and conditional chemical source term, respectively, and they can be modelled in various ways using the information from the fluid and scalar fields [67]. Here, the modelling of the closures for conditional scalar dissipation, hNf | f i
and the mixture fraction PDF, Pf, are on focus. In a sufficiently dilute spray, the
assumption of single droplet evaporation and combustion can be invoked and the local flame surrounding the droplet is similar to a counterflow diffusion flame. Scalar dissipation and PDF can be typically modelled by closures that have been derived and tested for single phase non-premixed combustion (see e.g. [9,21,67,68,91,96]). For instance, the conditional moment hNf | f i in one CMC cell is computed by the
statistics of filtered scalar dissipation eNf from the LES cells. The filtered scalar
dissipation, eNf, in one LES cell is modelled by a sum of the resolved and sub-grid
components e Nf = (D + CNDt) ∇ ef 2 , (1.2)
where CN is a modelling constant and Dt is the turbulent diffusivity that depends
on the turbulent models. The PDF is commonly presumed by a β- PDF or clipped Gaussian function. Some studies suggested modifications for the standard β-PDF for turbulent spray combustion modeling [48, 120]. This modelling strategy for scalar dissipation and PDF has been validated for the LES-flamelet and LES-CMC computations of dilute sprays [5, 19, 27, 31, 38, 60, 82, 107, 128, 129, 131]. How-ever, the omission of the effect of evaporation on the mixing is highly questionable
1.2. STATE OF THE ART 4
in moderately dense sprays. Here, the droplet collisions are neglected correspond-ing to the definitions of the term ”dense” in [39, 56, 64] where the spray volume fraction is of the order of or larger than 10−3. The conditions can be found e.g. in internal combustion engines or gas turbines after jet break-up. The combus-tion and evaporacombus-tion of different droplets interact and are addicombus-tionally affected by turbulence. These interactions can strongly influence the mixing characteristics in inter-droplet space and thus combustion. One fact is that group combustion rather than single droplet combustion is a predominant combustion mode in practical ap-plications [58, 81]. The similarity to counterflow diffusion flames is compromised and the current sub-grid mixing models cannot generally be applied to dense spray combustion.
The necessary details of experimental data - i.e. high resolution, instantaneous and simultaneous measurements of all major gas phase species, droplet size, velocity and temperature - are not yet available for spray flames. Measurements in turbulent spray flames do not resolve the details of the interactions between droplet evapo-ration, combustion and turbulence that occur at the micro scale in the immediate vicinity of the droplets [78, 84]. Some relevant experimental studies were limited to explore the correlations between the droplet evaporation rate and turbulence inten-sity [17, 53] and characterize the structure of evaporating vapour fields and flames in moderately dense sprays [71, 136].
At present, theoretical analysis and direct numerical simulations (DNS) are needed to elucidate the scalar mixing in inter-droplet space [87, 118]. Particle-turbulence interactions are frequently studied (e.g. [40, 45, 79]) but evaporation and combustion are usually omitted. Chiu et al. [24, 25] defined the group combus-tion number to distinguish different combuscombus-tion modes in sprays, the modes being identified by a series of so-called carrier-phase DNS (CP-DNS, the momentum and scalar transport are resolved but the droplets are treated as Lagrangian source point particles) of a laminar counter flame in [88, 90]. Most of the CP-DNS studies fo-cused on spray evaporation and combustion in computational configurations such as shear layers [86, 113, 154], isotropically decaying turbulence [104, 105, 114, 146] and spatially decaying turbulence [35, 93, 140]. The DNS data were collected “as is” and some of the studies used these data to evaluate the validity of the existing sub-models for conditional mixing statistics which are commonly employed for non-premixed single phase combustion [114, 123, 146]. All of them neglect, however, the effects of locally inhomogeneous fields and small scale mixing at length scales ranging from the droplet diameter to the inter-droplet spacing. The interpretation
of these effects may initiate more physical sub-grid mixing models for LES or RANS of turbulent spray combustion [13, 60, 118, 128]. Shinjo et al. [115] demonstrated for quite realistic turbulent configurations that droplets with sizes comparable to the turbulence scales affect the mixing characteristics. However, simple configura-tions such as droplet arrays are needed to improve our understanding of the more complex interactions between turbulence, combustion, evaporation and mixing in inter-droplet space. Imaoka and Sirignano [55, 56, 57] performed DNS where the interface between two phases is fully resolved and all relevant processes such as tem-perature and species gradients, evaporation, mixture preparation and combustion are directly simulated. They investigated the evaporation rate of droplet arrays with varying droplet number densities in a quiescent environment and found the inner droplets of a 53-matrix evaporate around 5000 times more slowly due to local cooling and local saturation of the surrounding mixture. Similar conclusions hold for earlier studies by Stapf et al. [124]. They reported a strong dependence of the evaporation rate and of the species concentrations on the position within the droplet cloud. Wu and Sirignano [150, 151, 152] and Zoby et al. [158] extended a similar analysis to convective environments. They studied effects of droplet spacing, relative velocity and ambient pressure on flame structure transition as well as surface temperature and burning rates of droplets. Cho et al. [26] used a similar method to study the effect of oxygen concentration and geometrical arrangements of droplets on burn-ing characteristics in the RANS context. However, the instantaneous interactions between droplets and turbulence were not captured.
The mathematical descriptions of the fuel vapour mass fraction or mixture frac-tion distribufrac-tion around an isolated droplet have been derived and reviewed for decades in some seminal papers [42, 75, 117, 149]. Bilger [13] extended the numer-ical analysis and derived new formulae for the mixture fraction conditional scalar dissipation and its PDF for isolated quiescent droplets which may be of interest for spray modelling. However, the interactions between the mixing fields of different droplets and the additional effect of turbulent convection present in most practical spray combustion were neglected. The scaling laws derived by Klimenko and Bilger [67] provide an alternative description of the mixing fields that may hold in cases where the mixing fields of multiple droplets interact. Following their ideas, two different zones are distinguished in inter-droplet space: a near droplet zone (NDZ) that consists of the boundary layer surrounding the droplet and a quasi-laminar wake region behind the droplet and the Kolmogorov scale zone (KSZ) that is lo-cated beyond the near droplet zone. An important difference between the two zones
1.3. OBJECTIVES 6
is the relative importance of the turbulence and thus, the determining parameters for fuel mixing in the zones: the small scale turbulent structures tend to be larger than the quasi-laminar wake region, they move the wake around but do not enhance mixing within this wake. Mixing in the near droplet zone is therefore determined by diffusion while turbulence dominates mixing in the Kolmogorov scale zone. Zoby et al. [159] used fully resolved DNS to investigate droplet evaporation in turbulent convective flows and provided an evaluation for the scaling laws for mixture fraction conditional scalar dissipation and its PDF in the near droplet zone. The near droplet zone is limited to a thin wake in the immediate neighbourhood of the droplets and does not dominate the entire statistics of mixture fraction space [144]. The mixing in the zone dominated by turbulence may significantly affect the estimate of the sub-grid scalar dissipation and PDF within one LES cell.
1.3
Objectives
The primary objective of the this study is to characterize the effects of turbulence and evaporation on mixing quantities such as mixture fraction scalar dissipation and the mixture fraction PDF in moderately dense sprays. The relevant scientific questions can be expressed as follows:
• What are the determining quantities for mixing in the Kolmogorov scale zone and how does the evaporation and turbulence affect the scalar dissipation? • Is there any mathematical correlation between the PDF and scalar dissipation
in inter-droplet space?
• Where is the location of the transition between the near droplet zone and Kolmogorov scale zone?
• Are the scaling laws valid in the fully resolved droplet arrays and what are the limitations?
• What are the similarities and differences between the fully resolved DNS and CP-DNS?
• How to use the CP-DNS data to evaluate the scaling laws?
• Complement the derivations of the scaling laws from [67] for the region domi-nated by turbulence [143].
• Use fully resolved DNS to investigate droplet array evaporation and combus-tion in convective turbulent flows and generate the DNS database for a range of investigated parameters [142, 143, 145].
• Quantify the transition location between the near droplet zone and Kol-mogorov scale zone, and suggest suitable fitting functions for unknown scaling constants in the scaling laws [145].
• Use CP-DNS where droplets move freely subject to turbulent advection to evaluate the validity of the scaling laws in the Kolmogorov scale zone for a wider range of investigated parameters and for inhomogeneous distributions of droplet characteristics [144].
1.4
Thesis Outline
Chapter2reviews the basic concepts of the key parameters determining the mixing. The characteristics of turbulent flows are described at first. A brief introduction of droplet evaporation is followed. Finally, the characteristic mixing quantities are discussed and the scaling laws for mixing in inter-droplet space are listed.
Chapter 3 presents the detailed derivations of the scaling laws for spatial distri-butions of mixture fraction, the scalar dissipation and PDF in the Kolmogorov scale zone. When droplets move relative to the surrounding turbulent flow, a wake-like structure develops. The Kolmogorov scale zone (KSZ) is located beyond the quasi-laminar region (near droplet zone) where the diffusion effect is weak and turbulence determines the mixing. The determining parameters are the droplet evaporation rate, turbulent energy dissipation and the relative velocity between the droplet and its surrounding phase. The mixture fraction conditional scalar dissipation is derived by applying non-dimensional analysis (Π-theorem). The mixture fraction PDF is obtained by a quasi-steady-state relationship with the scalar dissipation. This re-lationship is derived by the integral of continuity and mixture fraction transport equations for the droplet evaporation in a spherically symmetric coordinate system using Robin boundary conditions at the droplet surface.
Chapter 4 introduces the computational frameworks of fully resolved DNS and CP-DNS. Fully resolved DNS will form the basis for the detailed analysis of all essential local interactions within the gas phase including the boundary layer at
1.4. THESIS OUTLINE 8
the droplet surface and the smallest turbulence scales. The standard conservation equations for mass, momentum, energy, species and an additional transport equation for the conserved scalar mixture fraction are solved simultaneously for the gas and liquid phases. They are coupled by boundary conditions at the droplet surface including the mass (flux) conservation, heat conduction and species diffusion to ensure the correct coupling. However, the random character of the droplet position is neglected in the fully resolved DNS and the CP-DNS can be used to overcome this limitation. The standard governing equations in the fully resolved DNS are extended by the Lagrangian formulation where the droplets are treated as point sources of mass, momentum and energy in the CP-DNS. The inflow turbulence generation method and the discretization schemes for solving the governing equations are also presented.
Chapter5presents the results of fully resolved DNS of droplet array evaporation and combustion in turbulent convective flows and modelling for mixing fields in inter-droplet space. A validation of the necessary mesh resolution and domain size is performed to ensure a good balance between computational cost and solution accuracy. The newly derived scaling laws are tested for the non-reacting cases under different turbulent intensities. The analysis is then extended to the reacting flows with a wider range of turbulent scales and investigated parameters. The transition between the near droplet zone and the Kolmogorov scale zone can be determined by the evaluation of variation of mixture fraction in axial and radial directions in inter-droplet space. After the Kolmogorov scale zone is located, the scaling laws for mixture fraction conditional dissipation and its PDF are assessed. Suitable modelling constants in the scaling laws are extracted from the DNS and functional dependencies of the parameters are suggested.
In Chapter6, spray evaporation in spatially decaying turbulence is simulated by a series of CP-DNS. The CP-DNS overcomes a limitation of fully resolved DNS of regular droplet arrays where the random character of droplet position is neglected. The CP-DNS also covers a much wider parameter range than that was used to calibrate scaling laws in fully resolved DNS. The error introduced by the CP-DNS method is estimated by a comparison of the CP-DNS with results from fully resolved DNS as conducted in Chapter 5. It is concluded that CP-DNS data needs to be selected from beyond the insufficiently resolved quasi-laminar wake. The scaling laws are assessed by comparison with the CP-DNS statistics of the characteristic mixing quantities in the region where the small scale interactions between turbulence and evaporation determine the mixing.
In Chapter 7, carrier-phase direct numerical simulations (CP-DNS) are per-formed for turbulent spray combustion. The turbulent spray flame is statistically stabilized in the domain by dynamically controlling the mean velocity of the turbu-lent inflow and the fuel injection rate. Following the conclusions from Chapter 6for purely evaporating sprays, the DNS statistics are still taken beyond the unresolved quasi-laminar wake of the droplet for the assessment of the scaling laws for turbulent micro-mixing in spray combustion.
Scalar Mixing in Turbulent Sprays
The basic concepts on the mixing modelling in inter-droplet space are reviewed in this chapter. The characteristics of turbulent flows such as turbulent length scales, turbulent fluctuation, turbulent kinetic energy and its dissipation are described at first. A brief introduction of droplet evaporation is followed in the second section and the derivation of a single droplet evaporation model is also reviewed there. Then, the definitions of mixture fraction, scalar dissipation and PDF in inter-droplet space are discussed, and the key parameters involving the mixing are illustrated. In the last section, the existing scaling laws for the near droplet zone and the newly derived scaling laws for the Kolmogorov scale zone are listed.2.1
Turbulent Flows
The computational frameworks with different degrees of turbulence modelling are used to simulate turbulent flows. In the context of large eddy simulation (LES), large turbulent scales containing the most energetic eddies are resolved but small scale fluctuations are filtered and treated with models. Direct numerical simula-tion is required to resolve all the flow scales ranging from the Kolmogorov length scale and even smaller scales, e.g. the boundary layer at the droplet surface in the present study, to turbulent integral length scales. This leads to extremely high computational cost and DNS is usually limited to low Reynolds number flows (Re ∼ 100). However, the ability of current DNS has been demonstrated to present the LES(RANS) sub-grid scale velocity and scalar fields in various turbulent non-reacting or non-reacting flows [134,138]. Considerable simplifications can be formulated if a turbulent flow reaches a statistically stationary state after a sufficiently long transient period [101]. The turbulent velocity can be decomposed into a constant
2.1. TURBULENT FLOWS 12
temporally independent mean velocity and statistically isotropic velocity fluctua-tions. The average magnitude of turbulent fluctuation, U0, is characterized by the turbulent kinetic energy which is given by [22,101, 106, 126]
U0 = r
2k
3 . (2.1)
The turbulent kinetic energy, k, is supplied at the large turbulent scales by gradients of the mean velocity, followed by energy decay due to the break-up of large eddies into smaller ones, until the energy is finally dissipated into heat at the smallest (Kolmogorov) scales. This process is known as the principle of the energy cascade [106]. The large eddy length scale, lt, is of the order of the physical width of the
flow and its more rigorous definition is the integral length scale. For an isotropic turbulent flow, the integral length scale can be calculated by [22, 101]
lt=
Z ∞
0
Rii(rii) drii, (2.2)
where Rii is the two-point spatial correlation tensor and rii denotes the distance
of two different locations in the flow field. The two-point correlation is defined as [22, 101]
Rii=
u0i(x) u0i(x + rii)
u0i2
, (2.3)
where u0i(x) represents the fluctuating velocity at a location, x and rii denotes a
vector connecting two points. The large eddy (turn-over) time scale, which is also known as the time scale for the turbulent energy cascading from the large to the small eddy, is defined as [22, 101, 106,126]
τt=
lt
U0. (2.4)
The rate of energy cascading is equal to the rate of energy that is dissipated into heat at the smallest scales. This rate is known as the turbulent kinetic energy dissipation rate, ε, which can be estimated as being of the order of k/τt and expressed as
[22, 101, 106, 126]
ε ∼ U
03
lt
Note that the energy dissipation rate can be directly computed in DNS of reacting flows as [23] ε ≡ 1^ ρτij ∂u0i ∂xj . (2.6)
Here, τij denotes the viscous stress of a Newtonian fluid that is given by [22,101,126]
τij = µ ∂ui ∂xj +∂uj ∂xi − 2 3µ ∂uk ∂xk δij, (2.7)
where µ denotes the dynamic viscosity, δij represents the Kronecker delta that is
written as [101] δij = 1, if i = j, 0, if i 6= j. (2.8)
According to Kolmogorov’s first similarity hypothesis [69], the smallest (Kol-mogorov) length (η) and time (τη) scales can be uniquely determined by the viscosity,
ν, and energy dissipation, ε, viz.
η ≡ ν3/ε1/4, (2.9)
τη ≡ (ν/ε)1/2. (2.10)
2.2
Droplet Evaporation
Turbulent spray combustion is the predominant combustion mode in industrial ap-plications such as diesel and gas turbine engines. The liquid fuel is injected from a nozzle into a usually turbulent environment at high speed leading to a rapid dis-integration of the spray into droplets and subsequent evaporation [77, 127]. We focus on the subsequent droplet evaporation and the mixing between fuel vapour and surrounding gas phase.
The widely accepted simple single droplet evaporation model was derived and reviewed for decades in some classical literatures [2,42,121]. Here, a brief derivation based on fuel species diffusion will be presented. A more in-depth description can be found in [73,127]. Not that the model is employed for modelling the evaporation of Lagrangian point source particles in the carrier-phase DNS of spray combustion in Chapters6 and 7. Before the derivation, some rational assumptions are listed as
2.2. DROPLET EVAPORATION 14
follows:
• The fuel is single-component.
• The droplet is kept spherical and stagnant during evaporation in a quiescent infinite environment. Hence, a one-dimensional spherically symmetric coordi-nate system can be assumed.
• Gravity, other forces and radiation are neglected.
• The droplet temperature is uniform.
• The surrounding gas cannot be dissolved in the liquid fuel.
• Lewis number is assumed unity.
• The thermodynamic properties of the gas mixture are constant.
According to Fick’s law of diffusion, the net diffusion of fuel vapour in a spheri-cally symmetric coordinate system can be written as
˙
m00 = YFm˙
00
− ρDdYF
dr , (2.11)
where ˙m00 is the mass flux of evaporation, r is the radial distance from the center of droplet, YF is the fuel vapour mass fraction, ρ and D are gas density and diffusivity,
respectively. The overall mass conservation in the system is expressed as
Jm= 4πr2m˙
00
= constant, (2.12)
where Jmis the droplet evaporation rate. Rearrange Eq. (2.11) with the substitution
of Eq. (2.12) yields 1 r2dr = − 4πρD Jm 1 1 − YF dYF. (2.13)
The respective boundary conditions at the droplet surface and infinity are given by
YF(r = rs) = YF,s, (2.14)
YF(r = ∞) = YF,∞, (2.15)
