TH `ESE pr´esent´ee pour obtenir

Nod’Ordre : 2337

TH `

ESE

pr´esent´ee pour obtenir

LE TITRE DE DOCTEUR

DE L’INSTITUT NATIONAL POLYTECHNIQUE

DE TOULOUSE

´

Ecole doctorale : TyFEP Sp´ecialit´e : Dynamique des Fluides

Par M. Gabriel Staffelbach

Simulation aux grandes ´echelles et analyse acoustique de turbines

`a gaz industrielles multi-bruleurs

Soutenue le 12 Mai 2006 devant le jury compos´e de Denis Veynante Rapporteur

Pierre Comte Rapporteur

Epaminondas Mastorakos Examinateur Sebastien Ducruix Examinateur

Jim Kok Examinateur

Peter Kaufmann Examinateur Thierry Poinsot Directeur de th`ese

R´esum´e

Des mesures de plus en plus restrictives sur les ´emissions polluantes poussent les constructeurs de turbines `a gaz `a d´evelopper de nouvelles technologies ainsi qu’`a faire travailler les machines dans des conditions op´erationnelles in´edites. Il arrive que ces conditions op´erationnelles pro-duisent des effets ind´esirables qui d´eclenchent des instabilit´es de combustion. Seuls des tests pouss´es depuis la phase de conception `a la mise au point des derniers r´eglages permettent de pr´evoir ces instabilit´es. Mais ces tests sont quasiment impossibles sur une vraie turbine et sou-vent une version simplifi´ee est utilis´ee pour les essais exp´erimentaux ainsi que dans les ´etudes num´eriques. Une simplification habituelle est alors de n’´etudier que l’un des brˆuleurs de la turbine compl`ete (qui peut en compter plusieurs dizaines) et d’extrapoler ces r´esultats. Cette d´emarche rend impossible l’´etude de deux m´ecanismes: l’interaction entre brˆuleurs voisins et le d´eveloppement de modes acoustiques azimutaux dans des chambres annulaires. Dans cette th`ese nous utilisons la Simulation aux Grandes Echelles (SGE) pour ´etudier ces deux effets et de fac¸on plus g´en´erale pour comprendre la structure des flammes dans ces foyers et leur stabil-isation.

Premi`erement la stabilisation de flamme `a l’aide de flammes pilote est ´etudi´e dans une con-figuation de laboratoire. Apr`es, l’influence de modes azimutaux sur des turbines `a gaz annu-laires est ´evalu´ee en utilisant une m´ethode num´erique d´evelopp´ee pendant cette th`ese qui permet d’´evaluer l’impact de ces modes sur une turbine en n’´etudiant que un seul brˆuleur. Puis une SGE d’un brˆuleur triple est pr´esent´ee. L’impact des brˆuleurs lat´eraux sur le brˆuleur central est ´evalu´e. En dernier, la faisabilit´e d’une simulation compl`ete de chambre annulaire est d´emontr´ee.

MOTS CLES: Simulation aux Grandes Echelles, SGE, Combustion partiellement pr´em´elang´ee, Interaction Brˆuleur/Brˆuleur, Modes acoustiques azimutaux, R´eponse de flamme.

Pollutant emissions restrictions have driven gas turbine manufacturers to employ new technolo-gies and to operate these systems in extreme operating conditions. These operating conditions produce in some cases combustion instabilities which can have dramatic effects for the turbine. Extended experimental and numerical studies are then required to analyze the possible behavior of the end design. Unfortunately tests on the real set-up are not possible and simplified cases are used. Although gas turbines can contain up to 30 burners blowing into the same annular chamber, tests in laboratories are often performed on one single burner. This simplification obviously suppresses two mechanisms: burner/burner interaction and the possibility of acoustic azimuthal modes of the full annular chamber.

The objective of this thesis is to use Large Eddy Simulation (LES) to investigate whether these mechanisms are important or not. An additional issue considered is the effect of the pilot fuel injection on flame stability.

First, the influence of a pilot flame on the flame stabilization of a laboratory scale burner is stud-ied. Then, using a numerical method developed during this thesis, the impact of an azimuthal mode on a turbine is analyzed using single burner LES. A LES of a triple burner configuration is presented and the impact of the side burners on the central burner analyzed. To finish, the feasibility of full chamber LES is demonstrated.

Keywords: Large-Eddy Simulation, Turbulent lean partially premixed combustion, Com-bustion instabilities, Burner-burner interaction,Azimuthal acoustic modes, Flame response.

Note sur la forme du manuscrit

Afin de faciliter la relecture de ce manuscrit au membres du jury et aussi pour faciliter la r´edaction, ce document est r´edig´e en majeure partie en anglais. Pour des raisons d’accessibilit´e, une r´esum´e ´etendu est pr´esent´e en franc¸ais dans l’annexe. Les articles publi´es ou en attente de publication sont ´egalement inclus dans l’annexe .

Contents

Introduction

13

1 Numerical simulation for industrial gas turbines 19

1.1 Numerical tools for unsteady combustion in gas turbine burners . . . 19

1.2 Instantaneous Navier Stokes equations . . . 20

1.2.1 The governing equations . . . 20

1.2.2 Thermodynamical variables . . . 22

1.2.3 The equation of state . . . 23

1.2.4 Conservation of Mass: Species diffusion flux . . . 23

1.2.5 Viscous stress tensor . . . 24

1.2.6 Heat flux vector . . . 25

1.2.7 Transport coefficients . . . 25

1.2.8 Kinetics . . . 26

1.3 Governing equations for LES . . . 28

1.3.1 The LES Concept . . . 28

1.3.2 The Governing Equations for Non-Reacting Flows . . . 29

1.3.3 Models for the subgrid stress tensor τi jt: . . . 33

1.3.4 Modelling for reactive flows . . . 34

1.4.1 Cell-Vertex Discretization . . . 39

1.4.2 The numerical schemes . . . 42

1.4.3 Artificial Viscosity Models . . . 42

1.4.4 Boundary Conditions . . . 47

1.4.5 Non-Characteristic boundary conditions . . . 48

1.4.6 Characteristic boundary conditions . . . 48

1.5 An acoustic tool for industrial configurations . . . 54

1.5.1 Methodology: the Helmholtz equation . . . 54

1.5.2 Initial and Boundary conditions . . . 55

1.5.3 A test case: a box shaped combustion chamber . . . 56

2 Specificities of real gas turbine geometries: Implications for LES 59 2.1 Range of scales to resolve . . . 59

2.2 Combustion in industrial gas turbines . . . 60

2.2.1 Diffusion flames . . . 61

2.2.2 Perfectly/ Partially premixed combustion . . . 61

2.2.3 Pilot flames . . . 62

2.3 Acoustics . . . 62

2.3.1 Acoustics in reacting flows . . . 63

2.3.2 Flame transfer function . . . 63

2.3.3 Acoustics in an annular chamber: rotating and standing modes . . . 66

3 Influence of flame piloting 81 3.1 Simulation parameters . . . 82

3.2 The NDP1 test rig . . . 82

3.3 Results and discussion . . . 84

CONTENTS

3.3.2 Flame regimes . . . 88

3.4 Links between flame stabilization and PVC . . . 91

4 Using single burner simulations to study azimuthal modes 95 4.1 Single burner LES for studying azimuthal modes (ESBAC approach) . . . 96

4.2 Burner characteristics . . . 96

4.3 Flame response . . . 97

4.3.1 Burner type 1 . . . 98

4.3.2 Burner type 2 . . . 102

4.3.3 Comparison of type 1 and type 2 burners . . . 103

4.4 Conclusion . . . 103

5 LES of multi-burner chambers 105 5.1 Context: the EU projet DESIRE . . . 106

5.2 Study of multi-burner combustors: WP3 . . . 106

5.3 Geometry, regimes and Boundary conditions . . . 109

5.4 Cold flow results . . . 111

5.5 Reacting flow results . . . 113

5.6 Acoustic study of the set-ups . . . 117

5.7 Flame response . . . 120

5.8 Towards full burner simulations . . . 127

Conclusion

131 A Pr´esentation ´etendue de la th`ese 133 B Characteristic Wave Decomposition 141 B.1 Basic definitions . . . 141

B.2 Governing equations . . . 143 B.3 Some hints on the actual coding . . . 157 B.4 Link between AVBP formulation and original NSCBC . . . 159

C Publications 163

C.1 Large Eddy Simulation and acoustic analysis of multi-burner combustors . . . 163 C.2 Large Eddy Simulation of piloting effects on turbulent swirling flames . . . 178 C.3 HIGHLY PARALLEL LARGE EDDY SIMULATIONS OF MULTIBURNER

CONFIGURATIONS IN INDUSTRIAL GAS TURBINES . . . 194

REMERCIEMENTS

Remerciements

Je tiens `a remercier Thierry Poinsot pour son encadrement pendant ces ann´ees ainsi que pour sa confiance et sa bonne humeur. Il a pris le temps de relire et corriger les articles et ce manuscrit. Il n’a jamais d´esesp´er´e devant la tˆache ardue de corriger mon franc¸ais. De mˆeme je remercie Laurent Gicquel qui depuis le d´ebut m’a accompagn´e dans cette aventure et m’a ´egalement aid´e a parfaire mes diff´erents rapports. Merci aussi `a B´en´edicte Cuenot dont l’apport scientifique `a mon travail ainsi que son sourire permanent ont ´et´e pr´ecieux.

Je remercie les rapporteurs Denis Veynante et Pierre Comte pour avoir accept´e d’´evaluer mon travail ainsi que les autres membres du jury d’avoir accepter de se d´eplacer pour ma soutenance. Toute ma gratitude `a l’´equipe CSG (Isabelle Dast, Gerard Dejean, Fabrice Fleury, Nicolas Monnier, Patrick Laporte) pour leur aide tout au long de la th`ese et pour avoir support´e mes lubies de passionn´e d’informatique.

Merci `a Marie sans qui l’´equipe ne tournerait pas. L’humeur serait bien plus morose sans son ´energie et sa gentillesse.

Merci `a Michelle, Brigitte, Lydia et Chantal pour leur travail et leur gentillesse.

Une liste exhaustive de tous les gens que j’aimerai remercier en plus est impossible, je citerai sans ordre de pr´ef´erence:

C´eline Pri`ere qui malgr´e m’avoir support´e pendant plusieurs ann´ees continue encore `a sup-porter mon entˆetement de doctorant.

Karine Truffin pour sa gentillesse, sa bonne humeur et son aide quand je me trouvais devant un dilemme insolvable.

Ekaterina Makarova et Christophe Braga qui ont support´e mes d´elires de combustioniste au del`a du devoir et de l’amiti´e.

Antoine Dauptain sans qui ma th`ese n’aurait pas eu le mˆeme goˆut. Yannick Sommerer qui m’a soutenu `a chaque baisse de forme.

El´eonore Riber dont les coups de gueule et le sourire ont illumin´e des journ´ees sombres. Ma famille qui m’a permis d’effectuer ces ´etudes et m’a soutenu tout le long.

Merci `a Isaac Asimov et Terry Pratchett. Avec qui j’ai pass´e bien des soir´ees et m’ont permis de me d´etendre et d’appr´ecier la science.

Introduction

Context: energy production by combustion

The potential for further development of any country is directly linked to its energy production capabilities. Industry as well as individuals require ever increasing amounts of power. On all accounts, electricity is the favored form of energy because it can be easily transformed into any other kind of power (mechanical, thermal, etc ...). Electricity production methods are very diverse, from power dams to solar cells. Although it is universally agreed that fossil fuels will not last forever, they are still the dominant form of raw energy and combustion still provides close to 80 percent of the energy consumed on earth. At the current time no single practical alternative to combustion exists:

• Hydro electric power ( river or ocean dams, etc .. ) is reaching its maximum deployment. • Solar power efficiency is still very low to be commercially viable.

• The increasing technical and public relation problems that arise from nuclear fission have out-weighted its efficiency advantage. Furthermore, the development of multiple power plants at a fast pace is impossible in practice.

• Nuclear fusion is still a theoretical concept.

• Wind power can not be used as the main source of electricity because of its intermittent nature.

• Geothermal power requires specific geological locations and scientists fear for long term effects of the crust cooling (increased earthquakes and eruptions).

Burning fossil fuels is therefore difficult to avoid and the key question is to burn these fuels with high efficiencies and reduced pollutant levels. Presently many new large electric power-plants use gas turbines and burn natural gas. Even though it is one of the most efficient power sources

(up to 57% efficiency for present designs), increasingly stringent legislations have forced man-ufacturers to increase the turn-over of the research and development phases in order to have cleaner, cheaper and more efficient designs. Empirical formulas and experimental studies were used in the past for the design phase. Today, numerical simulation, in combination with these tools, is widely used for research and designs purposes.

Simulation methods: status of classical codes and limitations

Numerical studies are more and more used in industry to understand the operation of gas tur-bines. All gas turbine companies use commercial and in-house codes to predict the mean flow fields found for cold and reacting flows within gas turbines and these codes are essential building blocks for design. The majority of these codes use RANS (Reynolds Averaged Navier Stokes) equations [46, 77, 69] where only the averaged equations of motion are solved for. Even though this approach has been used for a long time, it is reaching limits now for the following reasons:

• RANS submodels for turbulent combustion are difficult to derive and often lack accuracy. Despite the numerous efforts devoted to improving RANS submodels for chemistry and flame/turbulence interaction (see for example the Symposiums on Combustion, Combus-tion and Flame in the last 20 years or typical reviews in [8, 39, 50, 69]), it is fair to say that RANS is not delivering the precision which was expected.

• RANS codes cannot predict unsteady phenomena: ignition, quenching, instabilities are phenomena which cannot be captured by RANS which intrinsically averages phenomena and cannot be applied to unsteady mechanims. Moreover, the numerical solvers used for RANS are usually not adequate for unsteady computations which require dedicated solvers. Such solvers are more inspired by the DNS (Direct Numerical Simulation) com-munity than by the RANS comcom-munity [62, 64].

Now the recent needs of the end users have pushed combustor technologies to cases where these two drawbacks become significant. The averaged flow field must be known and predicted with a high precision and RANS codes are limited by the accuracy of their submodels. More-over, while pushing the combustor technologies, many unsteady phenomena appear: burners cannot be ignited anymore; they can also quench at unexpected values or start oscillating, pro-ducing undesired noise and vibration levels.

INTRODUCTION

The Large Eddy Simulation (r)evolution

Progress in computer science as well as turbulence and combustion modeling have made Large Eddy Simulation (LES) a sound alternative for numerical studies [70, 77]. The progress brought by LES for cold flows is now accepted by everyone and demonstrated in multiple studies [90, 53, 54] where LES was shown to capture not only the mean flow correctly but also the flow unsteadiness, opening new paths for numerical flow prediction and control, aeroacoustics, etc... A significant part of this progress was made possible by advances in DNS [62, 61, 81, 80]

The extension of LES to reacting flows is the focus of numerous research groups all over the world. The fact that LES works well for reacting flows is still to be demonstrated even though the few existing examples on real configuration seem to show that this is the case [14, 25, 28, 38, 43, 70, 71, 73, 88, 91, 102, 106]. Having LES models which work for all flame regimes for example is still a difficult task. But even before reaching this stage, LES has already demonstrated a potential, not only to predict mean flows but also to address questions linked to unsteadiness. There is no discussion that these methods represent the future of CFD in combustion. Research on RANS modeling has declined dramatically in the last five years while LES studies grow very rapidly. New experiments are being developed to validate LES and industry is turning more and more towards LES methodologies.

The limits of LES

Having said and proved in a few cases that LES is the best method for the future is insufficient. Today LES faces significant challenges before becoming the reference method for combustion: • First, all ’academic’ problems related to the interaction of flames and turbulence must be solved and the large number of papers on LES approaches for turbulent combustion shows that this is not yet the case. These questions are not the main focus of the present work and are discussed in multiple research groups over the world.

• Second, going from laboratory-scale experiments to real industrial gas turbines raises a series of additional questions which are extremely difficult to solve and could prove to be bottlenecks of the whole approach. In other words, extrapolating LES technologies which work for small and well-defined experiments, to real configurations might be difficult. The objective of this work is to investigate these specific questions.

Reaching this objective is not only a question of computing power. It involves many aspects linked to modeling. The jump from small experimental devices to larger, real burners involve multiple new phenomena and changes which can be summarized as follows:

• First, of course, Reynolds numbers in real chambers will be much higher. In the labora-tory burner (at 1 bar) used by Roux for example [88], the Reynolds number based on the bulk velocity and the swirler diameter was 170.000. In a full V94.3 Siemens burner (at 17 bars), the Reynolds number based on the outer swirler diameter and the bulk velocity is 5.100.000. Models which proved acceptable in the first case might fail for the second one. A well known example of such difficulties is LES for aerodynamics: while LES works well to predict the flow in a boundary layer or a channel in laboratories, it still fails to predict the aerodynamics of a full aircraft simply because the jump in Reynolds number is too large.

• Second, real gas turbines use additional devices which make the combustion regime much more difficult to study. While most laboratory flames are either fully premixed [9, 41, 42] or fully non-premixed (like most flames used at Sandia for the TNF workshops; see http://www.ca.sandia.gov/TNF [18, 73]), gas turbine design often combines partially pre-mixed flames with pilot flames. These pilot flames are created by very small jets of pure fuel, thereby creating a combustion regime where all types of flamelets (partially pre-mixed and pure diffusion) are expected. More over, the pilot injectors are usually very small (a few millimeters) compared to the size of the chamber (a few meters) leading to a multiscale problem which is difficult to handle with LES.

• Third, most laboratory experiments are performed with a single burner. This burner may have a geometry which matches exactly the specifications of the burners used in the real machine. However, the real gas turbine can contain up to 24 burners arranged circumfer-entially around the turbine axis and blowing into the same annular chamber. Therefore, in the real gas turbine, burner / burner interactions are expected: flames issuing from one burner can interact with flames issuing from neighbouring burners leading to violent heat release. In the few cases where such interactions were studied, the effects were shown to be extremely strong [76, 86, 87, 116] and to lead to oscillatory modes which do not exist for a single burner. As a consequence, real gas turbine configurations can exhibit mecha-nisms such as burner / burner interactions which were unheard of in most laboratory scale experiments.

• Finally, acoustics is a critical phenomenon which scales with the configuration size and geometry. Acoustic / flame interactions have been ignored by many authors for a long time but there is a clear consensus now to say that these interactions are the source of mul-tiple problems in combustors and especially for combustion instabilities [77]. Acoustics are extremely dependent on the geometry and on the boundary conditions. Gas turbines come in many shapes. The present work focused on annular combustors in which all burners share the same combustion chamber. In this chamber many acoustic modes can develop. Some depend on only one burner but others rely on burner/burner interaction. For example, annular gas turbines exhibit azimuthal modes which are obviously never seen in single burner configurations. These modes can have frequencies of the order of

INTRODUCTION

100 Hz and interact significantly with combustion. Whether LES can handle them in a real configuration is an open question.

Objective of the present work

The objective of the present work was to push existing LES technologies to their limits by applying LES to real geometries corresponding to some of the largest gas turbines. The specific questions which were studied were the following:

• How does LES, initially developed and validated for small laboratory burners, extend to large-scale high-Reynolds gas turbines ?

• What are the mechanisms through which pilot flames in turbulent swirled burners help combustion to stabilize ?

• Is burner / burner interaction important when studying the mean flow in a gas turbine chamber ? Is it important when studying the unsteady activity ? These questions are crucial to determine whether combustion can be studied in a single burner and compared to the 24 burners of a real turbine.

• How mature is massively parallel computing for LES of combustion? Even though this question belongs more to the field of computational science and not to a Ph.D. in fluid mechanics, it is the key issue for studies of large-scale geometries. Such cases can not be computed without massively parallel machines and building LES codes for reacting flows which can exploit such architectures was a major work area during this Ph.D.

The present work was part of the European project DESIRE which includes Siemens PG, University of Twente, DLR Stuttgart, CIMNE and CERFACS. Siemens PG participated ac-tively to the project by providing the geometries and guiding the research in terms of applica-tions. Experimental results were obtained sometimes from real gas turbines (in which case the information was usually very limited) or from an experiment developed especially for the DE-SIRE project by Siemens and DLR. In particular a triple-burner set-up was built and operated in Germany. Building a 24 burner experiment is impossible: however, a triple burner case was feasible and comparing single- and triple- burner experiments was a logical first step.

All numerical studies were performed with the LES-code AVBP [64] co-developed by Cer-facs (Centre europ´een de recherche et de formation avanc ´Ee en calcul scientifique) and IFP (Institut franc¸ais du p´etrole). A Helmholtz solver developped at Cerfacs [3, 4] on the basis of

the work of C. Nottin [67] in laboratory EM2C of Ecole Centrale Paris was also used to com-pute all acoustic modes of the burners and help identify modes which appear either in the LES or in the experiment.

Chapter 1 describes the LES tool used for this study and the submodels used for all con-servation equations. Chapter 2 provides more details on the physical mechanisms which re-quire specific attention in a gas turbine: pilot flames, burner / burner interactions and azimuthal modes. This chapter presents a simple approach called ESBAC for ”Extreme Single Burners in Annular Combustor”. The ESBAC approach allows to evaluate the response of burners in a full annular combustor to an azimuthal mode by computing two single burner cases only. When the combustion chamber of an annular gas turbine is submitted to an azimuthal standing mode, cer-tain burners experience pressure oscillations only (at velocity nodes) while others are submitted to transverse velocity oscillations only (at pressure nodes). The ESBAC method proposes to study an azimuthal mode by performing LES on a single burner in this two extreme situations. Chapter 3 presents results obtained for flame piloting effects on a single burner. An analysis of the mechanism behind flame attachment with the use of pilot flames is performed.

Chapter 4 and 5 focus on burner/burner and flame/acoustic interactions in annular combustion chambers. In chapter 4 the ESBAC model is applied to evaluate a burner’s response to an az-imuthal mode and the impact of the mode in terms of heat release.

Chapter 5 presents studies of the effects of burner/burner interaction on a three-burner LES. Special interest is directed to the influence of the side burners on the central burner compared to a single periodic burner. An acoustic analysis as well as the response of the three burner set-up to acoustic excitations are performed. Finally, the possibility of extending the present work to full burner LES is demonstrated through the use of the LES code on Top500 machines, especially the Bluegene computers in the USA.

Chapter 1

Numerical simulation for industrial gas

turbines

1.1

Numerical tools for unsteady combustion in gas turbine

burners

This chapter presents the techniques used to study turbulent combustion in this thesis. Two main techniques have been used. Both start from the Navier Stokes equations but use different paths to solve them :

• The first method is Large Eddy Simulation (LES) where only small scales are modelled and both turbulent and acoustic fluctuations are accounted for.

• The second one is an acoustic Helmholtz solver which solve the linearized Navier Stokes equations in Fourier space. Turbulence is neglected.

In terms of CFD solvers, the thesis is devoted Large Eddy Simulation and the presentation is limited to this technique (no RANS methods are described).

The manuscript begins with a description of the instantaneous equations for reacting flows (section 1.2). In AVBP, in opposition to codes using passive scalar formulations of G-equations, the reaction rates are explicitly solved for. This implies that thermodynamic quantities must also be accounted for with precision to obtain final flame temperatures for example : changes of heat capacities with temperature and composition must be taken into account. This makes the solver much more complex because simple operations like finding temperature from energy are no longer explicit [64]. Moreover boundary conditions must also account for variable sound speed [64, 77].

Section 1.3 describes the filtering of these instantaneous equations. Most of these operations are standard except for the treatment of the reactive cases where the Thickened Flame model (TFLES) requires a specific description (section 1.3.4).

The cell-vertex method used to solve for the filtered equations is then presented in section 1.4 along with a presentation of the two main Schemes (Lax Wendroff and TTGC) and of the boundary condition treatment.

A second numerical approach needed to complement LES is acoustic tools. Section 1.5 shows how the linearized reactive equations are used to predict the frequencies and structure of modes appearing in complex geometries. The acoustic solver AVSP is built on the same architecture as AVBP and both codes are used simultaneously for all cases in this work.

1.2

Instantaneous Navier Stokes equations

This section describes the instantaneous compressible Navier-Stokes equations (as found in text books such as [1, 35, 77, 115]).

1.2.1

The governing equations

The set of conservation equations for reacting flows can be written: ∂ w

∂ t + ∇ · F = s (1.1)

where w is the vector of the conservative variables: w = ( ρu, ρv, ρw, ρE, ρk)T

with respectively ρ, u, v, w, E, ρk the density, the three cartesian components of the velocity vector ~V = (u, v, w)T, the energy per unit mass and ρk = ρYk for k = 1 to N ( N is the total number of species). The source term s is decomposed for convenience into a chemical source term and a radiative source term such that: s = sC+ sR.

It is usual to decompose the flux tensor F = F(w)I+F(w, ∇w)V into an inviscid and a viscous component. The three spatial components of the inviscid flux tensor F(w) are:

1.2. INSTANTANEOUS NAVIER STOKES EQUATIONS Inviscid terms:       ρ u2+ P ρ uv ρ uw ρ uv ρ v2+ P ρ vw ρ uw ρ vw ρ w2+ P

(ρE + P)u (ρE + P)v (ρE + P)w

ρku ρkv ρkw       (1.2)

where the hydrostatic pressure P is given by the equation of state for a perfect gas (Eq. 1.13). Viscous terms:

The components of the viscous flux tensor F(w, ∇w)V take the form:

      −τxx −τxy −τxz

−τxy −τyy −τyz

−τxz −τyz −τzz

−(uτxx+ vτxy+ wτxz) + qx −(uτxy+ vτyy+ wτyz) + qy −(uτxz+ vτyz+ wτzz) + qz

J_x,k J_y,k J_z,k       (1.3) J_kis the diffusive flux of species k and is presented in subsection 1.2.4 (Eq. 1.22). The stress tensor τi j is explicited in subsection 1.2.5 (Eq. 1.23). Finally, subsection 1.2.6 is devoted to the heat flux vector q (Eq. 1.26).

Although Eqs. 1.1 to 1.3 are suitably written in a symbolic form and directly used as such for the development of numerical schemes, theoretical descriptions and modelling of the physics are usually presented based on their indicial form. The equations then take the form:

∂ ρ ui ∂ t + ∂ ∂ xj (ρ uiuj) = − ∂ ∂ xj [P δi j− τi j], (1.4) ∂ ρ E ∂ t + ∂ ∂ xj (ρ E uj) = − ∂ ∂ xj [ui(P δi j− τi j) + qj] + ˙ωT + Qr, (1.5) ∂ ρk ∂ t + ∂ ∂ xj (ρkuj) = − ∂ ∂ xj [J_j,k] + ˙ωk, (1.6)

where a repeated index implies summation over this index (Einstein’s rule of summation). Note also that throughout the document, the index k is reserved to refer to the kth species and does not follow the summation rule (unless specifically mentioned).

1.2.2

Thermodynamical variables

The standard reference state used in AVBPis P0 = 1 bar and T0 = 0 K. The sensible mass en-thalpies (hs,k) and entropies (sk) for each species are tabulated for 51 values of the temperature (Tiwith i = 1...51) ranging from 0K to 5000K with a step of 100K. Therefore these variables can be evaluated by:

h_s,k(Ti) = Z T_i T0=0K C_p,kdT = h m s,k(Ti) − hms,k(T0) W_k , and (1.7) s_k(Ti) = sm_k(Ti) − sm_k(T0) W_k , with i= 1, 51 (1.8)

The superscript m corresponds to molar values. The tabulated values for hs,k(Ti) and sk(Ti) can be found in the JANAF tables [107]. With this assumption, the sensible energy for each species can be reconstructed using the following expression :

es,k(Ti) = Z T_i

T0=0K

Cv,kdT = hs,k(Ti) − rkTi i= 1, 51 (1.9)

Note that the mass heat capacities at constant pressure cp,k and volume cv,k are supposed constant between Ti and Ti+1 = Ti+ 100. They are defined as the slope of the sensible en-thalpy (Cp,k=

∂ hs,k

∂ T ) and sensible energy (Cv,k = ∂ es,k

∂ T ). The sensible energy henceforth varies continuously with the temperature and is obtained on each interval [Ti, Ti+1] by using a linear interpolation:

e_s,k(T ) = es,k(Ti) + (T − Ti)

e_s,k(Ti+1) − es,k(Ti) Ti+1− Ti

(1.10)

The sensible energy and enthalpy of the mixture may then be expressed as:

ρ es= N

∑

k=1 ρkes,k= ρ N

∑

k=1 Y_ke_s,k (1.11) ρ hs= N

∑

k=1 ρkhs,k= ρ N

∑

k=1 Y_khs,k (1.12)

1.2. INSTANTANEOUS NAVIER STOKES EQUATIONS

1.2.3

The equation of state

The equation of state for an ideal gas mixture writes:

P= ρ r T (1.13)

where r is the gas constant of the mixture dependant on time and space: r = _WR where W is the mean molecular weight of the mixture:

1 W = N

∑

k=1 Y_k W_k (1.14)

The gas constant r and the heat capacities of the gas mixture depend on the local gas composi-tion as: r=_WR = ∑Nk=1 Yk WkR = ∑ N k=1Ykrk with Cp = ∑N_k=1YkCp,k and C_v = ∑N_k=1YkCv,k (1.15)

whereR = 8.3143 J/mol.K is the universal gas constant. The adiabatic exponent for the mixture is given by γ = Cp/Cv. Thus, the gas constant, the heat capacities and the adiabatic exponent are no longer constant. Indeed, they depend on the local gas composition as expressed by the local mass fractions Y_k(x,t):

r= r(x,t), Cp= Cp(x,t), Cv= Cv(x,t), and γ = γ (x, t) (1.16) The temperature is deduced from the the sensible energy, using Eqs. 1.10 and 1.11. Finally boundary conditions make use of the speed of sound of the mixture c which is given by:

c2= γ r T (1.17)

1.2.4

Conservation of Mass: Species diffusion flux

In multi-species flows the total mass conservation implies that: N

∑

k=1

Y_kV_ik= 0 (1.18)

where V_ik are the components in directions (i=1,2,3) of the diffusion velocity of species k. They are often expressed as a function of the species gradients using the Hirschfelder Curtis approx-imation:

X_kV_ik= −D_k∂ Xk ∂ xi

, i= 1, 2, 3 (1.19)

where Xk is the molar fraction of species k : Xk = YkW/Wk. In terms of mass fraction, the approximation 1.19 may be expressed as:

Y_kV_ik= −Dk W_k W ∂ Xk ∂ xi, i= 1, 2, 3 (1.20)

Summing Eq. 1.20 aver all k’s shows that the approximation 1.20 does not necessarily com-ply with equation 1.18 that expresses mass conservation. In order to achieve this, a correction diffusion velocity ~Vc is added to the convection velocity to ensure global mass conservation (see [77]):

∑N_k=1(V_ik+V_ic) ·Yk= 0 or Vic= − ∑Nk=1Vik·Yk= ∑Nk=1DkW_Wk∂ X_{∂ x}k_i, i= 1, 2, 3 (1.21) and computing the diffusive species flux for each species k as:

J_i,k= −ρ  D_kWk W ∂ Xk ∂ xi −Yk V_ic  , i= 1, 2, 3 (1.22)

Here, Dkare the diffusion coefficients for each species k in the mixture (see subsection 1.2.7). Using Eq. 1.22 to determine the diffusive species flux implicitly verifies Eq. 1.18.

1.2.5

Viscous stress tensor

The stress tensor τ is given by the following relations: τi j= 2µ(Si j− 1 3δi jSll), i, j = 1, 3 (1.23) and S_{i j}= 1 2( ∂ uj ∂ xi +∂ ui ∂ xj ), i, j = 1, 3 (1.24)

1.2. INSTANTANEOUS NAVIER STOKES EQUATIONS

τxx=2µ₃ (2∂ u_{∂ x}−∂ v_{∂ y}−∂ w_{∂ z}), τxy= µ(∂ u_{∂ y}+∂ v_{∂ x}) τyy=2µ₃ (2∂ v_{∂ y}−∂ u_{∂ x}−∂ w_{∂ z}), τxz= µ(∂ u_{∂ z}+∂ w_{∂ x}) τzz= 2µ₃ (2_{∂ w}∂ z −∂ u_{∂ x}−∂ v_{∂ y}), τyz= µ(∂ v_{∂ z}+∂ w_{∂ y})

(1.25)

where µ is the shear viscosity (see subsection 1.2.7).

1.2.6

Heat flux vector

For multi-species flows, an additional heat flux term appears in the diffusive heat flux. This term is due to heat transport by species diffusion. The total heat flux vector then writes:

q_i= −λ ∂ T ∂ xi | {z } Heat conduction −ρ N

∑

k=1  D_kWk W ∂ Xk ∂ xi −YkVic  h_s,k | {z }

Heat flux through species diffusion

= −λ ∂ T ∂ xi + N

∑

k=1 J_i,kh_s,k (1.26)

for i = 1, 2, 3 where λ is the heat conduction coefficient of the mixture (see subsection 1.2.7).

1.2.7

Transport coefficients

In CFD codes for multi-species flows the molecular viscosity µ is often assumed to be indepen-dent of the gas composition and close to that of air1so that the classical Sutherland law can be used. In a first step we propose to make the same assumption for the multi-gas AVBP, yielding:

µ = c1 T3/2 T+ c2 ·Tre f+ c2 T_{re f}3/2 (1.27)

where c1and c2must be determined so as to fit the real viscosity of the mixture. For air at Tre f = 273 K, c1 = 1.71e-5 kg/m.s and c2= 110.4 K (see [114]). A second law is available, called Power law:

µ = c1( T T_{re f})

b _(1.28)

with b typically ranging between 0.5 and 1.0. For example b = 0.76 for air.

The heat conduction coefficient of the gas mixture can then be computed by introducing the molecular Prandtl number of the mixture as:

λ = µCp Pr

(1.29) with Pr supposed as constant in time and space.

The computation of the species diffusion coefficients Dkis a specific issue. These coefficients should be expressed as a function of the binary coefficients Di j obtained from kinetic theory (Hirschfelder et al. [36]). The mixture diffusion coefficient for species k, Dk, is computed as (Bird et al. [6]):

D_k= 1 −Yk ∑N_j6=kXj/Djk

(1.30)

The Di j are complex functions of collision integrals and thermodynamics variables. For a DNS code using complex chemistry, using Eq. 1.30 makes sense. However in most cases, DNS uses a simplified chemical scheme and modeling diffusivity in a precise way is not needed so that this approach is much less attractive. Therefore a simplified approximation is used in AVBP for Dk. The Schmidt numbers Sc,k of species are supposed to be constant so that the binary diffusion coefficient for each species is computed as:

D_k= µ ρ Sc,k

(1.31)

Note that the Schmidt number for each species k is assumed to be constant in time and space. P_rand Sc,kmodel the laminar (thermal and molecular) diffusion.

1.2.8

Kinetics

The source term on the right hand side of Eq. 1.1 writes:

sC=       Su S_v S_w ˙ ωT ˙ ωk      

where ˙ωT is the rate of heat release and ˙ωkthe reaction rate of species k. The source terms Su, S_v, Sw are used to impose source terms on momentum components, for example an imposed pressure gradient in periodic flows. In most cases however, Su= Sv= Sw= 0.

1.2. INSTANTANEOUS NAVIER STOKES EQUATIONS

The combustion model of AVBPis an Arrhenius law written for N reactantsMk and for M reactions as: N

∑

k=1 ν_{k j}0 Mk j N

∑

k=1 ν_{k j}00Mk j, j= 1, M (1.32) The reaction rate of species k ˙ωk is the sum of rates ˙ωk j produced by all M reactions:

˙ ω_k= M

∑

j=1 ˙ ω_{k j}= W_k M

∑

j=1 ν_{k j}Qj (1.33)

where νk j= νk j00 − νk j0 andQjis the rate progress of reaction j and is written:

Qj= Kf, j N

∏

k=1 (ρYk W_k ) ν_{k j}0 _{− K} r, j N

∏

k=1 (ρYk W_k ) ν_{k j}00 (1.34)

Kf, j and Kr, j are the forward and reverse rates of reaction j:

Kf, j = Af, jexp(− E_{a, j}

RT) (1.35)

where Af, j and Ea, j are the pre-exponential factor and the activation energy. Kr, j is deduced from the equilibrium assumption:

Kr, j = K_{f, j}

K_eq (1.36)

where Keqis the equilibrium constant defined by Kuo [46]: K_eq= p0 RT ∑N_k=1νk j exp ∆S 0 j R − ∆H0_j RT ! (1.37) where p0 = 1 bar. ∆H0j and ∆S0j are respectively the enthalpy (sensible + chemical) and the entropy changes for reaction j:

∆H0_j = hj(T ) − hj(0) = N

∑

k=1 νk jWk(hs,k(T ) + ∆h0f,k) (1.38) ∆S0_j = N

∑

k=1 νk jWksk(T ) (1.39)

where ∆h0_f,kis the mass enthalpy of formation of species k at temperature T0= 0 K. The heat release can then be written as:

˙ ωT = − N

∑

k=1 ˙ ωk∆h0f,k (1.40)

1.3

Governing equations for LES

1.3.1

The LES Concept

Large Eddy Simulation (LES) [80, 89], is an intermediate approach between DNS and the more classical Reynolds Averaged Navier-Stokes (RANS) methodologies. The derivation of the gov-erning equations for LES is obtained by introducing operators to be applied to the set of com-pressible Navier-Stokes equations. Unclosed terms arise from these manipulations and require models. The major differences between RANS and LES come from the operator employed in the derivation. In RANS the operation consists of a temporal or ensemble average over a set of realizations [80]. The unclosed terms are representative of the physics taking place over the entire range of frequencies present in the ensemble of realizations under consideration. In LES, the operator is a spatial filter of given size, 4, to be applied to a single realization of the stud-ied flow. Resulting from this ”spatial average” is a separation between the large (greater than the filter size) and small (smaller than the filter size) scales. The unclosed terms in LES are representative of the physics associated with small structures (with high frequencies) present in the flow. Figure 1.1 illustrates the conceptual differences between (a) RANS and (b) LES when applied to a homogeneous isotropic turbulent field.

(a) (b)

Figure 1.1: Conceptual representation of (a) RANS and (b) LES.

Due to the filtering approach, LES allows a dynamic representation of the large scale motions whose contributions are critical in complex geometries. The LES predictions of complex turbu-lent flows are henceforth closer to the physics since large scale phenomena such as large vortex

1.3. GOVERNING EQUATIONS FOR LES

shedding and acoustic waves are embedded in the set of governing equations [77]. Therefore, LES has a clear potential in predicting turbulent flows encountered in industrial applications. Such possibilities are however restricted by the hypothesis introduced while constructing LES models.

This chapter describes the equations solved for LES of reacting flows in AVBP. First, the filtered equations solved by AVBP for a turbulent non-reacting flow are described (subsection 1.3.2). Subsection 1.3.3 presents the models used for turbulent viscosity.Subsection 1.3.4 de-scribes specifically the models for flame/turbulence interactions (the TF and DTF models).

1.3.2

The Governing Equations for Non-Reacting Flows

The filtered quantity f is resolved in the numerical simulation whereas f0= f − f is the subgrid scale part due to the unresolved flow motion. For variable density ρ, a mass-weighted Favre filtering is introduced such as:

ρ ef = ρ f (1.41)

The balance equations for LES are obtained by filtering the instantaneous balance equations 1.1:

∂ w

∂ t + ∇ · F = s (1.42)

where F is the flux tensor. The flux F can be divided in three parts: the inviscid part FI, the viscous part FV and the subgrid scale turbulent part Ft:

F = FI+ FV+ Ft

The filtered source term s contains the combustion model, for the standard AVBPmodels for flame / turbulence interactions (TF and DTF), a specific description is given in subsection 1.3.4. The cut-off scale corresponds to the mesh size (implicit filtering). It is commonly assumed that the filter operator and the partial derivative commute.

Inviscid terms:

The three spatial components of the inviscid flux tensor are the same as inDNS but based on the filtered quantities:

FI =       ρu_e2+ P ρu_eev ρ_euw_e ρu_eev ρ_ev2+ P ρ_evw_e ρu_ew_e ρv_ew_e ρw_e2+ P ρ eEu_e+ P u ρ eEv_e+ P v ρ eEw_e+ P w ρ_ku_e ρ_kev ρ_kw_e       (1.43)

Viscous terms:

The components of the viscous flux tensor take the form:

FV =       −τxx −τxy −τxz

−τxy −τyy −τyz

−τxz −τyz −τzz

−(u τxx+ v τxy+ w τxz) + qx −(u τxy+ v τyy+ w τyz) + qy −(u τxz+ v τyz+ w τzz) + qz

J_x,k J_y,k J_z,k       (1.44) Filtering the balance equations leads to unclosed quantities, which need to be modeled. Subgrid scale turbulent terms:

The components of the turbulent subgrid scale flux take the form:

Ft=       −τxxt −τxyt −τxzt −τxyt −τyyt −τyzt −τxzt −τyzt −τzzt q_xt q_yt q_zt J_x,kt J_y,kt J_z,kt       (1.45)

Although Eqs. 1.42 to 1.45 are suitably written in a symbolic form and directly used as such for the development of numerical schemes, theoretical descriptions and modelling of the physics are usally presented based on their indicial form. The equations then take the form:

∂ ρu_ei ∂ t + ∂ ∂ xj (ρueiuej) = − ∂ ∂ xj [P δi j− τi j− τi jt], (1.46) ∂ ρ eE ∂ t + ∂ ∂ xj (ρ eEu_ej) = − ∂ ∂ xj [ui(P δi j− τi j) + qj+ qjt] + ˙ωT + Qr, (1.47) ∂ ρ eYk ∂ t + ∂ ∂ xj (ρ eY_k e u_j) = − ∂ ∂ xj [Jj,k+ Jj,k t ] + ˙ω_k, (1.48)

Repeated index implies the use of Einstein’s rule of summation over the subscript with the exception of index k (kth species) unless stated otherwise specifically.

The filtered viscous terms in non reactive flows

1.3. GOVERNING EQUATIONS FOR LES

• the laminar filtered stress tensorτ is given by the following relations:_e τi j = 2µ(Si j−1₃δi jSll), ≈ 2µ(eS_{i j}−1₃δi jSe_ll), (1.49) and e S_{i j} =1 2( ∂u_ej ∂ xi +∂uei ∂ xj ), (1.50)

Eq. 1.49 may also be written: τxx≈2µ₃ (2∂ue ∂ x− ∂ev ∂ y− ∂we ∂ z), τxy≈ µ( ∂eu ∂ y+ ∂ve ∂ x) τyy≈ 2µ₃ (2∂ev ∂ y− ∂eu ∂ x− ∂we ∂ z), τxz≈ µ( ∂ue ∂ z+ ∂we ∂ x) τzz≈ 2µ₃ (2∂we ∂ z − ∂ue ∂ x− ∂ev ∂ y), τyz≈ µ( ∂ev ∂ z+ ∂we ∂ y) (1.51)

• the diffusive species flux vector in non reacting flows follows: J_i,k = −ρD_kWk W ∂ Xk ∂ xi −YkVi c ≈ −ρD_kWk W ∂ eXk ∂ xi −YekVei c , (1.52)

where higher order correlations between the different variables of the expression are as-sumed negligible.

• the filtered heat flux is :

qi = −λ∂ T_{∂ x} i+ ∑ N k=1Ji,khs,k ≈ −λ∂ eT ∂ xi+ ∑ N k=1Ji,keh_s,k (1.53)

These forms assume that the spatial variations of molecular diffusion fluxes are negligible and can be modelled through simple gradient assumptions.

Subgrid scale turbulent terms for non-reacting LES

As highlighted above, filtering the transport equations yields a closure problem evidenced by the so-called ”Sub-Grid Scale” (SGS) turbulent fluxes (see Eq. 1.3.2). For the system to be solved numerically, closures need to be supplied. Details on the forms and models available in AVBPare given in this subsection.

• the Reynolds tensor is :

τi jt is modeled by:

τi jt= 2 ρ νt(eSi j− 1

3δi jSell), (1.55)

The model for the turbulent viscosity νtis explained in subsection 1.3.3. • the subgrid scale diffusive species flux vector:

J_i,kt= ρ (gu_iY_k−u_eiYek), (1.56) J_i,kt is modeled as:

J_i,kt = −ρ D_ktWk W ∂ eXk ∂ xi −Ye_kVei c,t ! , (1.57) with Dt_k= νt St_c,k (1.58)

The turbulent Schimdt number is the same for all species St_c,k= 1. • the subgrid scale heat flux vector:

qit= ρ( fuiE−ueiEe), (1.59) where e is the sensible energy. In the source code, the modelisation forq_etis written :

qit = −λt ∂ eT ∂ xi + N

∑

k=1 Ji,kteh_s,k, (1.60) with λt = µtCp P_rt . (1.61)

The turbulent Prandtl number is fixed in AVBP. Usually, P_rt = 0.9.

The correction diffusion velocities are henceforth obtained in AVBPfrom:

e V_ic+ eV_ic,t = N

∑

k=1 µ ρ Sc,k + µt ρ St_c,k ! W_k W ∂ eXk ∂ xi , (1.62)

1.3. GOVERNING EQUATIONS FOR LES

1.3.3

Models for the subgrid stress tensor τ

i jt

:

In this work, only the smagorinsky model was used, other models exist (Wall Adapting Linear Eddy (WALE) model, Filtered Smagorinsky model, Dynamic Smagorinsky model, etc ... ) but will not be detailed here.

LES models are derived on the theoretical ground that the LES filter is spatially and tempo-rally invariant. Variations in the filter size due to non-uniform meshes or moving meshes are not directly accounted for in the LES models. Change of cell topology is only accounted for through the use of the local cell volume, that is 4 = V_cell1/3. The filtered compressible Navier-Stokes equations exhibit sub-grid scale (SGS) tensors and vectors describing the interaction between the non-resolved and resolved motions. The influence of the SGS on the resolved mo-tion is taken into account in AVBP by a SGS model based on the introduction of a turbulent viscosity, νt. Such an approach assumes the effect of the SGS field on the resolved field to be purely dissipative.

The previous hypothesis is essentially valid within the cascade theory of turbulence. The notion of turbulent viscosity can therefore be introduced and yields a general model for the SGS which reads τi jt = −ρ (u_giuj−euiuej) = 2 ρ νt Sei j−1₃ τlltδi j, (1.63) with e S_{i j} = 1 2  ∂u_ei ∂ xj +∂uej ∂ xi  −1 3 ∂u_ek ∂ x_k δi j. (1.64)

In Eqn. (1.63) Ti j is the SGS tensor to be modeled, νt is the SGS turbulent viscosity,uei is the Favre filtered velocity vector (compressible flows) and eS_{i j} is the resolved strain rate tensor. A lot of LES models only differ through the estimation of νt.

Smagorinsky model

νt = (CS4)2 q

2 eSi jSei j, (1.65)

where 4 denotes the filter characteristic length (cube-root of the cell volume), CSis the model constant set to 0.18 but can vary between 0.1 and 0.18 depending on the flow configuration. The Smagorinsky model [104] was developed in the sixties and heavily tested for multiple flow configurations. This closure has the particularity of supplying the right amount of dissipation of kinetic energy in homogeneous isotropic turbulent flows. Locality is however lost and only

global quantities are maintained. It is known as being ”too dissipative” and transitioning flows are not suited for its use [89].

1.3.4

Modelling for reactive flows

The fore-mentioned methodology for LES allows to solve any given problem without further modeling provided the basic phenomena behind it has a ”big enough” scale ( larger than the filter size).

Therefore a difficult problem is encountered for LES of premixed flames: the thickness δ_l0 of a premixed flame is generally smaller than the standard mesh size ∆x commonly used as filter for LES. For this reason, the Thickened Flame (TF) model has been developed so as to resolve the flame fronts on a LES mesh. However, in turbulent flows, the interaction between turbulence and chemistry is altered: eddies smaller than F δ_l0 do not interact with the flame

any longer. As a result, the thickening of the flame reduces the ability of the vortices to wrinkle the flame front. As the flame surface is reduced, the reaction rate is underestimated. In order to correct this effect, an efficiency functionE has been developed [21] from DNS results (see Fig. 1.2) and is explained in the next subsection.

Figure 1.2: Direct Numerical Simulation of flame/turbulence interactions by Veynante ([2, 77]). Left: non thickened flame, right: thickened flame (F = 5).

1.3. GOVERNING EQUATIONS FOR LES

The combustion subgrid scale model:E

A complete description of the efficiency function can be found in [20]. The underlying model philosophy can be summarized through 3 main steps:

• The wrinkling factor of the flame surface Ξ is estimated from the flame surface density Σ, assuming an equilibrium between the turbulence and the subscale flame surface:

Ξ ' 1 +α∆e

s0_l haTis (1.66)

where haTisis the subgrid scale strain rate, ∆eis the filter size and α is a model constant. • ha_Ti_s is estimated from the filter size ∆e and the subgrid scale turbulent velocity u0_∆e: haTis= Γu0_∆e/∆e. The function Γ corresponds to the integration of the effective strain rate induced by all scales affected by the artificial thickening, i.e. between the Kolmogorov ηK and the filter ∆escales (see also [59]). Γ is written as:

Γ ∆e δ_l1 ,u 0 ∆e s0_l ! = 0.75 exp   − 1.2  u0 ∆e/s 0 l 0.3     ∆e δ_l1 2₃ (1.67)

Finally, the efficiency function is defined as the wrinkling ratio between the non-thickened reference flame and the thickened flame:

E = Ξ(δl0) Ξ(δ_l1) = 1 + αΓ∆e δ_l0, u0_∆e s0_l _u0 ∆e s0_l 1 + αΓ  ∆e δ_l1, u0_∆e s0_l _u0 ∆e s0_l (1.68)

s0_l and δ_l0are the laminar flame speed and the laminar flame thickness, respectively, when F = 1 and δ1

l =F δ 0 l .

It is possible to show thatE varies between 1 (weak turbulence) to Emax 'F2/3 _(large wrinkling at the subgrid scale). In turbulent premixed zones, the efficiency function is determined to ensure that the turbulent flame speed will be E S0_L = ST. The efficiency function is required when the vortex size r is defined by δ_l0> r > δ_lc for a real flame and by δ_l1= βF δ_l0> r > δ_lcfor a thickened flame. δ_lcis a cut-off length scale: for vortices lower than δ_lc, the flame remains unaffected. δ_lcis defined in [20], Eq. 31.

• The filter size ∆ecorresponds to the greatest scale affected by the flame thickening, that is to say δ_l1. In practice, ∆e= 10∆xwith ∆x=

3

√

voln, where voln is the cell volume. The subgrid scale turbulent velocity u0_∆

rotational of the velocity field to remove the dilatational part of the velocity which must not be counted as ”turbulence”. A Laplacian operator is directly applied to the velocity :

u0_∆

e= c2∆

3

x|∇2(∇ × u)| (1.69)

with c2≈ 2.

Estimation of the model constant α

The model constant α in Eq. 1.66 is estimated to match the asymptotic behavior of the wrinkling factor Ξ versus RMS velocity u’ for thin flames when ∆e goes to the integral length scale lt, the flame wrinkling Ξ goes to Ξmaxdefined by:

Ξmax= 1 + β u0/s0l (1.70)

with u’ the velocity at length scale lt. α is then deduced from Eq. 1.70: α = β 2 ln(2)

3cms[Re1/2_t − 1] (1.71)

where Ret=u

0_l t

ν is the turbulent Reynolds number and cms= 0.28. The reader is referred to [20] for more details.

Other forms of efficiency function have been derived by Charlette and Meneveau [16, 15].

Implementation of the standard Thickened Flame (TF) model

The filtered equations for total energy and for species (Eq. 1.42) must be modified in reactive flows when the TF (or the DTF model see below) is used. In this case, only the filtered equations for velocities (eq. 1.42) are used. For the species and energy, the filtered equations are replaced by the thickened equations as follows:

Viscous terms

• the filtered diffusive species flux vector is given by: J_i,k= −E F µ Sc,k W_k W ∂ eXk ∂ xi + ρke V_ic, (1.72) with e V_ic=E F N

∑

k=1 µ ρ Sc,k W_k W ∂ eXk ∂ xi , (1.73)

1.3. GOVERNING EQUATIONS FOR LES

• the filtered heat flux is:

qi= −E F µCp P_r ∂ eT ∂ xi + N

∑

k=1 Ji,keh_s,k, (1.74)

The source term

The filtered source term vector of Eq. 1.41 is written:

s =        0 0 0 E ˙ωT(eYk, eT) F E ˙ωk(eYk, eT) F        , (1.75)

where ˙ωT(eYk, eT) and ˙ωk(eYk, eT) are reaction rates computed with the Arrhenius expression and the filtered values of Ykand T .

Use of the TF model implies the following relation for the correction diffusion velocities:

e V_ic+ eV_ic,t = N

∑

k=1 E F µ ρ Sc,k W_k W ∂ eXk ∂ xi , (1.76)

The Dynamically Thickened Flame (DTF) model for LES

The TF model is adequate to simulate perfectly premixed flames. For partially premixed cases, this model is not suitable and must be ajusted for different reasons:

• In non reactive zones, where only mixing takes place, the molecular and thermal dif-fusions are overestimated by a factor F. In these zones, the thickening factor should be corrected to go to unity.

• In the flame zone, the thickening allows to resolve the diffusion and the source terms. Thus, the subgrid scale turbulent terms can be set to zero.

In other words, the TF model can remain unchanged in the flame zone but must be adapted outside the flame region. The DTF model has been developed to take into acount these points (Legier et al. [49]). Its application is addressed on Fig. 1.3. The thickening factor F is not

Figure 1.3: Schematic representation of the different regions found in a partially premixed flame and as defined for the DTF model.

1.4. NUMERICAL METHOD

a constant any more but it goes to Fmax in flame zones and decreases to unity in non reactive zones. This is obtained by writing:

F = 1 + (Fmax− 1)S (1.77)

whereS is a sensor depending on the local temperature and mass fractions. S = tanh(β0 Ω

Ω0

) (1.78)

where Ω is a sensor function detecting the presence of a reaction front. One possible method to construct this sensor is to use the kinetic parameters of the fuel breakdown reaction:

Ω = Yν 0 F F Y ν_O0 O exp(−Γ E_a RT) (1.79)

Even though Ω has the functional form of a reaction rate, it is not. This form is only one convenient way to identify the flame zone but other functions could be used as long as they track correctly the zones where combustion occurs.

Γ is used to start the thickening before the reaction, that is why Γ <1 (usually Γ = 0.5). The β0factor is set in AVBP (β0= 500). S varies between 0 in non reactive zones to 1 in flames.

Ω0is specified by the user by measuring it on a 1D premixed non-thickened flame. Thickening factor depending on local resolution

It is also possible to adapt the thickening factor to the local mesh spacing. This is achieved by calculatingFmax in Eq. 1.77 via the following formula:

Fmax =Nc ∆x

δ_L0 (1.80)

where Nc is the number of cells used to resolve the flame front (set by the user). Of course the proper value for δ_L0(the flame thickeness) is required.

1.4

Numerical Method

In the following section, the numerical method will be highlighted. Please note that the Ein-stein’s summation rule is no longer applied here unless stated otherwise.

1.4.1

Cell-Vertex Discretization

The flow solver used for the discretization of the governing equations is based on the “finite volume” (FV) method. There are two common techniques for implementing FV methods: the

so called cell-vertex and the cell-centered formulation. In the latter, not used in AVBP, discrete solution values are stored at the center of the control volumes (or grid cells), and neighbouring values are averaged across cell boundaries in order to calculate fluxes. The alternative cell-vertex technique, used as underlying numerical discretization method of AVBP, the discrete values of the conserved variables are stored at the cell vertices (or grid nodes). The mean values of the fluxes are then obtained by averaging along the cell edges.

Weighted Cell Residual Approach

For the description of the weighted cell-residual approach the laminar Navier-Stokes equations are considered in their conservative formulation:

∂ w

∂ t + ∇ · ~F = 0, (1.81)

where w is the vector of conserved variables and ~F is the corresponding flux tensor. For convenience, the latter is divided into an inviscid and a viscous part, ~F = ~FI(w)+ ~FV(w,~∇w). The spatial terms of the equations are then approximated in each control volume Ωjto give the residual R_Ωj = 1 V_Ωj Z ∂ Ωj ~ F ·~n dS , (1.82)

where ∂ Ωjdenotes the boundary of Ωj).

This cell-vertex approximation is readily applicable to arbitrary cell types and is hence straightforward to apply for hybrid grids. The residual (1.82) is first computed for each ele-ment by making use of a simple integration rule applied to the faces. For triangular faces, a straightforward mid-point rule is used, which is equivalent to the assumption that the individual components of the flux vary linearly on these faces. For quadrilateral faces, where the nodes may not be co-planar, in order to ensure that the integration is exact for arbitrary elements if the flux functions do indeed vary linearly, each face is divided into triangles and then integrated over the individual triangles. The flux value is then obtained from the average of four trian-gles (two divisions along the two diagonals). This so-called ’linear preservation property’ plays an important part in the algorithm for ensuring that accuracy is not lost on irregular meshes. Computationally, it is useful to write the discrete integration of (1.82) over an arbitrary cell as

R_Ωj = 1 N_dV_{Ωj i∈Ω}

∑

j

~

Fi· ~dSi, (1.83)

where ~Fiis an approximation of ~F at the nodes, Ndrepresents the number of space dimensions and {i ∈ Ωj} are the vertices of the cell. In this formulation the geometrical information has

1.4. NUMERICAL METHOD

been factored into terms ~dSithat are associated with individual nodes of the cell but not faces; ~

dS_i is merely the average of the area-weighted normals for triangulated faces with a common node i, i ∈ Ωj. Note, that for consistency one has ∑i∈ΩjdS~ i=~0. A linear preserving

approxi-mation of the divergence operator is obtained if the volume VΩj is defined consistently as

V_Ωj= 1 N_d2_i∈Ω

∑

j

~x_i· ~dS_i, (1.84)

since ∇ ·~x = Nd.

Once the cell residuals are calculated, one may then define the semi-discrete scheme dw_k dt = − 1 V_{k j|k∈Ω}

∑

j Dk_Ω jVΩjRΩj, (1.85) where Dk_Ω

j is a distribution matrix that weights the cell residual from cell center Ωj to node k

(”scatter operation”), and Vk is a ‘control volume’ associated with each node. Conservation is guaranteed if ∑k∈Ω_jDk

Ωj = I. In the present context, (1.85) is solved to obtain the steady-state

solution using explicit Euler or Runge-Kutta time–stepping.

The family of schemes of interest makes use of the following definition of the distribution matrix: Dk_Ω j = 1 n_n(I +C δ tΩj V_Ωj ~ AΩj· ~dSk), (1.86)

where nnis the number of nodes of Ωj and ~A is the Jacobian of the flux tensor. The simplest ‘central difference’ scheme is obtained by choosing C = 0 and is neutrally stable when com-bined with Runge-Kutta time-stepping. A Lax-Wendroff type scheme may also be formulated in which case C is chosen to be a constant that depends on the number of space dimensions and the type of cells used — it may be shown that this takes the simple form C = n2_v/2N_d. If one replaces the cell ‘time-step’ δ t_Ωj by a matrix Φ_Ωj with suitable properties, one may also ob-tain an SUPG-like scheme which has slightly better convergence and shock-capturing behavior, however, at some extra computational cost.

Computation of gradients

In order to recover the nodal values of the gradients ~∇w a cell approximation~∇w 

Ωj

is first calculated and then distributed to the nodes. The cell-based gradient is defined in a manner similar to the divergence (1.83) so as to be transparent to linear solution variations:

 ∂ w ∂ x  C ≈ 1 VC Z Z ∂ ΩC w ·~n∂ S (1.87)

which leads to the approximation ~_∇w Ωj = 1 V_Ωji∈Ω

∑

j widS~ i, (1.88)

A nodal approximation of the gradient is then obtained using of a volume-weighted average of the cell-based gradients:

~_∇w k= 1 V_{k j|k∈Ω}

∑

j V_Ωj(~∇w)Ωj, (1.89)

1.4.2

The numerical schemes

The details and implementation of the numerical schemes available in the code will not be detailed in the present document. Only their main characteristics are shown below:

• Lax-Wendroff [35, 47]: Finite Volume scheme, 2nd _{order in time (uses a one step} Runge-Kutta time integration), 2nd order centered in space scheme. Very fast for computations but more dissipative than the alternative.

• TTGC [21]: Finite element scheme, 3rd order in time and space. This is a very low dissipation scheme with high order accuracy, however it requires more computer power. Previous tests [101, 88, 82] have shown that the Lax-Wendroff scheme is sufficient for most tasks in LES with complex geometries. The extra expenditure in computer power involved with the use of TTGC (up to three times more) justifies its use only on specific tasks that require such a high order of accuracy.

1.4.3

Artificial Viscosity Models

Introduction

The numerical discretization methods in AVBP are spatially centered. These types of schemes are known to be naturally subject to small-scale oscillations in the vicinity of steep solution

1.4. NUMERICAL METHOD

variations. This is why it is common practice to add a so-called artificial viscosity (AV) term to the discrete equations, to avoid these spurious modes (also known as “wiggles”) and in order to smooth very strong gradients. We describe here the different AV methods used in AVBP. These AV models are characterized by the “linear preserving” property which leaves unmodified a linear solution on any type of element. The models are based on a combination of a “shock capturing” term (called 2nd order AV) and a “background dissipation” term (called 4th order AV). In AVBP, adding AV is done in two steps:

• first a sensor detects if AV is necessary, as a function of the flow characteristics,

• then a certain amount of 2nd _{and 4}th _{AV is applied, depending on the sensor value and on} user-defined parameters.

The sensors

A sensor ζΩj is a scaled parameter which is defined for every cell Ωj of the domain that takes

values from zero to one. ζ_Ωj= 0 means that the solution is well resolved and that no AV should be applied while ζ_Ωj= 1 signifies that the solution has strong local variations and that AV must be applied. This sensor is obtained by comparing different evaluations (on different stencils) of the gradient of a given scalar (pressure, total energy, mass fractions, . . . ). If these gradients are identical, then the solution is locally linear and the sensor is zero. On the contrary, if these two estimations are different, local non-linearities are present, and the sensor is activated. The key point is to find a suitable sensor-function that is non-zero only at places where stability problems occur.

Two sensors are available in AVBP: the so-called ‘Jameson-sensor’ (ζ_ΩJ

j) [37] and the

‘Colin-sensor’ (ζ_ΩC

j) [19] which is an upgrade of the previous one.

The Jameson sensor

For every cell Ωj, the Jameson cell-sensor ζ_ΩJ_j is the maximum over all cell vertices of the Jameson vertex-sensor ζ_kJ:

ζ_ΩJ_j= max k∈Ωj

ζ_kJ (1.90)

Denoting S the scalar quantity the sensor is based on (usually S is the pressure), the Jameson vertex-sensor is: ζ_kJ= |∆ k 1− ∆k2| |∆k 1| + |∆k2| + |Sk| (1.91) Where the ∆k₁and ∆k₂functions are defined as:

where a k subscript denotes cell-vertex values while Ωjis the subscript for cell-averaged values. (~∇S)kis the gradient of S at node k as computed in AVBP.

∆k₁measures the variation of S inside the cell Ωj(using only quantities defined on this cell). ∆k₂ is an estimation of the same variation but on a wider stencil (using all the neighbouring cell of the node k).

For example, on a 1D uniform mesh, of mesh size ∆x and for the cell [k∆x; (k + 1)∆x], the ∆k₁ and ∆k₂functions are estimated as follows:

∆k₁=∆x 2 S_k+1− S_k ∆x ∆ k 2= ∆x 2 S_k+1− Sk−1 2∆x (1.93)

The numerator of eq. (1.91) is then |∆k₁− ∆k₂| = ∆x 2 4 | S_k+1− 2Sk+ Sk−1 ∆x2 | = ∆x2 4 |∆ FD k,∆xS| (1.94)

∆FD_k,∆x is exactly the classical FD Laplacian operator evaluated at vertex k and of size ∆x. The Jameson sensor is thus proportional to the second derivative of S, which is zero when S is linear and which is maximum when the gradient of S varies rapidly. This is what happens for example on each side of a front or when wiggles occur.

It is important to note that this sensor is smooth: it is roughly proportional to the amplitude of the deviation from linearity.

The Colin sensor

As said above, the Jameson sensor is smooth and was initially derived for steady-state com-putations. For most unsteady turbulent computations it is however necessary to have a sharper sensor, which is very small when the flow is sufficiently resolved, and which is nearly maximum when a certain level of non-linearities occurs.

This is the aim of the so-called Colin-sensor, whose properties can be summarized as follows: • ζC

Ωj is very small when both ∆

k

1 and ∆k2 are small compared to SΩj. This corresponds to

low amplitude numerical errors (when ∆k₁and ∆k₂have opposite signs) or smooth gradients that are well resolved by the scheme (when ∆k₁and ∆k₂have the same sign).

• ζC

Ωj is small when ∆

k

1 and ∆k2have the same sign and the same order of magnitude, even if they are quite large. This corresponds to stiff gradients well resolved by the scheme. • ζC

Ωj is big when ∆

k

1and ∆k2have opposite signs and one of the two term is large compared to the other. This corresponds to a high-amplitude numerical oscillation.

• ζC

Ωjis big when either ∆

k

1or ∆k2is of the same order of magnitude as SΩj. This corresponds