TH `ESE pr´esent´ee pour obtenir

Nod’ordre: 2234

TH `

ESE

pr´esent´ee pour obtenir

LE TITRE DE DOCTEUR

DE L’INSTITUT NATIONAL POLYTECHNIQUE

DE TOULOUSE

´

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

Par M. Patrick Schmitt

Simulation aux grandes ´echelles

de la combustion ´etag´ee dans les turbines `a gaz

et son interaction stabilit´e - polluants - thermique

Soutenue le 29 Juin 2005 devant le jury compos´e de MM. Thierry Poinsot Directeur de th`ese

Johannes Janicka Rapporteur Denis Veynante Rapporteur Christian Angelberger Examinateur

Peter Flohr Examinateur

Iskender G¨okalp Examinateur

R´esum´e

La combustion partiellement pr´em´elang´ee en r´egime pauvre est utilis´ee dans les turbines `a gaz modernes afin de r´eduire les ´emissions d’oxydes d’azote. Le travail pr´esent´e propose de montrer que la simulation aux grandes ´echelles permet de pr´edire la formation de ces polluants dans un brˆuleur de turbine `a gaz. La quan-tit´e d’oxydes d’azote produite d´epend principalement des pertes thermiques, de la qualit´e du m´elange, de la combustion et de la stabilit´e thermo-acoustique de la configuration.

Un mod`ele qui suppose que le gaz est optiquement mince permet de prendre en compte le rayonnement. Une nouvelle loi de paroi thermique bas´ee sur la loi logarithmique permet de quantifier les pertes thermiques par convection. Une cin´etique chimique simple `a deux ´etapes coupl´ee `a une troisi`eme r´eaction mod´elise respectivement la combustion du m´ethane et la formation des oxydes d’azote. En-fin, le mod`ele de flamme ´epaissie, qui int`egre les effets de la turbulence, est adapt´e `a la combustion partiellement pr´em´elang´ee en r´egime pauvre.

Dans la configuration industrielle pr´esent´ee, les injections de carburant ainsi que le refroidissement par air froid sont pris en compte. Dans le cas du calcul non-r´eactif, les champs de vitesses et le m´elange sont compar´es avec succ`es aux donn´ees exp´erimentales. La position de la flamme et les champs de vitesses associ´es sont les principaux crit`eres de validation du calcul r´eactif. On montre que les pertes thermiques et les conditions aux limites acoustiques influencent grandement la pr´ediction des ´emissions d’oxydes d’azote. En particulier, une forte instabilit´e de combustion due au couplage entre l’acoustique et la formation du m´elange air-carburant est mise en ´evidence. Sans cette instabilit´e, les ´emissions d’oxydes d’azote sont r´eduites de 75%. N´egliger les pertes thermiques et le refroidissement conduit `a d´ecupler la production d’oxydes d’azote.

Discipline: Dynamique des Fluides

Mots cl´es: Simulation aux Grandes ´Echelles, Combustion pauvre partielle-ment pr´em´elang´ee, Rayonnepartielle-ment, Lois de paroi dynamiques et ther-miques, ´Emissions d’oxydes d’azote, Instabilit´es de combustion, Turbines `a gaz.

Intitul´e et adresse du laboratoire: CERFACS

42, av. G. Coriolis 31057 Toulouse Cedex 1 France

Danksagung

Mein gr¨oßter Dank gilt Thierry Poinsot, der mit seiner Zuversicht und enormen Erfahrung mir ein idealer Doktorvater war. Er ließ mir die n¨otige Freiheit, war jedoch bei Fragen immer zur Stelle und verstand es meiner Arbeit entscheindende Impulse zu geben.

Andr´e Kaufmann, Charles Martin, Eleonore Riber und Thilo Sch¨onfeld m¨ochte ich herzlich f¨ur die wissenschaftliche und auch private Unterst¨utzung danken. Desweiteren gilt mein Dank Franck Nicoud1, f¨ur die Hilfe bei den Wandfunktio-nen und Klaus-Peter Geigle2 sowie Bruno Schuermans3 f¨ur die Messdaten und Informationen zum ev7is.

Cerfacs stellt ein aussergew¨ohlich angenehmes und stimulierendes Umfeld f¨ur junge Wissenschaftler dar und ich m¨ochte speziell Laurent Benoit, Marta Garcia, Laurent Gicquel, Ghislan Lartigue, Vincent Moureau, Yannick Sommerer, Lau-rent Selle und Karine Truffin f¨ur all die Diskussionen und Hilfestellungen danken. Mein Dank gilt auch dem ganzen Sekretariat von Cerfacs, allen voran Marie Laba-dens und auch der Rechneradministration: Isabelle d’Ast, G´erard Dejean, Fabrice Fleury, Patrick Laporte und Nicolas Monnier.

Mein besonderer Dank gilt den Mitgliedern der Jury f¨ur ihre Bereitschaft meine Doktorarbait zu beurteilen. Zuallererst Denis Veynante4 und Johannes Janicka5 f¨ur die Beurteilung des Manuskripts aber auch Iskender G¨okalp6, Christian An-gelberger7 und Peter Flohr3. Letzterem m¨ochte ich auch f¨ur die Initiative zur Zusammenarbeit von Cerfacs und Alstom Power danken, die den Grundstein f¨ur diese Doktorarbeit legte.

Note sur la forme du manuscrit

Afin de faciliter la relecture de ce manuscrit au membres du jury et aussi pour fa-ciliter la r´edaction, ce document est r´edig´e en majeure partie en anglais. Pour des raisons d’accessibilit´e, l’introduction et la conclusion sont pr´esent´ees en franc¸ais dans l’annexe (B) `a partir de la page no191.

Les planches en couleur se trouvent dans l’annexe (A) `a partir de la page no179.

1_{Institut de Math´ematiques et de Mod´elisation, Universit´e de Montpellier II (F)}

2_{Institut f¨ur Verbrennungstechnik, Deutsches Zentrum f¨ur Luft- und Raumfahrt, Stuttgart (D)} 3_{Alstom Power, Baden (CH)}

4_{Laboratoire E.M2.C, ´Ecole Centrale Paris (F)}

5_{Fachgebiet Energie- und Kraftwerkstechnik, Technische Universit¨at Darmstadt (D)} 6_{Laboratorie de Combustion et Syst`emes R´eactifs, Orl´eans (F)}

7_{Institut Franc¸ais du P´etrole, Rueil-Malmaison (F)}

Extended Abstract

Modern gas turbines use turbulent lean partially premixed combustion in order to minimise nitrous oxide (NOX) emissions while ensuring flashback safety. The

Large-Eddy Simulation (LES) of such a device is the goal of this work. Focus is laid on correctly predicting the NOX emissions, which are influenced by four

factors: heat transfer, mixing quality, combustion modelling and thermo-acoustic stability.

As NOX reaction rates are strongly influenced by temperature, heat transfer by

radiation and convection is included. Radiation is predicted by a model, which assumes that the gases are optically thin. Convective heat transfer is included via a newly developed and validated wall-function approach based on the logarithmic law of the wall for temperature.

An optimised 2-step reduced chemical reaction scheme for lean methane com-bustion is presented. This scheme is used for the LES in conjunction with an additional third reaction, fitted to produce the same NOX reaction rates as in the

complete reaction mechanism. Turbulence is accounted for with the thickened flame model in a form, which is optimised for changing equivalence ratios and mesh-resolutions.

Mixing is essential not only for predicting flame stabilisation, but also for pollu-tant emissions as NOX reaction rates depend exponentially on equivalence ratio.

Therefore the full burner geometry, including 16 fuel injections is resolved in LES. Additionally, effusion cooling and film cooling is accounted for in a sim-plified manner. The non-reacting flow is extensively validated with experimental results.

As mixture-fraction fluctuations do not only arise from turbulence, but also from thermo-acoustic instabilities, care was taken to provide acoustic boundary con-ditions that come close to reality. The resulting LES shows a strong thermo-acoustic instability, comparing well with experimental observations. By making the boundaries completely anechoic it is shown that when the instability disap-pears, the NOX levels are reduced by 75%. Additionally, neglecting all heat

trans-fer, effusion and film cooling, the NOX levels are increased again by one order of

magnitude.

Keywords: Large-Eddy Simulation, Turbulent lean partially premixed com-bustion, NOX emissions, Radiation, Dynamic and thermal

wall-functions, Combustion instabilities, Gas-turbine combustion. 6

Contents

Nomenclature 9

Introduction

15

1 Large Eddy Simulation 21

1.1 Introduction . . . 21

1.2 Filtering the Navier-Stokes Equations . . . 26

1.3 Turbulence Closure . . . 29

1.4 Numerics . . . 33

1.5 Post-processing LES . . . 39

2 Wall-Layer Modelling 45 2.1 Turbulent Boundary Layers . . . 45

2.2 Modelling Strategies . . . 52

2.3 Equilibrium Law for Cell-Vertex Formulations . . . 54

3 Performance of the Wall-Function 61 3.1 A RANS Simulation using the LES Implementation . . . 61

3.2 LES of a Turbulent, Plane, Infinite Channel . . . 65

3.3 Complex Geometries . . . 81

4 Reacting Flows 83 4.1 The Multi-Species Navier-Stokes Equations . . . 83

4.2 Chemistry and Flame Properties . . . 86

4.3 A 3-step Scheme Including NONONO and COCOCO . . . . 91

5 Turbulent Combustion 99 5.1 Turbulent Combustion Regimes . . . 99

5.2 Modelling Approaches Premixed Combustion . . . 102

5.3 Application of the TFLES model . . . 107 7

6 Acoustics 113

6.1 Wave propagation . . . 113

6.2 Acoustics and Flames . . . 120

7 Non-reactive Simulations of the ev7is 123 7.1 Burner Description . . . 123

7.2 LES and Experimental Setup . . . 126

7.3 Qualitative Analysis of Unsteady Flow . . . 133

7.4 Velocity Measurements . . . 136

7.5 Mixing . . . 143

8 Reactive Simulations of the ev7is 149 8.1 Combustion Setup . . . 149

8.2 Combustion in Anechoic Chamber . . . 155

8.3 Combustion with partially reflecting outlet . . . 162

Conclusion

177 A Colour Plates 179 Precessing Vortex Cores . . . 180

Mixing . . . 181

Snapshots of the Reactive Simulations . . . 184

Phase Averages of the Flame . . . 188

B R´esum´e en franc¸ais 191 Introduction . . . 192

Conclusion . . . 197

Bibliography 199

Nomenclature

α boundary condition relax αF TFLES model constant α1...n Runge-Kutta coefficients

αst fuel staging ratio

β Reflection coefficient for

frequen-cies > α

βF constant for local thickening βj temperature exponent

∆H0_j enthalpy change for reaction j

∆S0_j entropy change for reaction j

∆t time step

∆tr time between two data samples δ t_Ωj characteristic cell time

∆y distance between boundary node and first interior node

∆ LES filter width

δ flame thickness

δν the viscous length-scale ∆c mesh spacing

δc channel half-width ∆e filter size for TFLES

δw gap between boundary node and

physical boundary

δi j Kronecker symbol δ_L0 thermal flame thickness

˙

ωk species source term (due to

com-bustion) ˙

ωT heat source term (due to

combus-tion) ˙

ωNO,pr prompt NO reaction rate

˙

ωNO,th thermal NO reaction rate

˙

ωNO total NO reaction rate

˙

ω_T,(1) fluctuating part of the heat-release

˙

mst1 fuel mass fluxes through stage 1

˙

mst2 fuel mass fluxes through stage 2 ε dissipation of turbulence kinetic

energy

ηk Kolmogorov length scale

dwwwk

dt nodal residual

Γ integration of the effective strain rate

γ ratio of specific heats

γs constant for local thickening

κ von K´arm´an constant λ thermal conductivity

λ1 thermal conductivity of the fresh gases

haTis residual strain rate L1 incoming wave

L5 outgoing wave

Qj rate of progress of reaction j Sd

i j traceless symmetric part of the

square ofg_fi j µ molecular viscosity µm mean viscosity µt turbulent viscosity µw viscosity at the wall

µk j exponents for rate of progress ν kinematic viscosity

νk j molar stoichiometric coefficient Ω modified reaction rate of the fuel

breakdown reaction

ω angular frequency (2π f )

Ω0 maximum ofΩfor a laminar flame

Ωj computational cell τ total shear stress

a Reynolds average of a

a(F) Favre average of a

U bulk velocity

W mean molecular weight of mix-ture

− →

AΩj Jacobian of the flux tensor

− →

F inviscid flux tensor

− →

Fl inviscid flux at face −

→

G flux tensor of the diffusive terms

− →

Gl viscous flux at the boundary face −

→

dSi node normal −→

dSl boundary face normal − →_x i node coordinates φ equivalence ratio φ+ phase of p+_c φ− phase of p−_c

φg global equivalence ratio

R

RR_Ωj cell residual

w

ww vector of conservative variables

ℜ(·) real part of complex number

ρ density ρ(0) mean density σ Stefan-Boltzmann constant τ optical thickness τc chemical timescale τk Kolmogorov time

τt turbulence integral time scale τw wall shear stress

NOMENCLATURE 11

τac acoustic time

τar minimum acoustic recording time τcv convective time

τi j viscous stress tensor τ_{i j}R residual-stress tensor τ_{i j}r anisotropic residual-stress τtc cut-off time-scale

τti integral turbulence time-scale b

a filtered quantity a

e

a Favre filtered quantity a

f

gi j filtered velocity gradient tensor f

Si j filtered rate of strain

Ξ wrinkling factor of the flame sur-face

ξ isothermicity parameter

A boundary surface

a0 fluctuating part of a

A+ amplitude and phase of p+_c

A− amplitude and phase of pc−

Ag pre-exponential constant of global

reaction

An flame surface normal to the flow

Af j pre-exponential constant

Ain surface of the principal inlet

ap,k Planck mean absorption

coeffi-cient

arms standard deviation of a

a(F)rms density weighted standard

devi-ation of a

Atotal total flame surface

B constant for friction law

B1 constant for friction law

c sound speed

c1, c2 constants for Sutherland’s law

Cf skin-friction coefficient

Ck k-equation constant

cp specific heat at constant pressure

CS Smagorinsky constant

Cw WALE constant

c₍₀₎ mean speed of sound

Cε k-equation disspation constant cab correlation coefficient of a and b

c_ab density weighted correlation co-efficient of a and b

Cf ,comp compressible skin-friction

coef-ficient

CFL Courant-Friedrichs-Levy number

D burner exit diameter

d turbulence motion of size d

Dk Diffusivity of species k

Dt species turbulent diffusion

Dk_Ω

j distribution matrix Da Damk¨ohler number

E total non-chemical energy

E1 volume-integrated acoustic energy

E_d turbulence kinetic energy asso-ciated to the turbulence motion of size d

EF efficiency function for the TFLES

model

Ef activation energy of the fuel

break-down reaction

Ej activation energy

Eκ integration constant for the

loga-rithmic law of the wall

F factor for artificial flame thick-ening

f modification factor for Cf

F1 total acoustic flux through bound-ary

fw law of the wall for velocity

Fκ integration constant for the

loga-rithmic law of the wall

FLES filter

gw law of the wall for temperature

h total heat exchange coefficient

hk specific enthalpy of species k

I unit matrix

i imaginary unit

i, j indexes for vectors and summa-tion convensumma-tion

k node index

k wave number

kf filtered turbulence kinetic energy

kr residual turbulence kinetic energy

kt total number of nodes

Kf j forward rate of reaction

Kr j reverse rate of reaction

Ka Karlovitz number

l characteristic length

l face index

lc characteristic length of the

do-main

lc cut-off length scale

lk k-equation length scale

LP Planck mean absorption length

ls Smagorinsky length scale

lt turbulence integral length scale

lw WALE length scale

N estimation for the number of grid points needed in DNS

n number of the resonant mode

Nc number of computational cells needed

to resolve a flame

NOMENCLATURE 13

nn number of nodes for each cell

Nu Nusselt number

p pressure

pa reference pressure

pc complex valued pressure

p+_c acoustic wave travelling in the pos-itive x direction

p−_c acoustic wave travelling in the neg-ative x direction

pt boundary target pressure

pw pressure at the wall

p₍₀₎ mean pressure

p₍₁₎ pressure fluctuation due to acous-tics

Pr Prandtl number

q ratio of cut-off length scale to tur-bulence integral length scale

qi heat flux

qR_i residual energy fluxes

Qr volumetric heat source term

qw wall heat-flux

QPOPE_LES Pope’s LES resolution criterion

QRANS_LES LES resolution criterion based on τxy

R perfect gas constant

r specific gas constant

R(www) nodal residual of www

r+ amplitude of p+_c

r− amplitude of pc−

ra distance from the flow axis

Rl reflection coefficient of the left

boundary

Rr reflection coefficient of the right

boundary

Re Reynolds number

Reτ friction Reynolds number Ret turbulence Reynolds number

S flame sensor for local thickening

s mass stoichiometric ratio

S1 volume-integrated Rayleigh source term

sL laminar flame speed

sT turbulent flame speed

sR_k,i residual species fluxes

Sck Schmidt number of species k

Sct turbulence Schmidt number

T temperature

t time

T+ non-dimensional temperature

T1 temperature at boundary node

T2 temperature at first interior node

Tτ friction temperature Tm mean flow temperature

Ts temperature of the surroundings

Tw temperature of the wall

Tad adiabatic flame temperature

Tav averaging timespan

Tin temperature of the fresh gases

u velocity

u0_c velocity fluctuation associated to the cut-off length scale

u0_d velocity fluctuation associated to the turbulence motion of size d

u0_k velocity fluctuation associated to the Kolmogorov length scale

u_t0 velocity fluctuation associated to turbulence integral length scale

u, v, w velocity in x, y, z

u+ non-dimensional velocity

u1 velocity at boundary node

u2 velocity at first interior node

uϕ circumferential velocity

compo-nent

uc complex valued velocity

ui velocity

uR_i residual velocity fluctuation

US spatial unmixedness

UT temporal unmixedness

u(0) mean velocity

u(1),n normal component of u(1)

u₍₁₎ velocity fluctuation due to acous-tics

u0_∆

e velocity fluctuation associated to

∆e

uτ friction velocity

Uin mean normal velocity at the

prin-cipal inlet

ure f mean velocity at the burner exit

V volume of the computational do-main

Vk,i diffusion velocity of species k

Wk molecular weight of species k

X characteristic dimension of the en-closure

x coordinate

xi coordinates

Xk molar fraction of species k

y+ wall distance measured is viscous lengths

y1 y-coordiante of boundary node

y2 y-coordiante of first interior node

Yk species k

yw distance of first interior node to

physical wall

Z mixture fraction

Zst stoichiometric mixture fraction

V_k nodal volume

Introduction

Combustion of Fossil Fuels

While it is considered as a fact that fossil fuels will run out one day, this day is still in a distant future. Therefore, one of the immediate concerns is how to use the still available resources in the most effective way and how to minimise the damage done to our environment thereby.

Combustion of hydrocarbon fuel and air produces necessarily water and carbon dioxide. As carbon dioxide is contributing to the green-house effect, its produc-tion must be limited by using highly efficient power plants. Today, the most effi-cient power plants running on fossil fuel have close to 60% electrical efficiency. They consist of gas-turbines combined with steam turbines (combined cycle) and run on natural gas. If also waste heat is used (cogeneration), the fuel utilisation rate can rise up to 90% [3].

A major concern influencing the design of modern gas-turbines is the production of other pollutants, such as nitrous oxide (NOX). It is not only dangerous for plant

life but also creates photochemical smog and is one of the main precursors for ozone, which is a major problem in today’s cities [49]. NOX is formed through

different chemical mechanisms. Close inspection of those mechanisms lead to the conclusion that high temperature combustion has to be avoided in order to reduce

NOX concentration. Today, the by far most popular method for achieving this in

stationary gas-turbines is lean premixed combustion. An industrial gas-turbine burner, which is operating in this regime is the focus of this work. It is the small-est version of Alstom’s ev burner [21] (where “ev” stands for environmental). The standard size ev burner is used in gas turbines like the GT26, which is shown in Figure (1). It consists of a low and high pressure compressor, the ev burner and its combustion chamber, a high pressure turbine, the sev burner (the “s” stands for sequential) and its combustion chamber and finally a low pressure turbine. In this work, the ev burner is looked at separately without taking into account com-pressor, turbine, sequential combustion or even the original combustion chamber. Their influence is either neglected or included via appropriate boundary condi-tions.

ev burner turbine (hp) sev burner turbine (lp) compressor (hp) compressor (lp) air inflow exhaust

direction of principal flow

Figure 1: Schematic illustration Alstom’s GT26.

Large-Eddy Simulation

Understanding all physical processes related to turbulent lean premixed combus-tion is a very difficult task. In addicombus-tion to experiments involving sophisticated diagnostic equipment, more and more numerical simulations are carried out. This is due to the progress not only made in computing power but also in combustion and turbulence modelling. Both the flow patterns and the combustion process in modern gas-turbine burners are of unsteady nature and call for time resolved simulations. A computationally accessible and theoretically sound approach is Large-Eddy Simulation (LES), where the large flow structures are resolved and the smaller ones are modelled [79].

Cerfacs has a leading role in the development and application of LES to reacting flows in complex geometries. Most of the work carried out at Cerfacs on reacting flows is in relation with the LES-code AVBP [66], co-developed by Cerfacs and IFP. The present work concentrates on necessary developments of AVBP for the simulation of the ev burner and the resulting simulations. It is part of the Euro-pean Framework Programme 5 project FuelChief, which includes Alstom Power, Ecole Centrale Paris, DLR Stuttgart, DLR Cologne and Cerfacs.

INTRODUCTION 17

prompt

(Fennimore) + Nitrous Oxide

NO

_X4

fuel

thermal

(Zeldovich)

X

(gaseous fuel) exponential dependence on temperature exponential dependence on equivalence ratio equivalence ratio fluctuations* turbulence1,5 instabilities6,*

+

+

Chamber temperature equivalence ratio augmentation

+

+

+

+

no radiation4 _or

convection2,3 _{heat loss}

+

Figure 2: Mechanisms that influence NOX production in a lean partially premixed

combustion system. The superscript numbers refer to the corresponding chapters of the present thesis. The stars in superscript mark the factors influenced by fuel staging.

Influencing Factors on NO

NO

NO

_XXX

Production

Tackling all aspects of the combustion process of the ev burner is impossible in one single thesis due to the inherent complexity. This work concentrates there-fore on the major aspects influencing nitrous oxide formation. A summary of the involved processes is given in Figure (2). Nitrous oxides are formed by differ-ent chemical reactions [30, 64, 68]. It is convenidiffer-ent to group them in three parts: thermal NOX, fuel NOX and prompt NOX:

Thermal NONONOXXX (Zeldovich mechanism) is formed everywhere where oxygen and

nitrogen are present and temperatures are sufficiently high. It has an expo-nential dependence on temperature.

Fuel NONONOXXX has its origin in nitrogen, which is bound to the hydrocarbon fuel. As

only pure methane combustion is considered in this work, this mechanism is not important.

Prompt NONONOXXX (Fennimore mechanism) is formed in the flame by the intermediate

of hydrocarbon radicals. It is convenient to group this mechanism with the Nitrous Oxide mechanism as both are active close to the flame front and have an exponential dependence on equivalence ratio.

It is therefore necessary, that the reaction kinetics used for this study correctly pre-dict thermal and prompt NOX production. Then, the NOX formation in this

con-figuration is influenced by equivalence ratio, equivalence ratio fluctuations (due to the exponential dependence [27]) and chamber temperature (as shown in Fig-ure (2)). Considering a configuration with a fixed overall equivalence ratio, this observation directly leads to two main topics of this thesis: thermal modelling and prediction of the fuel-air mixing. The important thermal processes (excluding the flame) are radiation and heat losses at the walls. These have to be accounted for by appropriate models.

The fuel-air mixing, is influenced by various factors. In order to reproduce the spatial and temporal mixing quality, the fuel and cooling air injections have to be resolved explicitly. Turbulence obviously contributes to NOX formation: by

changing mixing, it modifies local equivalence ratio and temperature and there-fore NOX as indicated in Figure (2). Predicting large scale structures is needed to

predict NOX. Turbulence, however, is not the only factor which controls NOX

for-mation through mixing. Recent studies show that acoustic interactions can have a major impact on fuel-air mixing [88]. An eigenmode of the system is capable of modifying the fuel-air mixing because of the different acoustic properties of the air and fuel alimentation. If such a perturbation causes a change in heat-release after an appropriate time delay, the eigenmode will receive additional acoustic en-ergy. This closes a resonant loop and the result is a combustion instability [45]. It may create quite strong equivalence ratio fluctuations and has to be taken into account because it also changes NOX emissions.

A particularity of the burner considered in this study is the staging concept. It allows for the modification of the fuel distribution in the burner. This proved to have an impact not only on NOX formation, but also on combustion instabilities.

Its validation is the ultimate goal of the FuelChief project but out of the scope of this thesis.

Generally speaking, this thesis focuses on what seems to be the centre of many modern gas turbine designs: finding a trade-off between NOX emissions and

INTRODUCTION 19

Outline

This thesis deals in detail with the topics discussed above and is divided into the following chapters8:

Chapter (1) introduces turbulence and the concept of LES. Also, related aspects

like residual stress modelling and post-processing are described. Further-more, the fully compressible solver AVBP, used in this thesis to be able to predict reactive LES including acoustics-combustion coupling is described.

Chapters (2) and (3) deal with the weakest part of many LES approaches: walls.

First, an introduction to the modelling of turbulent boundary layers is given and a new “law of the wall” model, developed during this thesis, is de-scribed. Then, a turbulent channel flow simulation is presented and vali-dated using experimental data and correlations for velocity and heat trans-fer.

Chapter (4) introduces to modelling of reacting flows and presents a model for

NOX chemistry. Additionally, a radiation model, based on the local gas

composition is presented and its influence on the NOX emission predictions

is shown.

Chapter (5) presents turbulent combustion and its modelling. The thickened

flame approach, used to represent flame-turbulence interactions is described in more detail and examples of its practical application are given.

Chapter (6) introduces to acoustics. In particular, acoustic outlet boundary

con-ditions for LES are described and a basic description of combustion insta-bilities is given.

Chapter (7) details the ev burner and its staging principle, which is capable of

in-fluencing emissions and combustion pulsations by the modification of fuel-air mixing. The results of various non-reacting simulations are presented. Not only the velocity fields, but also the mixing predictions are validated against experimental data.

Chapter (8) shows the results of the reactive simulations. The comparison of

two simulations, one without thermal modelling, another including all ther-mal models shows the impact of cooling on the NOX emissions.

Simula-tions with different outlet reflection coefficients show the impact of thermo-acoustic instabilities on the NOX emissions.

Chapter 1

Large Eddy Simulation

1.1

Introduction

1.1.1

Turbulent Flow

Flows encountered in technical applications are rarely laminar. To be laminar, the fluid must be either very viscous, very slow or flowing in a very narrow duct. This observation is usually quantified by the ratio between inertia and viscous forces also called the Reynolds number:

Re =u · l

ν (1.1)

with u : characteristic velocity l : characteristic length

ν : kinematic viscosity.

Therefore, laminar flows are characterised by low Reynolds numbers. An exact limit between laminar and turbulent flow is hard to define because other parame-ters such as wall roughness and upstream flow-conditions may influence the tran-sition from laminar to turbulent flow. For instance, a simple channel flow changes from laminar to turbulent in the vicinity of Re ≈ 1500 [81]. For most other flows, similar values are found.

In contrast to laminar flows which are normally steady in time and spatially quite “smooth”, turbulent flows are unsteady in time and spatially very irregular (also often referred to as chaotic, see Figure (1.1)). This makes the simulation of turbu-lence much more demanding and its interpretation a lot more complicated. However, there is also an important advantage to turbulence: it is several orders of magnitude more diffusive than laminar flow. The turbulent eddies (which make up turbulence) are highly effective in exchanging mass, momentum and heat. This has a crucial importance for reacting flows. Not only the mixing of reactants is

Figure 1.1: Turbulent jets at different Reynolds numbers; left: relatively low Reynolds number; right: relatively high Reynolds number (adapted from a film sequence by R.W. Stewart, 1969, taken from the book by Tennekes and Lum-ley [97]).

highly enhanced but also the chemical reactions are strongly accelerated. This was recognised very early in combustion research as illustrated by Figure (1.2), where the combustion time (the time needed to reach maximum pressure in a closed vessel) is plotted as a function of the proportion of fuel in the mixture for a case with and without turbulence. It is seen that in this case, turbulence accel-erates combustion by at least a factor of two (unfortunately this factor is far from universal). Most combustion devices in use today rely on turbulence and would not work without it (for example the piston engine).

0.30 0.25 0.20 0.15 0.10 0.05 0.00 Combustion Time [s] 13 12 11 10 9 8 7 6 CH4 Volume Fraction [%] Laminar Case Turbulent Case

Figure 1.2: Combustion time plotted as a function of the proportion of fuel in the mixture for a case with and without turbulence [105].

1.1. INTRODUCTION 23

1.1.2

Turbulent Scales and Energy Spectrum

As already stated above, laminar or turbulent flow regimes are determined by the Reynolds number. But even above transition, the Reynolds number still has a major influence on the flow. It is a measure for the difference in length scales present in a turbulent flow. Figure (1.1) presents two turbulent jets at different Reynolds numbers. The higher Reynolds number jet has the same large-scale structure as the lower Reynolds number jet, but the smallest scales are finer. The largest scale is called the integral length scale lt, the smallest one is called the

Kolmogorov length scale ηk.

The integral length scale is usually close to the characteristic size of the geometry. For example, the flow through a duct has an integral length scale of the order of the duct diameter. The Reynolds number formed with the integral length scale and its associated velocity fluctuation u_t0is called the turbulence Reynolds number:

Ret =

u0_t· lt

ν (1.2)

This Reynolds number normally is several orders of magnitude higher than 1 (100 to 2000 in most combustion devices [79]). This means that the integral length scale is practically not affected by viscous dissipation and dominated by inertia. For turbulence which is homogeneous and locally isotropic (statistics must be invariant under translation and locally invariant under rotation), the energy of the large scales flows to the smaller scales through the Kolmogorov cascade [35]. This cascade is based on the fact, that the eddies associated to the large scales are unstable and break-up to form smaller and smaller eddies, where the energy flux from one scale to another is constant along scales. It is given by the dissipation of the turbulence kinetic energy. This dissipation ε is estimated as the turbulence kinetic energy Ed= u02d divided by its associated time-scale d/u0d:

ε = u 02 d d/u0_d = u03_d d (1.3)

where u0_d is the velocity fluctuation of the motion of size d.

Along the cascade, the Reynolds number formed with u0_d and d goes down from the turbulence Reynolds number to values close to one, where turbulence is dis-sipated by molecular viscosity. The associated length scale is the Kolmogorov length scale. Therefore, the turbulent Reynolds number formed with ηk and its

corresponding velocity fluctuation u0_k must be equal to one:

Rek=

u0_k· ηk

Combining Equations (1.3) and (1.4) gives a relation for the Kolmogorov length scale: ηk= _ν3 ε 1/4 (1.5) The ratio of the integral length scale lt to the Kolmogorov length scale ηk,

com-paring the largest with the smallest turbulence eddies can be expressed by using Equations (1.2), (1.3) and (1.5):

lt ηk

= Re_t3/4 (1.6)

This illustrates the wide range of turbulence scales. Figure (1.3) shows the tur-bulence kinetic energy Ed, which is plotted as a function of the inverse length

scale (which is proportional to the associated frequency, normally expressed by the wave number k = 2π/d). It can be seen that the large turbulent structures carry the main part of the turbulence kinetic energy which flows through the Kol-mogorov cascade and is dissipated at the smallest scales.

log(1/d) log(1/l )t

Computed in LES Modeled in LES

Computed in DNS Modeled in RANS Turbulence spectrum log(E ) η_k log(1/ ) d

Figure 1.3: Turbulence kinetic energy spectrum plotted as a function of the in-verse length scale (proportional to the wavenumber). RANS, LES and DNS are summarised in terms of spatial frequency range.

1.1.3

Simulation Approaches

The Navier-Stokes equations completely describe turbulent flows. Therefore a Direct Numerical Simulation (DNS) of turbulence does not need any modelling of

1.1. INTRODUCTION 25

turbulence. However, as outlined above, turbulent flows are intrinsically unsteady and involve various length scales. Therefore an accurate simulation must provide sufficient spatial and temporal resolution. An estimate of the necessary spatial resolution is possible when assuming that the total number of necessary grid points N must at least be equal to the ratio of the integral turbulent length scale to the Kolmogorov length scale:

N > lt

ηk

(1.7) With Equation (1.6) and a three-dimensional grid (which is essential, since im-portant processes in turbulence are indeed three-dimensional), this results in the following estimation for the number of grid points:

N >Re3/4_t 3= Re9/4_t (1.8)

For a turbulence Reynolds number of 1000, Equation (1.8) suggests 1 billion grid points. It has to be kept in mind, that this is only for a domain size correspond-ing to the integral length scale and that an equivalent temporal resolution has to be provided too. The high computational cost for a DNS limits this approach at present to purely academic configurations.

A computationally very accessible approach to turbulence simulation is to solve for the time-average solution. The Navier-Stokes equations are averaged in time resulting in the Reynolds Averaged Navier-Stokes (RANS) equations. They semble the original Navier-Stokes Equations (all instantaneous quantities are re-placed by average quantities) without time dependence and with several additional terms to account for turbulence. These terms have to be provided by additional models. Its computational costs are low: the simulation is steady in time, spatially only mean gradients have to be resolved and even two-dimensional simulations are meaningful. The problem of RANS-simulations is that none of the existing turbulence models is suitable for a wide range of applications. It proved to be too difficult to model all involved processes’ influence on the mean flow. Particularly the influence of large scale structures on the mean flow is nearly impossible to model (as occurring in flow instabilities or combustion instabilities).

An intermediate approach for turbulence simulation is to include the time de-pendence in the simulations and to average spatially [86]. Since the averaging is not carried out over the whole domain, but just locally, it is actually a filter-ing procedure with a filter size smaller than the turbulence integral length scale. The filtered Navier-Stokes equations resemble strongly the original Navier Stokes equations. The filtering introduces additional terms, that are easier to model than the respective terms in the RANS equations, since only a smaller, more universal part of turbulence has to be accounted for. This approach is called Large Eddy Simulation (LES). To estimate its computational costs, the Kolmogorov scale in

Equation (1.7) is replaced by a cut-off scale lc= lt/q with q > 1. This results in

the grid resolution becoming independent of the turbulence Reynolds number:

N > lt lt/q

3

= q3 (1.9)

Therefore this approach is computationally significantly less expensive than DNS, but more expensive than RANS simulations. However it yields results which are vastly superior to RANS and if care is taken they can even rival DNS results. The three approaches for turbulence simulations explained above are illustrated with the help of the turbulence kinetic energy spectrum in Figure (1.3). The re-solved and modelled parts of the turbulence length scales for the different models are indicated in the upper part of the figure.

Only the LES approach will be further detailed in the following. For details about DNS, see for example the review by Leonard [52]. For RANS see for example the book by Ferziger and Peri´c [26].

1.2

Filtering the Navier-Stokes Equations

1.2.1

The Compressible Navier-Stokes Equations

The standard Navier-Stokes equations consist of one equation for mass conserva-tion and one equaconserva-tion for momentum conservaconserva-tion in each coordinate direcconserva-tion. An additional equation for the conservation of total non chemical energy is added to be able to account for heat exchange and viscous heating. The final set of equa-tions, with the convective terms on the left hand side and the diffusive terms on the right hand side reads:

∂ ρ ∂ t + ∂ (ρ ui) ∂ xi = 0 Mass (1.10) ∂ (ρ uj) ∂ t + ∂ (ρ uiuj) ∂ xi + ∂ p ∂ xj = ∂ τi j ∂ xi Momentum (1.11) ∂ (ρ E) ∂ t + ∂ [ui(ρE + p)] ∂ xi = ∂ uiτi j ∂ xj −∂ qi ∂ xi Energy (1.12)

with ρ : density t : time

ui : velocity (i = 1, 2, 3) xi : coordinates (i = 1, 2, 3)

p : pressure τi j : viscous stress tensor

1.2. FILTERING THE NAVIER-STOKES EQUATIONS 27

Before detailing the diffusive terms, some important quantities of the fluid have to be defined:

• The molecular viscosity µ depends only on temperature T and may be

cal-culated using Sutherland’s law [95]:

µ (T ) = c1

T3/2 T + c2

(1.13) The constants for air are: c1= 1.458 · 10−6 kg/(ms

√

K) and c2= 110.4 K. The relation between kinematic viscosity ν and molecular viscosity µ is:

ν = µ

ρ (1.14)

• The Prandtl number Pr is comparing molecular diffusion of momentum to

molecular diffusion of heat:

Pr = cp µ

λ = 0.72 for air at standard conditions (1.15)

where λ is the thermal conductivity and cp the specific heat at constant

pressure (cp= 1004.5 J/(kgK) for air at standard conditions).

Pressure is computed from the equation of state for a perfect gas:

p = ρ(γ − 1)(E −1

2u 2

i) (1.16)

where γ the ratio of specific heats (γ = 1.4 for air).

The viscous stress tensor in Equations (1.11) and (1.12) is defined as:

τi j= − 2 3µ ∂ uk ∂ xk δi j+ µ _{∂ ui} ∂ xj +∂ uj ∂ xi  (1.17) where δi j is the Kronecker symbol, which is equal to one for i = j and zero for

i 6= j. This definition is valid for compressible, Newtonian fluids. Note that the

trace of the so defined stress-tensor is zero. The heat flux qifound in Equation (1.12) is:

qi= −λ ∂ T ∂ xi

(1.18) where temperature is computed from

T = γ − 1 r (E − 1 2u 2 i) (1.19)

with r the specific gas constant (r = 287 J/(kgK) for air). It is the ratio of the perfect gas constant (R = 8.314 J/(moleK)) and the mean molecular weight W .

1.2.2

The Filtering Operation

As stated above, filtering in physical space corresponds to a weighted-average over a certain volume. A quantity a, filtered in space by a filter FLES results in the

filtered quantitya [51]:_b

b

a(x,t) =

Z ∞

−∞FLES(r)a(x − r,t)dr (1.20)

with x : spatial coordinate.

For variable density flows, mass-weighted filtering (called Favre filtering [79]) is used, in order to avoid modelling of additional terms introduced by density fluctuations: e a(x,t) = 1 b ρ Z ∞ −∞FLES(r)ρ(x − r,t)a(x − r,t)dr (1.21)

with a_e : Favre filtered quantity.

Examples for filter shapes are the box filter or the Gaussian filter. The filters have to be normalised so thatR∞

−∞FLES(r)dr = 1, and their size corresponds in general

to the computational grid spacing.

However, the shape of the filter is not necessarily defined explicitly, since the filter itself does not appear any more in the filtered Navier-Stokes equations. It often serves rather as a mathematically sound framework for LES than for actual sim-ulations. Thus, when no explicit filtering is used, the filter shape is implicitly in-cluded in the turbulence closure and can be determined a posteriori. It is needed to post process experimental or DNS results that are compared to LES computations.

1.2.3

The Filtered Conservation Equations

Filtering the Navier-Stokes equations is simple when the filter size is constant over the whole domain. Then, filtering and derivative operators are commutative and filtering is straightforward. Once the filter has been applied to the conservation equations, additional diffusive terms appear and the existing diffusive terms are modified. The convective parts of the equations stay untouched and are regrouped on the left hand side. The right hand side shows the modified molecular diffu-sion terms after the equal sign. The additional terms (framed) are the unclosed

1.3. TURBULENCE CLOSURE 29 turbulent terms: ∂ρb ∂ t + ∂ (ρbuei) ∂ xi = 0 Species (1.22) ∂ (ρ_bu_ej) ∂ t + ∂ (ρ_bu_eiuej) ∂ xi + ∂bp ∂ xj =∂τci j ∂ xi − ∂ ∂ xi [ρ (_b u_giuj−ueiuej)] Momentum (1.23) ∂ (ρ ebE) ∂ t + ∂ [u_ei(ρ ebE +p)]b ∂ xi =∂ueicτi j ∂ xj −∂qbi ∂ xi − ∂ ∂ xi [ρ ( f_b uiE −ueiE)]e Energy (1.24) Note that Equation (1.24) contains already two assumptions:

∂u_di p ∂ xi ≈ ∂uei bp ∂ xi and ∂ [uiτi j ∂ xj ≈ ∂ueicτi j ∂ xj (1.25) A very detailed analysis of the filtered energy equation can be found for example in the thesis of Vreman [101]. Here it is only stated, that the above assumptions are made and only the most important terms are retained for modelling. For all those right hand side terms, an appropriate model has to be found. The unframed terms are normally modelled by using the filtered variables directly in their lam-inar expressions (Equations (1.43) and (1.44)). The framed terms however need additional models based on the unresolved turbulence scales.

1.3

Turbulence Closure

1.3.1

Unresolved Reynolds Stresses

The framed term in Equation (1.23) describes the unresolved turbulence’s impact on the filtered velocity field. A simple way of modelling turbulence was intro-duced by Boussinesq [8]: the turbulent viscosity hypothesis. If turbulence is con-sidered a diffusive process similar to molecular diffusion, it should be sufficient to provide the value of a turbulent viscosity via a more or less sophisticated model. This is also the basis of most RANS models.

The residual-stress tensor, describing the unresolved turbulence’s influence on the filtered field is defined as:

τ_{i j}R=ρ (b ugiuj−ueiuej) =ρ ub R

iuRj (1.26)

with τ_{i j}R : residual-stress tensor uR_i : residual velocity

And the residual1turbulence kinetic energy is defined as: kr= 1 2τ R ii (1.27)

Based on the analogy to the molecular stress, the residual-stress is decomposed into the anisotropic residual-stress τ_{i j}r (which is trace-free) and the isotropic residual-stress 2₃kr:

τ_{i j}R = τ_{i j}r +2

3krδi j (1.28)

Defining the modified filtered pressurep as_e

e

p = _bp +2

3kr (1.29)

allows to simply add the turbulent viscosity µt (given by a turbulence model) to

the molecular viscosity (given by the temperature), since the anisotropic residual-stress tensor can now be written exactly as the viscous residual-stress tensor of Equa-tion (1.17): τ_{i j}r = 2µtSf_{i j} (1.30) with Sf_{i j} = 1 2 _∂ e ui ∂ xj +∂uej ∂ xi  −1 3 ∂u_ek ∂ xk δi j (1.31)

In the following, several popular models for the turbulent viscosity are described. Note that the modification of the equation of state induced by the modified pres-sure is neglected (the thesis of Ducros [24] gives more details on this assumption). The impact of this assumption on acoustic computations using LES is unknown at the moment.

Smagorinsky’s Model

Proposed by Smagorinsky et al. [92], the model is based on the analogy to the mixing-length hypothesis. The turbulent viscosity is modelled as the product of the density, the Smagorinsky length scale ls and the characteristic filtered rate of

strain: µt=ρ lb 2 s q 2fSi jSf_{i j} (1.32)

The Smagorinsky length scale is the product of the Smagorinsky constant Cs and

the filter width∆:

ls= Cs∆ with Cs= 0.18 (1.33)

1.3. TURBULENCE CLOSURE 31

The filter width is normally set equal to the mesh spacing∆c(characteristic length

of the computational cell). The value given above for the Smagorinsky constant was deduced theoretically by Lilly [55]. However, this constant was found to de-pend to some extent on the type of flow. This is not its only drawback. Also, the model does not predict zero turbulent viscosity at a solid boundary and it is not able to predict transition correctly. Furthermore, the turbulent viscosity, as predicted by Smagorinsky’s model never becomes negative, so there is no en-ergy transfer from the residual motions to the filtered motions (the so-called back-scatter).

Still, the model is widely used. Its biggest drawback, the non-zero turbulent vis-cosity at walls is easily corrected by damping functions or by replacing the char-acteristic filtered rate of strain by a more adapted quantity (as seen in the next paragraph). The dependence of Cs on the geometry and the lack of back-scatter

are often masked by numerical errors which in most LES codes are significant compared to the LES modelling approximation.

WALE model

To obtain the right scaling for the turbulent viscosity when approaching a solid boundary, the Van Driest damping function is often used [99]. A more ele-gant way is the WALE (Wall-Adapting Local Eddy-viscosity) model proposed by Nicoud and Ducros [69]. They replace the characteristic filtered rate of strain by a term that detects strong rates of deformation and/or rotation and not shear as in Smagorinsky’s model: µt =ρ lb 2 w (Sd_{i j}Sd_{i j})3/2 (fSi jSfi j)5/2+ (Sd_{i j}Sd_{i j})5/4 (1.34) Sd i j = 1 2(gfi j 2 +g_fji2) − 1 3δi jgfkk 2 (1.35) f gi j= ∂u_ei ∂ xj (1.36) This expression for µtallows for the right scaling of the turbulent viscosity, when

approaching walls and also the prediction of transition. The constant used to determine the length scale for the WALE-model has to be evaluated numerically in homogeneous isotropic turbulence and is given in the following:

lw= Cw∆ with Cw= 0.5 (1.37)

But still, like for Smagorinsky’s model, the model constant is not completely in-dependent of the type of flow.

k-Equation Model

Supposing the residual turbulence kinetic energy is known, it can be used to pro-vide a characteristic turbulent velocity√kr. In combination with a characteristic

length scale, a model for the turbulent viscosity is obtained [108]:

µt=ρ lb k p

kr (1.38)

lk= Ck∆ with Ck= 0.05 (1.39)

From the exact transport equation for the residual turbulence kinetic energy [61], a simple transport equation for kr can be derived:

∂ (ρ k_b r) ∂ t + ∂ (ρ_bu_eikr) ∂ xi = ∂ ∂ xi  µt ∂ kr ∂ xi  − τ_{i j}R∂uei ∂ xi −ρC_b ε p k3 r ∆ (1.40)

The convective part is grouped on the left-hand side of Equation (1.40). The right-hand side terms are (from left to right): turbulent diffusion, production of kr and

dissipation of kr. Cε is a constant and close to one.

This model behaves similarly to Smagorinsky’s model (including its drawbacks). In special cases like recirculating flows it has advantages over Smagorinsky’s model, since the residual stresses are no longer just dependent on the local veloc-ity gradients, but also on the transported value of kr. Furthermore, the availability

of kr through the model is useful for the evaluation of LES quality and as input

for combustion models.

Other Models

There are numerous other residual-stress models used for LES, which are less commonly used (as they do not have major advantages in actual applications). An example are the structure function models [53] which are based on theoretical considerations of the turbulence spectrum.

Dynamic Formulations

An important extension to LES-models is the so-called dynamic procedure [28], which supposes that the LES-model must still be valid when using a second, larger filter. As on this second filter level, the small-scale resolved turbulence is known, modelling constants (such as CSin Smagorinsky’s model) can be

deter-mined. LES-models including a dynamic procedure add considerable complexity to residual-stress modelling, since some sort of spatial averaging of the obtained constant is necessary. In many academical configurations, their superiority was proved. In industrial applications however, the complexity and associated compu-tational overhead is not always justified.

1.4. NUMERICS 33

1.3.2

Other Unclosed Quantities

Unresolved Energy Fluxes

For the effects of unresolved turbulence on the filtered energy, a gradient assump-tion in analogy to the heat fluxes for molecular diffusion is often used. The framed term on the right-hand side of Equation (1.24) is modelled as:

qR_i =ρ ( f_b uiE −ueiE) = −λe t ∂ eT ∂ xi

(1.41) where the thermal conductivity due to the residual turbulent motions λt is defined

from the turbulent viscosity and a turbulent Prandtl number:

Prt=

cpµt λt

= 0.9 (1.42)

Filtered Molecular Diffusion Fluxes

Assuming that molecular viscosity and thermal conductivity do not change across the filter width, both quantities can be extracted from the filtering operator. The stress tensor and heat fluxes can then be simply written as:

c τi j ≈ −2bµ fSi j (1.43) b qi≈ −bλ ∂ eT ∂ xi (1.44)

1.4

Numerics

1.4.1

Generalities

Numerical accuracy

It is important to stress that the dissipation in LES should only be due to the resid-ual and molecular stresses. As the filter width for LES is normally close to the cell size, the dissipation due to the residual-stress model acts mainly on the smallest structures resolved by the grid. Since numerical dissipation in this range is impor-tant for a non-centred space discretisation (as often used for robust RANS codes), centred space discretisation must be used.

In order to avoid stability problems when using centred space discretisation, so-called artificial viscosity schemes can be used. These have to be adapted to act only when extreme situations occur, in order not to deteriorate the whole simu-lation. Another way of ensuring the stability of centred schemes is to derive an

algorithm which conserves kinetic energy (which is only possible for incompress-ible flows).

Since important dispersion errors have to be avoided, the temporal scheme has to be at least second order. Even higher-order schemes are often preferred [66].

Numerical costs

As LES is very expensive in terms of computational resources, efficient numerical algorithms and parallelisation are very important. Depending on the application, the choice has to be made between the structured and the unstructured grid ap-proaches. For simple geometries, a structured solver is most effective and also easy to implement. When complex geometries are of interest, unstructured meth-ods are often preferred. They are computationally less efficient and harder to implement but constitute the most obvious choice for combustion chambers. Implicit temporal integration is in general less expensive than explicit temporal integration, since the computational overhead of implicit schemes is more than compensated by the greater time-steps. However, since the complexity of the implicit time-stepping becomes even greater when combustion is added, explicit time-stepping if often preferred. This question (implicit vs explicit) is still the centre of multiple discussions in the combustion community.

1.4.2

Cell-Vertex Method for the Euler Equations

The numerical approach described in the following is based on the unstructured compressible solver AVBP of CERFACS and IFP. It is parallel and highly scalable. It was developed on an initiative by Sch¨onfeld and Rudgyard [87].

Formulation

It is convenient to group the terms of the Euler Equations (left hand side of Equa-tions (1.10)-(1.12)) into two different parts. The vector of conservative variables

w

ww includes the five variables from the temporal derivatives. The flux tensor−→F

re-groups the remaining spatial derivatives (one column per spatial direction). Then, the Euler equations can simply be written as:

∂ www ∂ t +∇·

− →

F = 000 (1.45)

For a better understanding, it is useful to follow the discretisation in an exam-ple. Figure (1.4) shows a regular triangular mesh. The conservative variables

1.4. NUMERICS 35

F

A

C

E

D

B

                                    3 5 4 6 2 7 1

Figure 1.4: Control volume for the divergence of the Euler fluxes. The involved node-normals are shown only for cell A.

are stored at the nodes which are marked by numbers and normally referenced to by the index k (= 1, 2 . . . kt) . The cells that define the control volumes are

marked by letters and normally referenced to by the symbol Ω with the index

j (= A, B . . . nj). The nodes associated to a cell are referenced to by the index

i (= (1, A), (2, A) . . . (kt, nj)).

The node-normal vectors defined for each face are associated to the nodes. In the example of Figure (1.4), nodes (1, A), (2, A) and (3, A) each have a normal−→dSias

shown by the arrows. They are in fact normal to the opposed faces. Their length is such, that the result of Equation (1.46) corresponds to the cell volume V_ΩA:

V_Ωj= 1 N_d2

∑

i∈Ωj − →_x i· − → dSi (1.46)

with Nd : number of dimensions −→xi : node coordinates.

With the above definitions, the divergence for cell A, called the cell residual RRR_ΩA, is written: RRR_Ωj = 1 NdVΩj

∑

i∈Ωj − → Fi· − → dSi (1.47)

where the Euler fluxes−→Fi are constructed from the conservative variables at the

nodes. To be able to determine the node residual dwwwk

Equa-tion (1.45) in time, the nodal volume Vkhas to be defined: Vk=

∑

j|k∈Ωj V_Ωj nn (1.48)

where nnis the number of nodes for each cell. For example, the volume associated

with node number 1 is a third of the sum of cells’ A − F volumes. Now the node residual is written as:

R(www) =dwwwk dt = − 1 Vk

∑

j|k∈Ωj V_Ωj nn R RR_Ωj (1.49)

The control volume used for this residual is shown in grey in Figure (1.4). Now, the conservative variables can be updated for the new iteration (n + 1) from the preceding one (n):

www(n+1)= www(n)−∆t R(www(n)) (1.50)

where∆t is the time step. It is determined by the Courant condition with states

that for stability, information must not travel further than across one cell. As the Euler equations include acoustics, information travels at sound speed c plus the convective speed u:

∆t = CFL∆x

|u ± c| (1.51)

where ∆x is the characteristic length of the cell and CFL

(Courant-Friedrichs-Levy number) is a stability parameter which is normally lower than one. The sound speed c is:

c =pγ r T (1.52)

This is the simplest possible way to integrate the Euler equations in time and is called explicit Euler time-stepping with centred space discretisation. The de-scribed scheme is first order accurate in time and second order accurate in space.

Multi-stage Runge-Kutta

Runge-Kutta stepping is the extension of the simple explicit Euler time-stepping. An m-stage (low-storage) Runge-Kutta scheme simply writes:

www(0)= www(n) www(1)= www(0)− α1∆t R(www(0)) .. . = ... www(m)= www(0)− αm∆t R(www(m−1)) www(n+1)= www(m) (1.53)

1.4. NUMERICS 37

where α1 to αm are the Runge-Kutta coefficients. Unfortunately, the two-stage

scheme is unstable, so the first additional, useful scheme is the three-step one (with coefficients 0.5, 0.5, 1.0). It is third order accurate in time (spatial accuracy stays second order). However it is also three times more expensive than the explicit Euler time-stepping.

Lax-Wendroff scheme

A useful alternative is the Lax-Wendroff scheme. It is obtained by introducing the distribution matrix Dk_Ω j into Equation (1.49): dwwwk dt = − 1 Vk

∑

j|k∈Ωj Dk_ΩjV_ΩjRRR_Ωj (1.54) Dk_Ωj= 1 nn (I + n 2 n 2Nd δ t_Ωj V_Ωj − → AΩj· − → dSk) (1.55)

where δ t_Ωj is a characteristic cell time, I the unit matrix and−→A_Ωj the Jacobian of the flux tensor. The obtained scheme is second order accurate in time and space. Its computational cost is much closer to the explicit Euler scheme than to the three-step Runge-Kutta scheme.

Finite Element Schemes

It is nearly impossible to develop schemes of higher order (in space) on unstruc-tured meshes in a finite volume context. Fortunately, the cell-vertex formulation can be extended to a finite element approach, where higher order schemes are possible. Colin and Rudgyard [16] developed a two-step Taylor-Galerkin scheme (TTGC) that is third-order in space and time. It has a computational cost of ap-proximately 2.5 times Lax-Wendroff (which is slightly less than the three-step Runge-Kutta).

Achieving higher order in space is particularly useful for three-dimensional, un-steady simulations since it provides a much better accuracy on meshes already used for second-order simulations. This is considerably cheaper than refining the mesh for a second order method [66].

1.4.3

The Diffusive Part

The extension of the described solution procedure to the Navier-Stokes equations can be done by constructing as for the convective terms a flux tensor for the dif-fusive terms −→G of Equations (1.11) and (1.12). This requires the gradients of

F

                                                                                     

A

C

E

D

B

6 1 3 7 4 5 2

Figure 1.5: Control volume and normals for the divergence of the diffusive fluxes. the conserved variables. They are obtained by the same approach as used for the computation of the divergence of the Euler fluxes:

(−→∇www)_Ωj = 1 NdVΩj

∑

i∈Ωj w wwi· − → dSi (1.56)

So, the gradients are calculated at the cells, but not scattered back to the nodes. This is done to keep the control volume as small as possible. The viscous flux tensor is assembled at the cells, using cell-averaged values of the conservative variables where necessary. Then the second integration is carried out by using all normals attached to one node and the corresponding cell-fluxes:

dwww_k dt = 1 NdVk

∑

j|k∈Ωj − → GΩj· − → dS(i=k) (1.57)

The contribution to the residual of node 1 from cell A would only be the nor-mal at node (1, A) multiplied with−→G_ΩA. This is seen in Figure (1.5), where all involved normals are shown. Additionally the control volume is shown in grey. As the total residual at a node is now influenced by nodes just one edge away, this diffusive operator is called a 2∆-operator. This compact diffusive operator recognises the highest frequency mode which is an important property for LES calculations. Note that the extension of this approach to bilinear control volumes (such as quadrilaterals) involves some additional considerations which will not be treated here.