Scalar transport and flamelet modelling in turbulent jetdiffusion flamesCitation for published version (APA):Sanders, J. P. H. (1994). Scalar transport and flamelet modelling in turbulent jet diffusion flames Eindhoven:Technische Universiteit Eindhoven DOI: 10.6100/IR409238

DOI:10.6100/IR409238

Document status and date:Published: 01/01/1994

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Scalar Transport and Flamelet

ModeHing in Thrbulent Jet

Ditfusion Flames

Scalar Transport and Flamelet Modelling in Thrbulent Jet

Diffusion Flames

PROEFSCHRIFf

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. J.H. van Lint,

voor een commissie aangewezen door het College van Dekanen in het openbaar te verdedigen op

woensdag 26 januari 1994 om 16.00 uur

door

Johannes Peter Henricos Sanders

geboren te Roermond

druk: wibro dissertatîedrukkerij, helrnond,

Dit proefschrift is goedgekeurd door de promotoren prof.ir. J.K. Nieuwenhuizen en prof.dr. D.J.E.M. Roeka.erts

ISBN 90 386 01433

Contents

1 Introduetion 1.1 Prologue . . .... 1.2 Transport equations 1.3 Turbulence . . . . . .

1.3.1 Description 1.3.2 Turbulence modeHing .

1.4 Combustion ......... . 1.4.1 Types of diffusion flames . 1.4.2 Global jet diffusion ftame behaviour . 1.4.3 Chemica! kinetics . . . . . . 1.4.4 Cambustion models . . . . . . . . . .

1.5 Turbulence oombustion interactions . . . . . 1.5.1 lnfluence of turbulence on cambustion 1.5.2 lnfluence of cambustion on turbulence

2 Turbulence 2.1 Averaging ........... .

2.1.1 Conventional averaging . 2.1.2 Favre-averaging

2.2 The k c model . . . . 2.3 Discussion ...... .

2.3.1 The k c model 2.3.2 A veraging . . . .

2.4 Scalar transport: Standard model 2.5 Scalar transport: Non-equal scales model

2.5.1 Introduetion ........... . 2.5.2 Dissipation rate equations from the TSDIA. 2.5.3 Comparison with other models . 2.5.4 Condusions and discussion .

3 lsothermal Jets 3.1 Physical description. 3.2 Laminar round jets 3.3 Turbulent jets . . .

3.3.1 Simila.rity properties 3.3.2 Spreading rates ... 3.3.3 Modifications to the k- e; model

7 7 9

11 11 11 13 13 16 16 17 18 18 18

21 21 22 23 24 26 26 28 29 30 30 33 35 37

39 39 41 43 43 44 45

3.3.4 Comparison with experimental data . 3.4 Scalar transport: Non-equal scales model ..

3.4.1 Determination of the coefficients ... 3.4.2 Asymptotic behaviour, experimental data. 3.4.3 Predictions of constant density jets . . . 3.4.4 Discussion of the non-equal scales model 3.4.5 Predictions of variabie density jets

3.5 Conclusions . . . . . . . . . . . . . . . . .

4 Combustion 4.1 Some alternative methods 4.2 Flamelet method . . . .

4.2.1 Mixture fraction 4.2.2 Flamelet theory . 4.2.3 Counterflow diffusion flame

4.3 Results for counterflow Harnes . . . 4.3.1 Definition of the mixture fraction 4.3.2 Hydrogen ftame 4.3.3 Metharre fiame .

4.4 Statistles . . . . . . . . . 4.4.1 Mixture fraction 4.4.2 Non-equilibrium parameter.

4.5 Conclusions . . . . . . . . .

5 Turbulent Hydrogen Flames 5.1 Laminar ftamelet modelEng 5.2 Flame-isothermal jet charaderistics 5.3 Influence of coftow 5.4 Predictions of NO

5.4.1 Effects of strain 5.4.2 The choice of the small eddy strain rate 5.4.3 Inftuence of coflow on NO predictions.

5.5 Scalar transport . . . . . . . . . . . . 5.5.1 The time scale ratio 5.5.2 Comparison with experiments

5.6 Differentlal ditfusion effects . 5.7 Conclusions

6 Stabilized Natural Gas Flames 6.1 Stabilization . . . . . . . . 6.2 Radiation . . . . . . . . . . .

6.2.1 Empirica! method ... 6.2.2 Radlation modelling based on soot volume fractions

6.3 A stabilized metharre fiame . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .

49 50 51 .54 57 58 60 61

63 63 65 65 66 67 70 70 71 73 76 77 80 80

81 81 84 88 92 93 96 96 99 99

101 104 109

113 113 114 115 115 118 121

7 Modeling and Calculation of Turbulent Lifted Ditfusion Flames 123

2

8 Conclusions 137

A Favre-averaging 149

B Numerical Methods 151 B.l Finite Volume Method 151

B.l.l General equations . 151 B.l.2 Elliptic method .. 154 B.l.3 Parabolized method . 156 B.l.4 Computing times .. 156

B.2 Integration of the /1-function . 157 B.3 Newton iteration method ... 158

B.3.1 General description . . 158 B.3.2 Computational details 159

c Two-Scale-Direct-Interaction-A pproximation 161

D Transport Properties and Reaction Mechanism 165 D.l Di:ffusion veloei ties . . . . . . . ~ . . . . . 165 D.2 Thermadynamie and transport properties . 166 D.3 Reaction mechanism . . . . ~ . ~ ~ 167

E Definition of the Mixture Fraction 171

3

Roman symbols

a Laminar strain rate (s- 1 ) L Mean beam length ( m) ai i Number of atom i Le Lewis number (-)

in species j (-) m Mass production rate bij Anisotropy tensor (-) (kg

Cp Specific heat ( J kg- 1 K-1) M Molar mass (kg kmole- 1)

C2 P lanck 's secoud j\tt Chemica! symbol (-) constant ( m J() Ms Number of reactions (-)

c Turbulence N Number of species (-) coefficien t (-) p Static pressure (N m-2)

D Nozzle diameter (m) p Probability density D, Etfective nozzle diameter ( m) function D; Ditfusion coefficient ( m2 s-1

) p Turbulence e Intemal energy ( J kg- 1

) production rate (J s-1m-3 )

E Activation energy Yr Prandtl number (-) (J kmole-I) ij Heat flux vector

Eg Energy density (J fun ct ion ( m) Q Heat souree (J m-3 s- 1)

e Mass entrainment (m-1 ) r Radial distance ( m) f Mixture fraction (-) Rij Reynolds stress f Stream function ( m s-1 ) tensor (N m-2) fi..fs Similarity functions (-) Rr Time scale ratio (-)

r Specific body force Re Critica! time scale (N kg-1 ) ratio(-)

fv Soot volurne fraction (-) Ru Gas constant g Scalar varianee (-) (J kmole- 1K- 1 )

g Gravitational Re Reynolds number (-) acceleration (m2s- 1) s Turbulent strain rate ( s-1

)

h S pecific en thalpy ( J kg -l ) su;sf Spreadlng rates (-) h Liftotf (m) s Mean strain rate (s-1 )

I u Jet invariant (m2s-1 ) Sc Schmidt number (-) j 0: Planar; t Time (s)

1: Axisymmetric (-) T Temperature ( K) k Turbulent kinetic T A veraging time ( s)

energy (m 2s-2 ) u Axial velocity ( m s- 1)

k Rate coefficient V Radial veloei ty ( m s -l) (kmole V; Ditfusion velocity (m s-1)

kg Absorption coefficient x Axial distance ( m) (m-1) x Molar fraction (-)

J( Slope of inverse profile (-) y Transverse distance ( m) fu;ff Integral length scales ( m) y Mass fraction (-) tk Kolmogorov length ( m) z Mixture fraction (-)

4

Greek symbols Subscripts

6;j Kronecker delta (-) 0 Reference point; 6(R) Dirac delta function nozzle condition E Specific visrous 1 Fuel stream

dissipation rate (m2s-3) 2 Oxidizer stream ég Scalar dissipation rate (s-1) 1/2 Halfwidth f Emissivity ( ·) 1/100 Analogous to Eijk Alternating tensor ( ·) halfwidth for 1/100 Tf Boundary layer a Ambient

coordinate (-) ad Adiabatic 4> Dimensional coefficient b Burning part 4> Generalized variabie c Centreline value 4> Dimensionless cfl Co flow

streamfunction (·) e Edge ( oxidizer side) op Viscous dissipation rate efJ Effective

(J m-3s-t) ( molecular+ turbulent) I Turbulence constant (-) exp Experiment al t Intermittency factor (-) I Scalar turbulence x Scalar dissipation ra te (s-I ) F Fuel Xr Radiative fraction (-) Species ç Boundary layer k Reaction number

coordinate ( ·) m ~ixture averaged K Wavenumber (m-1) 0 Oxidizer K Soot related coefficient {-) p Production range Àti À2 TSDIA model constants (-) q Quenching À Thermal conductivity rad Radiation

(J m-1 K-1) s Scalar turbulence Ik Dynamic viscosity st S toichiometric

(kg m-1s-1) t Turbulent IJ Kinematic viscosity (m2s-1) Transfer range v[;vi' Stoichiometrie u ~echanical turbulence

coefficients (-) w ~olar production rate

(kmole m-3s-1) Superscripts w Vorticity ( s-1)

Conventional average w Density ratio Pal P1 (-) n Non-equilibrium Favre-average

parameter ( s-1) Conventional fluctuation 11 Favre-fluctuation p Density (kg m-3)

Time derivative a Stefan-Boltzmann coeflicient (J m-2K-1s-1) Velocity

I ~ixture fraction O"t Fixed turbulent Schmidt

or Prandtl numher (-) Tij Stress tensor (N m-2)

Ts Turbulent scalar flux (kgm-2çl)

Tu;TJ Turbulent time scales ( s) ( Dimensionless strain

rate (-)

.5

6

Chapter 1

Introduetion

1.1 Prologue

At the Eindhoven University of Technology, faculty of Mechanica! Engineering depart­ment WOC-WF, research is conducted in the field of laminar and turbulent flames. Turbulent flames constitute an important phenomenon in combustion from a scientific as wel! as from a practical point of view.

lndustrial applications involve large power. This implies high flow veloeities and therefore most industrial flames are turbulent. Turbulence increases the efficiency of mixing processes by factors of hundreds or thousands.

Premixed turbulent flames occur, for instance, inspark ignition engines. For safety reasons, however, most of the large flames are non-premixed, or dilfusion fla.mes. They occur in furnaces, in Diesel engines and in cambustion cha.mbers of gas turbines.

A principal difficulty in the modelling of turbulent flames is the adequate modelling of turbulence itself. Turbulent flows show fluctua.tions on smal! physical dimensions such that the ratio of the smal! to the large scales in the flow sca.les with the Reynolds number as Re-314 • Even in a constant density fluid this makes the numerical solution of the governing equations, i.e. the continuity a.nd the Navier-Stokes equations, not feasible at high Reynolds numbers. In a. compressible environment such as a turbulent fla.me the additional equa.tions of the large number of species mass fractions and of the enthalpy (temperature), in which transport properties depend on the t.emperature and on the composition, would lead to an even larger dema.nd on computer resources.

The solution for this problem is found in a.veraging the equa.tions with respect to time. However, due to the a.vera.ging process information conta.ined in the original equa.tions is lost. This informa.tion is rela.ted to the correlations of fluctua.ting variables a.nd it ha.s to be retrieved by ma.king assumptions a.bout the dyna.mics of these correla.tions. This is the modeHing of turbulence.

The a.vera.ging process a.lso leadstoa need for turbulent combustion models since it is found to be ha.rdly possible to solve all equa.tions for the mass fractions a.nd temperature in a straightforward ma.nner.

Finally, the influence of turbulence on combustion and vice-versa often calls for subtie modifications on combustion models and on turbulence models, respectively.

The flames diseussed in the present study are turbulent, free jet, dilfusion flames at atmospheric pressure. They arise if fuel issues at high velocity from a. nozzle into stagna.nt or coflowing air, and if somewhere in the resulting mixing la.yer the temper at ure is high

7

enough to initialize an exothermic reaction. For the flow to be turbulent the gas velocity must be high enough. The reaction between the fuel and the oxidizer occurs at those locations where the mixture is stoichiometrie, which means that the local mass ratio of fuel and oxidizer is optima! for combustion. This situation is achieved by ditfusion of the reactants. The speed of the ditfusion process limits the rate at which the consumption of reactants proceeds, and this explains the name 'ditfusion flame'.

Environmental problems due to high emissions of CO, C0 2 and NOx, demand re­search on how to perform combustion in a way least darnaging to the environment. The present study aims at gaining knowledge about predictive and computational methods for turbulent combustion processes and applying them to the case of turbulent ditfusion flames. Eventually the computational method may be used to develop software for the design of turbulent combustors.

In the present study numerical predictions of the detailed chemistry in turbulent ditfusion flames, which are essential for the development of clean burners, have been made with the la mi nar fiamelet concept. In this concept the turbulent flame is considered an ensemble of one-dimensionallaminar flames. Sirree it is possible to calculate laminar flames including detailed chemistry, the flamelet concept enables the calculation of the distributions of chemica! componentsin turbulent ditfusion as wel! as premixed flames.

The stability of ditfusion flames is also investigated with the flamelet concept, result­ing in an accurate prediction of the lift-off height. This is the distance from the burner to the plane where the lifted flame is stabilized.

A model has been developed for the turbulent transport of scalar variables, which does not rely on the assumption of equallength and time scales for mechanica! and scalar turbulence1 • This model wil! be called the non-equal scales model for scalar transport.

In the following sections of this chapter the governing fluid dynamic and thermo­dynamic equations as wel! as general background on turbulence and combustion are given.

In chapter 2 the mechanica! turbulence model used in the computational method for turbulent flame calculations is described and possibilities for improvement of this model are discussed. The non-equal scales model for scalar transport is derived and it is compared with models found in the literature.

In chapter 3 isothermal round jets, either with constant or with variabie density, are discussed. Computational results for laminar and turbulent jets are compared with analytica! solutions and measurements, respectively. Special attention is paid to the prediction of the spreading rate of the turbulent round jet. The merits of the non-equal scales model for scalar transport are illustrated with results for a turbulent round jet. The theoretica! problems regarding this model are discussed and possible solutions are given.

In chapter 4 the modelling of turbulent ditfusion flames is discussed with emphasis on the laminar flamelet concept [PET84]. Predictions of laminar flamelets are presented and some charaderistic values are compared with those found in the literature. The statistica! description of the the mixture fraction and the strain rate is given. These two variables determine the turbulent ditfusion flame structure and if their distribution is known, the laminar flame data can be used in the prediction of the turbulent flame.

In chapter 5 computational results of turbulent hydrogen ditfusion flames are com-

1 Here, the expression 'mechanica! turbulence' is used for the fluctuations of veloeities and other quantities that have a vectorial nature.

8

pared with measurements from the literature. The performance of the non-equal scales model is applied in these strongly non-constant density ftows. Sealing behaviour of NO with initial conditions, such as the Reynolds number, is investigated in 100 % hydro­gen flames using the flamelet concept. Finally, differential dilfusion effects and their consequences are investigated.

In chapter 6 the flamelet method is applied to stabilized natural gas flames, where flame radiation due to soot formation can be important. In a first attempt a model for radiation and soot formation is developed within the flamelet concept and it is applied to these flames.

In chapter 7 the problem of the calculation of the lift-off height is addressed. Some arguments are given against the generally accepted choice of the scalar dissipation rate as the variabie descrihing non-equilibrium effects in turbulent flames. It is argued and shown that for purposes of lift-off calculations the strain rate rather than the scalar dis­sipation rate has good sealing properties, which is in line with the arguments forwarded by Peters [PET92]. Calculations of a lifted turbulent dilfusion ftame are accomplished by introducing a probability density fundion for the ftuctuating flame base. It is found that accurate predictions of the lift-off height are possible if the strain rate rather than the scalar dissipation rateis used, in particular the strain rate associated with the small eddies. It is also found that cakulations of the lift-off height itself are possible on the basis of an isothermal jet, which gives support to common practice in the literature.

Finally, in chapter 8 the conclusions are summarized.

1.2 Transport equations

The transport of mass, momentum, and energy is described by the following set of transport equations. The equations consist of the continuity equation

op + v . (pu) = o Ot

(1.1)

with p the mass density and i1 repcesenting the velocity. The density is further defined in terms of the mass fractions Y; and densities p; for N species as

N

p=LPi i=l

(1.2)

with p; Yip. The mass-averaged velocity i1 is defined as i1 Y;v; where v; is the velocity of the i-th species. This leads to the mass dilfusion velocity i/; of species i: i/; v; u. The equation for the mass fraction of species i is obtained from

op; " ( ~) . fit + v · p;v; = p; (1.3)

with p; the production rate of species i. Using Fick's law Y;i/; D;\lY;, with D; the dilfusion coefficient of species i in the mixture, the equation for }'; is

o~~i + v. (pi1Y;) = v. (pDSli) + pY;. (1.4)

9

The momenturn equation in conservation form reads

apil " ( ~~) " " ~ - + V • puu = - V p + V • T + pg at (1.5)

in which pis the static pressure and §is the gravitational acceleration. Fora Newtonian fluid, neglecting the bulk viscosity, the stress tensor is related to the strain rate s;j as [BAT90b]

28 ~ T·· - 2ns··-- ··n" ·u •1 - r •1 3 •Jr v (1.6)

where fl is the molecular dynamic viscosity, Ó;j the Kronecker delta and s;i the strain rate tensor

Sij = ~( aui + auj ). 2 axj axi

(1.7)

The energy equation is most conveniently written in termsof the enthalpy h = e+; = I:~ Y;hi. Here e is the internal energy and h; is the enthalpy of species i

hi = h? + fT Cp,i(T')dT' lro

(1.8)

where T0 is a reference temper at ure and h? is the enthalpy of formation of species i. The energy equation then reads [KU086]

Dh Dp ~ . N ~ p--- =-V ·q+<l>+Q+pLY;fV:

Dt Dt i=l ' ' ' (1.9)

where the substantial derivative {51

is ft+ u· V. In Eq. (1.9) ij is the heat flux vector,

<I> is the viseaus dissipation rate, Q is an external heat souree or sink and the last term is due to body forces, including gravity.

At low Mach numbers viseaus dissipation is neglected 2 and pressure gradients may also be neglected. External heat sourees are excluded in the present study and conser­vative body forces can also be neglected. The heat flux vector ij contains several terms: conduction, with Fourier's law written as (-À VT) in which À is the thermal conduc­tivity; transport due to interdiffusional processes (PL~ hiY;\Ï;); the Dufour effect which is the heat flux produced by concentration gradients; and the flux due to radiation. The Dufour effect is negligible and the radiation flux (in gas phase cambustion without radiant transfer to solid surfaces) is also often neglected [WIL85]. However, in flames in which soot can be present, radiation has to be accounted for. In the present deriva­tion of the enthalpy equation, however, it is not considered. The interdiffusion term is written as pI:~ hiDi f?-. U nder additional assumptions of an equal diffusivity D for

all species and the Le:is number Le = Àj(pDCp,m) being equal to 1, the interdiffusion and conduction terms combine into a single diffusion term. The enthalpy equation then becomes

aph ~ 11 -+V· (puh) =V· (-Vh) at Pr (1.10)

2 Although viseaus dissipation is negligible in the enthalpy equation, which later on is used for cambustion purposes, it is essential in turbulence where it is called t: (pt: = <1>).

10

where Pr is the Prandtl number Pr Cp,m;t/ À with Cp,m the specific heat of the mixture

N

Cp,m(T) 2:: Y;cp,i (1.11) i=l

where cp,i denotes the specific heat of species i. Note that, contrary to the mass fraction Eq. (1.4), the enthalpy Eq. (1.10) has no

reaction souree term. This is used in the calculation of the turbulent diffusion flames. The set of equations is completed by the equation of state for an ideal gas

P pM (1.12) RuT

in which M equals the molar mass of the mixture and R,. is the gas constant.

1.3 Turbulence

1.3.1 Description

At a certain exit velocity of a laminar jet issuing from a nozzle the shear layer close to the nozzle will be subject to, so-called, Kelvin-Helmholtz instahilities [LES90]. These initia! instahilities are generated by the large velocity difference between jet gas and surrounding air. Molecular viscosity is responsible for the exchange of momenturn across the interface between the flows, thereby diminishing the velocity difference. Vortices most effectively transport momentum because the vortical structure increases the length of the interface. Further increasing the jet exit velocity leads to accordingly larger val u es of vorticity and eventually to a fully developed turbulent flow. In such a turbulent flow the velocity fluctuates in both space and time. Mathematically, the origin of the turbulence lies in the non-linear convective terms in the momentum equation. The interaction of vorticity with velocity gradients produces turbulence and this happens most effectively in the large scales of motion [TEN90]. In a developed three-dimensional turbulent flow the vortices are stretched by the mean strain rate, which implies a change of length scale. This mechanism is called vortex stretching. The energy is transported from larg to small scales by this mechanism. At the scale of the smallest eddies, i.e. the Kolmogorov scale, the energy is dissipated by viscous action through the interaction of the strain rate of the smallest eddies with the molecular viscosity. This process is called the turbulent energy cascade. It is useful to depiet this concept by consiclering the turbulent energy spectrum, see Fig. 1.1 The energy density E(K) is a fundion of the waverrumher r;;. The eddies associated with the large integrallength scale Lu contain most

· of the energy, while in the smal! staleeddies with the di mension of the Kolmogorov scale lk the energy is dissipated due to molecular viscosity. The total rate of dissipation of turbulent energy is denoted by ê and on dimensional grounds lk should be proportional to ( v3 / E) 114 where v is the kinematic viscosity. The shape of the intermediate spectrum between fu and ek is the so-called inertial range and is independent of the geometry of the flow.

1.3.2 Turbulence modelling

Modelling of turbulent flows can principally be done in two ways: deterministically and statistically.

ll

ln{ifl) u

Production

range

Slope-513 (lnertial range)

Dissipation

range

In (lC IJ

Figure 1.1: The scaled turbulence energy density E/(u2l,.) as a function of the scaled wavenumber 1d". Here, u is the rms value of the velocity fluetuations. The peak value of fhe Spectrum OCCUrS at Kl,. ~ 1 and it fal/s of fo 0 at Kf". ~ f"j fk.

Determinist ie

The most straightforward way is solving the Navier-Stokes equations directly: the so­called Direct Numerical Simulation (DNS). To resolve the small scales of size lk a very fine grid, and consequently a large computer memory, is needed. The required time resolution makes these calculations also very time consurning. Capabilities of current computers limit DNS calculations (without cornbustion) to low Reynolds numbers and smal! physicalsizes of the flow dornain (of the order of centimetres ). For instance, Spa] art [SPA88] calculated the turbulent boundary layer on a flat plate up to Re= 1410, where Re is based on the momenturn thickness. He used about 107 gridpoints. Since Reynolds nurnbers of turbulent diffusion flarnes are well above 2000 and their physical sizes can be meters, DNS is not practical at present. It must be noted that even these deterministic rnethods suffer from unpredictability for large times due to numerical inaccuracies and incomplete resolution of the srnall scales.

Large eddy sirnulations (LES) [LES90] separate the modeHing of the srnall and the large scales of motion. Large scales are treated 'exact' while the smal! scales are space­averaged and modelled. To obtain an accurate time resolution this method demands very large computer times. Reynolds numbers attainable with LES are high, but because of the rnadelling of the small scales where cambustion takes place, and the associated large computer times, LES do not provide an option for the prediction of turbulent diffusion flames at present.

C)laos theory is used in fluid mechanics mainly in some aspects of the transition to turbulence. How chaos concepts can be applied to fully developed turbulence is not clear at the moment [LES90].

12

Statistica!

All statistica! methods use some kind of an averaging technique. The complexity of the method depends on the amount of information that is desired. Two major approaches are in use. On the one hand correlations in space can be modelled: the so called two­point correlation models. On the other hand there are the one-point models which exclusively take correlations in one spatial position into account. These models are most widely used in modeHing of turbulence in engineering, and have the largest range of applicability. In two-point closures, for instance, mostly homogencity and sometimes also isotropy of the flow is assumed.

In the present study one-point models are used, in particular the k-E model [JON80] with k the turbulent kinetic energy per unit mass and E its dissipation. The turbulent energy is the integral of E( K) over all wavenumbers K. Contrary to the LES method, now all scales of motion are modelled thereby removing information over coherent structures, for example. Coherent structures can be defined as relatively large amounts of fiuid with a certain dynamica! charaderistic that are convected without significantly losing their identity duringa large eddy turnover time k/t:.

One of the basic assumptions in most current one-point closures is that all modelled processes are assumed to proceed with the sametime scale T,. and length scale R,.. On dimensional grounds it is assumed that these scales are proportional to Tu ~ k/E and Ru ~ k312 j E respectively. One further assumption in most of thesemodelsis that the time and length scales for scalar transport Ts and R. are equal totheir mechanica! counterparts Tu and Ru. This assumption is relaxed in chapter 2 and results of the non-equal scales model for scalar transport are presented in chapters 3 and 5.

1.4 Cambustion

1.4.1 Types of ditfusion flames

The most important configurations for diffusion fiames are coflowing and counterfiowing jets, see Fig. 1.2 and 1.3, respectively for a schematic representation. Turbulent flames in the axisymmetric coflowing contiguration are important because they represent the prototype of many flames in industrial applications. Laminar flames in the counterflow configuration, which are called fiamelets, are important because this geometry is suffi­ciently simple to allow computations including detailed chemica! kinetics. Furthermore in the laminar fiamelet model for turbulent combustion the hypothesis is made that they represent the local structures in a turbulent fiame.

Chemica! equilibrium

In most laminar and turbulent diffusion Harnes chemica! reactions proceed much faster than diffusion processes. In such cases the Damköhler number, which is the ratio of a charaderistic diffusion time to a chemica! reaction time, is very large. The fiame is approximately in chemica! equilibrium, a thin reaction zone separating fuel and oxidizer. The most obvious assumption then is to assume infinitely fast rcactions (instantaneous chemica! equilibrium), thereby uncoupling the llow problem from the chemica! problem. Temperature and mass fractions, which obey equations in which a chemica! souree term occurs, are coupled in the form of a coupling function. This coupling function, the so

13

Coflowing Jet

Coflow velocity profile

Nozzle

Figure 1.2: Schematic illustration of a coflowing jet dijJusion flame.

Counterflow Diffusion Flame

Streamline

"'"

Fuel

i 1

i• \ Stagnation r.x.Jint ~_streamline

~t<>nn<'lltlnn ~~

1 Oxid1zer

Figure 1.3: Schematic illustration of a counterflowing dijJusion flame.

14

Bum1ng branch

Tmax Q Unstable branch

lsothermal mix1ng branch

1taq 1/a or the Damkohier number

Figure 1.4: The maximum temperature as a function of the inverse of the strain rate a or the Damköhler number.

called mixture fraction, obeys a souree free convection-diffusion equation that is rela­tively easy to solve. The mixture fraction, which can be interpreted as an element mass fraction, is invariant under combustion.

Strictly it can be introduced only if diffusivities are all equal. If chemica! equilibrium is assumed, the mixture fraction suffices to describe mathematically the structure of a laminar diffusion flame. The first solution, assuming a one-step irreversible reaction mechanism, was given by Burke and Schumann and is called the Burke-Schumann so­lution [BUR28). Later this model is called the 'SCRS'. In a turbulent flame the same concept can be applied if the distribution fundion or the probability density function (pdf) for the mixture fraction is known.

Non-equilibrium effects

For smaller values of the Damköhler number the chemica! reactions are no Jonger ca­pable of maintaining chemica! equilibrium. In a laminar counterflowing diffusion flame (fiamelet) this happens if the velocity (gradient) at which the two streams approach each otl1er becomes large. An increasing velocity gradient (strain rate) decreases the residence time, which is interpreted as the inverse of a strain rate, and also the Damköhler num­ber. Firstly, the temperature wil! decrease and eventually, at a critica! strain rate aq, the flame will extinguish [LIN74). The maximum temperature in the flame as a fundion of the inverse of the strain rate a, or the Darnköhler number, gives the so called S-curve for a diffusion fiame, illustrated in Fig. 1.4. The maximum temperature in an unstretched fiame at high Damköhler numbers is called the adiabatic fiame temperature 3

. At the point I the fiame is ignited and at point Q, the fiame is quenched (extinguished) due to the high value of the applied strain rate. The non-equilibrium effects due to the velocity gradient are called "f!ame-stretch". The name refers to the picture that the f!ame surface is increased at increasing velocity gradients.

The strain rate can be interpreted as a turbulence intensity, and therefore in a tur-

only if all diffusivities are equal.

15

bulent flame non-equilibrium effects, such as localized flame quenching, are the result of too intense turbulence.

Using these features of laminar flames a turbulent flarne model, the laminar flamelet model, can be constructed which incorporates these non-equilibrium effects.

1.4.2 Global jet dilfusion Dame behaviour

If the fuel exit velocity in a laminar jet dilfusion flame is increased the flame length will increase. Here the flame length is defined as the distance from the nozzle to the point on the symmetry axis where the mixture is stoichiometrie.

If the flame at higher veloeities eventually becomes turbulent the flame length, nor­malized by the nozzle diameter, will be independent of the fuel exit velocity. At higher

. fuel exit veloeities the chemica! time scales wil! become of comparable order of magni­tude as the flow time scales, which can be interpreted as the large eddy turnover time kfe. Therefore, the Damköbler number is low, expressing that chemica! equilibrium is hard to achieve, and the flame will be locally extinguished. This leads to a decrea.<re of the mean temperature in the turbulent flame. If the fuel exit velocity is further in­creased, the localized extinctions will become so frequent that the flame willlift-off from the burner rim suddenly and become stabilized at some distance downstream. The axial distance from the nozzle to the point where the flame is stabilized is called the lift-off height. The lift-off height is a linear function of the fuel exit velocity until the total flame blows off. This latter phenomenon, which could be caused by passing of a large coherent flow structure [PET83], is not considered in the present study.

1.4.3 Chemica} kinetics

In chemica! kinetics the dynamica! behaviour of mass and molar fractions is studied. The molar fraction X; of component i, is defined by

X; Y;M/M; (1.13)

where M; is the molar mass of component i. The molar mass of the mixture M can be obtained by summing Eq. (1.13) with respect to i:

N

M l::XM; (L14) Î=l

The production rate of a component i, introduced in Eq. (1.4) as p}i and denoted with M;w; from now on, has to be specified in termsof the temperature and the concentrations of the species through the reaction mechanism. Suppose t.he mechanism is given by M. elementary reaction steps (see also appendix D.3)

N N

l::v:,"M;--+ :L:<"M;, k l, ... ,M. (1.1.5) i=l i=l

in which M; identifies the chemica! symbol for species i. The symbols vi," and v;:k are stoichiometrie coefficients. The production rate w; is the sum of the production rates in all reactions E~1 wi,k· The production rate of species i in reaction k can be written as

16

Wi,k = (vi:k- vi,k)wk. The reaction rate for the k-th reaction is written as (law of mass action)

N I

wk = kk ll x;•.•. (1.16) i=l

The rate coefficient kk for the k-th reaction is modelled according to the empirica! Arrhenius expression

(1.17)

where Ek is the activation energy for reaction kandAk is the frequency factor for reaction k. The frequency factor is approximated by Ak = BkTJ:, with Bk and O:k constants. A table of values for these reaction constants that are used in the present study is given in appendix D.3.

1.4.4 Combustion models

Turbulent flames are to he computed using equations for averaged variables. Gombustion models are necessary because the chemica! souree terms, w;, which also have to be aver­aged, are strongly non-linear with the temperature and with the species mass fractions. This non-linearity causes a problem si nee, for instance, exp( -EI RuT) -j. exp(-EI RuT) where the overbar denotes a time average. This illustrates that the souree term cannot be evaluated by simply inserting the mean variables. However, the souree termscan be averaged exactly in termsof a joint probability density function (joint pdf) P(6 ... Çn):

(1.18)

where 6 ... Çn are the variables on which the souree term depends. These are the tem­perature and the concentrations of many species. Pope [POP85, POP90] explained how to derive, model and solve the transport equation for this n-dimensional pdf, see also [ROE93]. For flames the dimensionality of the pdf prohibits a solution by way of finite difference techniques; a Monte Carlo method has to be used. In the case of hydrocarbon flames so many species occur that the direct application of the method is impossible. It has to be supplemented by carefully derived simplifications of the chemica! reaction mechanism and still, the computational costof the Monte Carlo simulations is very high. Furthermore, the available submodels for mixing on the smallest scales, including molec­ular diffusion, are weak points of the method. When reactions occur in the regime of distributed reaction zones, i.e. reactions are slow with respect to ftuid motions, molecu­lar diffusion does not have to be modelled. However, in flames the reactions are usually very fast compared to fluid motions and consequently the correct representation of local mixing is important.

In the laminar ftamelet model for turbulent combustion [PET84, PET86] a different approach is followed. Here, the emphasis is on the proper description of the local struc­tures in the turbulent flow, rather than on the solution of averaged transport equations for all species and the temperature. The turbulent ftarne is assumed to be an ensemble of laminar flames. For this model to be valid the ftamesurface must be thin4 . If this is the case, then numerical (or experimental) data from suitable laminar ftames can be incorporated in the turbulent flame calculations if they are averaged appropriately. In

4 Formally: thin in mixture fraction space [PET91].

17

this averaging procedure so-called 'presumed'-pdf's are used which are determined by their first two moments alone, contrary to the pdf-transport equation method. In the flamelet method the solution of the equations for the species mass fractions and the tem­pcrature are separated from the turbulent flow calculation. It provides an economical way to include the results of detailed chemica! kinetic mechanisms in the prediction of turbulent flame properties. It also contains a detailed representation of local diffusive mixing and is able to account for non-equilibrium effects such as flame extinction. The latter phenomenon can lead to lift-off or blow-olf of a turbulent flame. The simple chem­ica! equilibrium model (SCRS) discussed in section 1.4.1 can be obtained as a limiting case of the laminar flamelet model.

All these reasons have led to the choice for the laminar flamelet model, rather than the Monte Carlo pdf method for the turbulent flame calculations.

1.5 Turbulence combustion interactions

1.5.1 Inftuence of turbulence on combustion

Turbulence increases mixing by a amount while on the other hand, turbulence in­duces two effects that will ca.use tempera.tures in a. turbulent fla.me to begenerally lower than in a laminar flame. Firstly, turbulence causes flame stretch, and therefore non­equilibrium chemistry, leading to lower Secondly, turbulence introduces the phenomenon of unmixedness. This term reflects that if at some position in space the mean temporal concentration is stoichiometrie, the insta.ntaneous mixture wil! not always he stoichiometrie, due to turbulent fluctuations. Because the temperature at sto­ichiometrie conditions is highest, this implies that the mean temperature in a turbulent flame a.t mean stoichiometrie conditions is lower than the adiabatic flame temperature.

1.5.2 Inftuence of combustion on turbulence

Diminishing of turbulence due to combustion

In a gas the molecular viscosity v increa.ses with temperature, approximately like T 312 •

At increasing viscosity the Kolmogorov dissipation length scale R.k also increases. The turbulent Reynolds number Re~, which is proportional to (R.v./l~:) 413 , wil! decrease be­cause the integral length scale lv. is approximately equal to the physical boundaries of the flow and therefore constant.

In extreme situations a flame is laminarized if the corresponding isothermal flow is turbulent [ CHI79, T AK81]. La.minarization of a flame caused by the increase of molecular viscosity is extremely difficult to model. This effect wiJl be seen to play a role in the flames investigated in chapters 5 and 6.

Increase of turbulence due to combustion

Flames cause large density variations which may increase the turbulence intensity. This type of turbulence production by variabie density effects is called flame generated turbu­lence. It is different from the shear turbulence in a turbulent jet for instance.

18

Generation of turbulence by density fluctuations is caused by the baroclinic term (un­derlined) in the vorticity equation [BOR88, ORA81]

DiJ

Dt iJ. '\lü- ('\l . ii)w + ~'\lp x '\lp+ v'\l 2w.

p (1.19)

Although density gradients are large in turbulent diffusion flames, pressure gradients generally are not, so that the baroclinic term is neg]ected in the present study.

19

20

Chapter 2

Turbulence

In this chapter the turbulence modelling is outlined. The method of mass-weighted averaging (Favre-averaging) that is most often used in turbulent fiows with varying density, such as turbulent flames, is discussed. This mass-weighted averaging effectively removes all density correlation terms from the time-averaged equations. The averaged equations in a flame are essentially the same as for constant density flows, the main influence of the flame being captured by the variation of the mean density, as discussed in chapter 1. Since the Reynolds-stresses a.nd sca.lar diffusion fluxes are also equal to those for constant density flows the influence of combustion on the turbulence is restricted to the density variations.

The turbulence modelling is performed within the k - r; context. Some ways to overcome deficiencies of the k e model are presented and fina.lly a new model for the turbulent scalar transport is derived which allows different scales for mechanica] and scalar turbulence. This model will be called the non-equal scales model. It ma.inly consists of a new expression for the scalar dissipa.tion rate, lea.ding to a.n expression for the eddy-diffusivity that no Jonger contains a fixed turbulent Schmidt number.

2.1 A veraging

If the random variabie <P( x, t) is statistically sta.tionary, a conventiona.l time average can be introduced as

<i>( x, t) (2.1)

where T is much smaller than the time scale of the large scale (time dependent) motion. It will be shown later that if density fluctua.tions occur, such as in a turbulent fla.me, ma.ss or density-weighted averaging is more appropriate, i.e., insteadof Eq. (2.1)

~(i,t) l r+ lt-LT p(i,r)t/J(x,r)dr.

2

(2.2)

This type of a.veraging is usually called Favre-averaging [FAV69]. The turbulent jets and fla.mes in the present study are stationary with respect to

times much larger than the averaging time T, (j) and ~ are only functions of i.

21

2.1.1 Conventional averaging

Conventional averaging for flows with varying density leads to rhany density-velocity correlations, that have to be modelled. Although the method is not used in this study it is discussed here to illustrate the problem that emerges with this type of averaging.

The conventional averaging procedure consists of the so-called Reynolds decompo­sition of every variabie 4> in a conventionally time averaged part 4) and a fluctuation

4>' 4> 4> + </>'. (2.3)

The averaging technique is "idempotent" which means 7fo 4). Furthermore the time average of a fluctuation is zero: 4>' 0.

The governing equations for the averaged quantities are obtained by time averaging . the variables in the instantarreons equations. The averaged continuity equation becomes

\1 . (p1Ï) + \1 . (p'iP) = 0 (2.4)

where correlations emerge. These correlations, which are zero in an incompressible flow, are non-zero in a flame.

In the momenturn equation, time averaging of the non-linear convective terms gen­erate correlations of the type u~uj, which are called the Reynolds-stresses. They appear in the averaged momenturn equation for u;

a +-a (r;i

Xj p'u:uj) + pg; (2.5)

as apparent stresses. However, also correlations involving density fluctuations emerge. Forthese unclosed terms equations can be derived which, however, contain higher order correlations: the so-called third-order moments (because they involve three velocity ftuctuations) and so on. In the present study only second-order moments are modelled. This means that they are expressed in terms of lower order moments, which are the mean variables themselves.

In order to model the Reynolds-stresses, first their differential equations are obtained by multiplying the equation for u: by uj, adding this to the equation for uj multiplied by u: and averaging the resulting equation. For instance, the momenturn equation for u:, needed in this procedure, is

Du: Dt

in which the material derivative is

D

Dt

(2.6)

(2.7)

The Reynolds decomposition with conventional averaging is not further pursued because too many density-velocity correlations would appear in the equation for the Reynolds­stresses. To circumvent modeHing these terms the Favre-averaging technique is now introduced.

22

2.1.2 Favre-averaging

With mass-weighted averaging, the variabie </>is decomposed into a mass-weighted quan­tity ~ and the fiuctuation 1/>11

The mass-weighted quantity is, according to Eq. (2.2), defined as

- pi/> </>= -.

p

Some characteristics of Favre-averaging are (see appendix A):

ptj!' = 0; </>" = -;p ~

1-­+ -p'u"u" + p • J

p' t/>" --#0· p )

(2.8)

(2.9)

(2.10)

From these relations it is clear that, contrary to a conventional average the average of a Favre-ftuctuation t/>11 is not 0, and that density-velocity correlations are needed if the transformation from a Favre to a conventional average is to be made.

The merit of Favre-averaging is the absence of density fluctuation termsin the time averaged equations. The equations now read

V· (p~) = 0 (2.11)

V·(~~) -Vp+V·(T+R)+pg (2.12)

where the Reynolds-stress tensor is Favre-averaged R;i = -pu;'u'J. The overbar over the velocity corr_elation does not mean a conventional time average but denotes the range of

the tilde: u?u'j = pufu'J fp. The Reynolds-stress transport equation can be obtained by multiplying the momen­

turn equation for u; with uj, time averaging and adding the same equation with i and j interchanged:

(2.13)

The terms on the left hand side all originate from the non-linear convective termsin the momenturn equations. These are the convection of the Reynolds-stress, the production (two terms) and a third-ordt:,'f transfer term. On the right hand side appear the pressure terms, and the terms representing the interaction of the fiuctuating motion with the molecular stresses, leading to dissipation terms. The latter are rearranged by noticing that in a highly turbulent flow

(2.14)

23

Furthermore the stress tensor Tij will be written as Tij 2ps;j for this derivation of the dissipation terms. Then aftersome manipulations the last two termsin Eq. (2.13) become

_ fJu'[ fJu'j 11

fJ fJu~ 2pv--- + u·-(Jl-) +

fJxk fJxk 1 fJx; OXk (2.15)

The first term represents molecular diffusion of Reynolds-stress and the other terms are tbe dissipation of Reynolds-stress. The dissipation terms can be written in terms of a dissipation rate tensor E;1

(2.16)

in which the first term is the most important one. The second and third terms are called dilatation dissipation and are usually neglected. They are only important in supersonic flow, but in flames which are stuclied in this work their in~uence is negligible [ZHE92].

The equation for the turbulent kinetic energy k = ~ui'ui' is obtained by contraction (i= j) of Eq. (2.13):

1 {J --- --(-p u"u"u" + \7 · 11\7k

2 OXk ' ' k r (2.17)

In this equation the first term on the right hand side is production of turbulence by the interaction of the Reynolds-stress with the mean velocity gradient. This term, with an opposite sign appears also in the equation for the kinetic energy of the mean flow (not given). So clearly the turbulence is sustained by the mean flow gradients. The second term is due to pressure work, the third and fourth terms are transport of turbulent kinetic energy by turbulent and molecular diffusion. The last term represents dissipation of turbulent energy by viseaus motion and is called E.

2.2 The k- ê model

All correlations of the fluctuations in the preceding section have to be modelled. For the Reynolds-stresses several ( one-point) dosure models exist. The most generally applica­ble are the so-called two-equation modelsof the k-E type, and the so-called second-order models. The former category comprises, additional to the equations for the mean quan­tities, equations for the turbulent kinetic energy k and its dissipation ra te E. In the latter model transport equations for the 6 components of the Reynolds-stress tensor and the dissipation rate equation are solved. In these equations, however, third-order mo­ments, and other terms, have to be modelled and a large number of empirica! coefficients appear. Although these second-order models are more genera!, in the case of turbulent free jets the k- E model perfarms wel! [LAU74). In this model the Reynolds-stresses are expressed in analogy to the constitutive relation in laminar flow of a Newtonian fluid, in which the molecular viscosity is replaced by an effective viscosity:

(2.18)

24

with the effective viscosity fteff = ftt + ft· The turbulent viscosity ftt is usually much larger than the molecular viscosity, and therefore fteff ftt is a good approximation. The secoud and third termsin Eq. (2.18) ensure that 2k (in constant density as well as in variabie density fiows ).

Velocity-scalar correlations are modelled in an anaiogous manner by introducing an eddy-diffusivity ft•· It is often taken proportional to ftt, although this is not done in the non-equal scales model, to he discussed later. The constant of proportionality is called the turbulent Prandti or Schmidt number O"t,~ in analogy with larninar flow, i.e.

(2.19)

The turbulent viscosity Jlt can he defined by introducing a typical turbulent large scale length l,. "' P/2/6 and a large scale time scale Tu "' k/6, as Jlt "'pl:fr,. leading to

p CJl.p-.

6 (2.20)

Now the equation for the turbulent kinetic energy, Eq. (2.17), needs to he closed by introducing models for the various unknown terms. The pressure term is written as

-u(' 8p = -u~' Op - 8p'u:' + p' au;' ' 8x; ' 8x; 8x; axi.

The terms on the right hand side are a term representing transport due to a mean pres­sure gradient, the gradient of the pressure-velocity correlation and a pressure fiuctuation­dilatation term. The latter can safely he neglected [JAN92, ZHE92]. According to Strahle [STR90] the second term can he important but not in a free jet diffusion fiarne. Therefore this term is lumped together with the turbulent and molecular diffusion terms using a gradient transport hypothesis and a fixed Prandtl number O"k:

18-­---u"u~'u'! +V· ft'ii'k

2 axj ' ' J

Finally the term due to the mean pressure gradient is only non-zero in a variabie density flow. Then u;' is non-zero and it is modelled using Eq. (2.10) with p'u:' handled by a gradient transport assumption such as in Eq. (2.19), i.e.

11 Op 1 Jlt Op Op u-------' axi - p2 O"k 8x; 8x;.

The equation for turbulent kinetic energy in its final form then reads

(2.21)

where P~r -pui;u'JfJJf:; represents the production of turbulent energy. Note that the pressure gradient-density gradient term is zero in the absence of pressure gradients. In the free jet diffusion flarnes considered in the present study the pressure gradients are sufficiently small to negiect this term.

25

The k e model is complete if the equation for the dissipation rate e is known. A model e-equation is more difficult to derive from first principles and therefore this equation is based on physical reasoning, dimensional arguments and coordinate invari­ance [HAN72]. The resulting dissipation rate equation is analogous to the equation for the turbulent kinetic energy, Eq. (2.21 ). If the large eddy frequency e/ k is introduced to model the production and dissipation terms, and a gradient transport hypothesis is employed with a Prandtl number ere, the dissipation equation can be expressed as

V· (pile) =V· (/LeffVê) + ~(C1Pk C2pê). er, k

(2.22)

Here, the constants C1 and C2 are determined empirically by consiclering various equilib­rium turbulent boundary layers and homogeneons turbulence behind a grid, respectively.

The five constauts in the standard k-é model [JON72, LAU74] are given in Ta.ble 2.1.

Table 2.1: Standard model coefficients in the k- ê model.

2.3 Discussion

2.3.1 The k- E model

The standard k - ê model relies on two basic a.<>sumptions. Firstly, one single time scale kje for both turbulent transport and dissipation is used. Secondly, isotropy in the models for the Reynolds-stresses and scalar fluxes is assumed. Now some generalizations that have been proposed in the literature are discussed.

Multiple time scales. The k ê model, as well as the more elaborate second-order transport models, are all based on the assumption that turbulent transport and dissi­pation of turbulent energy proceeds with the sametime scale k/é. Consequently the small scale variations respond instantaneously to large scale variations, which means that the turbulence is in approximate equilibrium. In reality this appears to be often the ca.<>e, and the success of the k- ê model in engineering applications might be due to the fact that many of theseflowscan bedescribed with only one time scale [HAN79]. A new model with separate time scales for large and small scales of motion was proposed by Hanjalic et al. [HAN79] and more recently a refined version has been forwarded by Kim and Benson [KIM92]. In these roodels the turbulent energy spectrum is divided into several, usually two, zones which represent the large scale motion (production of turbulence) and the smallscale motion (transfer region, in equilibrium with the dissipa­tive region), in analogy with the Eddy Simulation (LES) technique. Associated with each zone is a separate time scale which can account for possible non-equilibrium effects in the turbulence energy spectrum. Non-equilibrium effects are here defined by inequality between the rates of production and dissipation.

26

The multiple ( two) time scales model mathematically consists of 4 instead of 2 equa­tions for turbulent energy and dissipation: two equations for every wavenumber region. For the production range these are most similar to the standard single time scale model, l.C.

'\7 . (pilk") ( j/,ef[ (2.23) = '\7. -'\?kp) + pk pêp Uk,p

'\7 . (pilêp) ( JLeff ) 1 C Pkêp ê2

= '\7 . -'\7 êp + Cp,L c - p (2.24) + p,2-k- p,3Pk· Ue,p p p p

The energy in the production range, kp is transferred by its dissipation rate êp into the transfer range with which the energy k1 and dissipation rate Et are associated. It is assumed that the transfer range is in equilibrium with the purely dissipative part of the spectrum, such that ê 1 equals the energy that is dissipated into heat in the smallest scales. The equations governing the transfer range are:

'\7 · ( JLeff '\7 kt) + pêp - Pêt uk,t j/,ef[ ê;

'\7 ·(-VEt)+ Ct,JPk + (J'E 1t t

(2.25)

(2.26)

The form of the equations for k" and k1 is physically evident. The dissipation equations have been derived applying physical dimensional analysis [KIM89] and each contain one extra term, compared with the dissipation equation in the standard k ê model.

These terros ( Cp,l ~:f!. and C1,1p.(,) are the so-called variabie energy transfer functions [KIM89]. If turbulence is in equilibrium both dissipation rates are equal (êp E-1). Also the production rate Pk of turbulent energy must then be equal to the dissipation rates.

In addition the eddy-viscosity must be quantified. It is defined analogously to the single-time scale model with ê replaced by êp· This reflects the idea that the turbulent length scale is associated with the large scale motion:

(2.27)

where k = kP + k1 is the total turbulent kinetic energy. The constauts in the model were determined by Kim and Chen [KIM88] and are

reproduced in Table 2.2. It can be noted that in the case of equilibrium turbulence in

Table 2.2: Model coefficients in the multiple time scale k- ê model.

which Pk = pêp p€1 the coefficients in the equations resembie those in the single-time scale model.

27

Anisotropy. The second assumption in the standard k c model is isotropy. This is related to the fact that the same eddy-viscosity is used to model all Reynolds-stresses. The non-linear model of Speziale [SPE87], basedon the analogy of non-Newtonian larn­inar flows and turbulent Newtonian flows, consists of anisotropic expressions for the Reynolds-stresses. This model comprises a non-linear model for the Reynolds-stress tensor R with higher order terms than given in Eq. (2.18). The non-linear terms involve higher derivatives of the velocity field and productsof velocity gradients. The exact form of the terms is determined by subjecting them to the demand of material frame indif­ference [SPE79]. Preliminary computational results with this model for the flow behind a backward facing step are promising with respect to predictions of Reynolds normal stresses and the size of the recirculation zone [SMA92]. It is impossible to predict these correctly with the linear version of the Reynolds-stress model Eq. (2.18). However, con­vergence rates of the numerical procedure are poor, and a converged salution can only be obtained if the velocity fields are filtered before higher derivatives are calculated. Fora turbulent jet the method did notshow a different behaviour. Moreover, since the most important Reynolds-stress in a jet is u'v', the impravement of the predietien of the normal stresses is probably only of marginal importance for this type of flow. Therefore this model is not used in turbulent combustion calculations.

2.3.2 Averaging

Favre-averaging is mathematically preferabie over conventionally time averaging and simply removing the density-velocity correlation terms. With Favre-averaging formally no terms are neglected and a system of compact equations is retained. Furthermore, the interpretation of terms involving density correlations is difficult in the conventional averaging technique. However, the modeHing relations for Reynolds-stresses such as Eqs (2.18), (2.19) and (2.20), are the sameasin constant density flows. The question whether this is justified should be answered. Furthermore, although Favre-averaged vari­ables eau readily be calculated, some experimental measurements give Favre-averaged quantities while other measuring techniques give conventional averages. Also this prob­lem needs some discussion.

ModeHing of Favre-averaged correlations

Firstly, it should be noted that the set of equations for Favre-averaged quantities is equal to the set for conventionally averaged quantities, in which all density fluctuation terms are removed, if the pressure gradient-density gradient term in Eq. (2.21) is 0. In free diffusion flames which will be considered only, pressure gradients are small and this term is negligible. In this case the equations for the Favre-averaged quantities are identical to those for conventionally averaged quantities with density fluctuation terms neglected, or in other words to equations for conventionally averaged quantities for constant density flows. For the moment it is assumed that modeHing assuinptions employed for constant density flows are transferable to the system of equations for Favre-averaged quantities.

Secondly, it must be noted that in the equations for Favre-averaged quantities still conventional time averages appear for the density and pressure, and effectively also for the molecular stress r, although in Eq. (2.14) the latter has been written in termsof a Favre-decomposition for simplicity. This mixture of terms is a fundamental problem [YOS92] but it is not of great practical importance because 1) the molecular stress is

28

much smaller than the Reynolds-stress, 2) p is not physically significant, and 3) density fiuctuations are assumed to be directly correlated with temperature fiuctuations, inde­pendent of pressure and therefore Favre-averaged and conventionally averaged pressures are equal.

The rnain reason for not altering modeHing assurnptions for terms in equations for Favre-averaged quantities is the crudeness of the constant density approaches them­selves. The inaccuracies which are irnplicitly accepted by introducing gradient transport concepts and eddy-viscosity and ditfusivity are probably too large to justify subtie Favre­rnodifications on them, [JON92, BIL80].

In higher order turbulence models such as a full second-order model, modeHing as­sumptions are somewhat less severe, thereby ma.king an explicit account of density fluc­tuations more appropriate [JAN81].

Measurements

In general optica! measuring techniques yield conventionally averaged quantities. With Laser Doppier Velocimetry the velocity of small (smaller than 0.5pm) seeding particles is measured. Favre-averaged veloeities can be obtained if the density is measured at the sarne time and position. The ditference between Favre and conventionally aver­aged veloeities is due to the term which is usually negative, rnaking Favre-averaged veloeities somewhat smaller than their conventional counterparts. However, the ditfer­ence between Favre and conventionally averaged veloeities is very small (less than 4%) [STA81, DRI82, DRA86].

Thermo-couples which are thinner than about 40~tm give conventionally averaged temperatures while Favre-averaged temperatures are obtained with couples thicker than 40~trn [BRA81]. Cornposition measurernents with sampling probes yield Favre-averaged mass fractions [BIL 77].

Predictions in the present study yield Favre-averaged variables ( except for the density and pressure), while for scalar quantities such as temperature, mixture fraction and its varianee and also composition conventional averages can be obtained, see chapter 4.

2.4 Scalar transport: Standard model

The enthalpy and mixture fraction are scalars which obey convection-ditfusion type transport equations which are completely similar. Derroting the scalar by ~ this (general) equation reads

- - tt -\J · (purp) = \J · ( -\Jrp) + \J · T 5 crq,

(2.28)

where O'rJJ is the molecular Prandtl or Schmidt number, and r, is the turbulent scalar ditfusion flux

(2.29)

the modeHing of which is normally done using Eq. (2.19). For the description of the unmixedness in a turbulent ditfusion fiame the level of

fluctuations i~ the mixture fraction must be quantified which can be done by consiclering

its varianee f" 2 = g. The corresponding transport equation is obtained by multiplying

29

the equation for the instantaneous mixture fraction with the mixture fraction itself and averaging. The result is

V-(pilg) ft V · (-V g) + V · rs + P9 O'g

(2.30)

which is the scalar analogon of the equation for k (2.17). In the derivation of this equation the notion that all gradients of the fluctuating mixture fraction are much larger than ~gradients of the mean mixture fraction is used. The turbulent flux rs stands for

-pui'g" and can be rewritten using Eq. (2.19). The production of the mixture fraction fluctuations is defined as

(2.31)

The scalar dissipation rate is defined by

(2.32)

and expresses the smoothing of scalar fluctuations on the smallest scales by the molec­ular diffusivity 1 . A classica! model for this variabie is based on the assumption of proportionality of time scales for mechanica! and scalar turbulence k/s"' gfs9 with a constant of proportionality equal to 2 [BEG78]. A new model for the scalar dissipation in which this assumption is not made is discussed in the next section.

In summary the standard scalar transport model, which from now on will be called the equal-scales model, consists of the following equations

V· (pilg) V . (-lleff_Vg) + :!.n'--'---"­ug

ê 2p-g

k

(2.33)

(2.34)

with u1,q, and u9 are fixed turbulent Schmidt numbers. For the mixture fraction equation a 1,q, = u1 . It is common practice to take the Schmidt numbers O'J and u9 both equal to 0. 7 if the k- e model is adjusted so as to improve agreement with experiments on round jets. If the standard model is used often 0.9 is taken.

2.5 Scalar transport: Non-equal scales model

2.5.1 Introduetion

TSDIA

The new scalar transport model is basedon the Two-Scale-Direct-Interaction- Approxi­mation of Yoshizawa., see appendix C. This statistkal theory uses the Direct Interaction Approximation (DIA) of Kraichnan [KRA59] which is a two-point turbulence theory, valid for the smallest sca.les of turbulence. The TSDIA provides the link to the larger

1 Whether the scalar dissipation lengt.h scale is larger or smaller than the Kolmogorov dissipation length scale depends on the molecular Schmidt number.

30

scales by performing a scale expansion on the transport equations. In this way it is possible to derive a turbulent constitutive equation and expressions for the scalar dif­fusion flux as an expansion in the mean field gradients. The non-linear terms in these expressions descri he anisotropies generated by gradients of the mean fields and they are similar to extensions on the linear model proposed by Speziale [SPE87] and Rubinstein et aL [RUB90]. Furthermore equations for the mechanica! and scalar dissipation rate .can he derived which resembie the standard empirica! equations closely.

The non-equal scales scalar transport model consists essentially of new expressions for the eddy-diffusivity and scalar dissipation rate which are to he applied to Eqs. (2.28) and (2.30). It is shownon dimensional grounds that these two parameters have an inter­relation. The TSDIA scalar dissipation rate equation is solved, thereby also providing an expression for the eddy-diffusivity. In this way the assumption of equallength scales and time scales for mechanica! and scalar turbulence, leading to a constant turbulent Schmidt number is no longer needed. The equation for the scalar dissipation rate is compared with other proposals for this equation in the literature, which always have been made in the context of second-order closures rather than the k - t: type of models.

Dimensional considerations

Turbulent scalar transport represented by the scalar ditfusion flux Ts,i given in Eq. (2.29) is to be modelled analogously to the Reynolds-stress. A scalar eddy-diffusivity !la must then he introduced. This turbulent exchange coefficient expresses the additional trans­port of mass or heat by the action of the turbulence. Associated with a diffusivity is a length scale f. f and a time scale Tf, which give the expression

_iJ /la rv p-. (2.35)

TJ

Now, the variables which govern the scalar transport are the scalar fluctuations g, the scalar dissipation t:g and the dissipation rate of turbulent energy t:, which accounts for the influence of the turbulent velocity field. This can be deduced from the inertial range (equilibrium) spectrum for scalar fluctuations E9 [TEN90]

(2.36)

This spectrum is valid for scales smaller than the integral scale f. f and for scales larger than the scalar dissipation scale because the diffusivity does not appear in Eq. (2.36) 2 The scalar energy contained in scales smaller than the dissipation scale is negligible. And if the spectrum of scalar energy is extended to the energy containing eddies an expression for f. 1 can be obtained by noting that

(2.37)

Evaluation of this integral gives

(2.38)

the Kolmogorov scale fk is larger than the scalar dissipation scale then the spectrum is only valid for scales large than ~'•·

31

The time scale TJ can only be g/t:g and consequently the eddy-diffusivity is modelled as

(2.39)

where the constant of proportionality appears. In the equal-scales model Eqs. (2.33) and (2.34) the scalarand mechanicallength or

time scales are proportional. This means f1 ~ lu or TJ ~ Tu. Proportionality of these scales results in the expression for Eg

€g ""gk. (2.40)

The eddy-diffusivity in Eq. (2.39) then equals the one given in Eq. (2.19), lts ~td(/t,<J>· The proportionality constant in Eq. (2.40) is mostly taken to be 2, which implies that the mechanica! to scalar time scale ratio, defined as

(2.41)

is constant and equal to 2. It is emphasized that a constant time scale ratio implies a constant turbulent Schmidt or Prandtl number. The interrelation between the two is given by Set,</>"" R;. Fixed turbulent Schmidt numbers are denoted by (Tt while variabie Schmidt numbers are denoted by Sc,_.

The main features of the equal and the non-equal scales scalar transport models are now briefly recapitulated. The eddy diffusivity lts that related the turbulent diffusional

flux u/<fo" to the scalar gradient á~jäx; is given as

Equal scales lts

Non equal scales lts

and the scalar dissipation rate is given as

Equal scales é:g

Non equal scales êg

ltt

(/!

g2E: Cf-e2

IJ

€ 2-g

k ,P/''€),2

(2.42)

(2.43)

(2.44)

(2.45)

where the scalar dissipation rate of the non-equal scales model already has been given, although the forma! discussion of this variabie is presented in the next section. It is emphasized that the expression for e9 in the equal-scales model relies on the assumption Rr = 2, which is not needed in the non-equal scales model.

Measurements on Rr

From measurements on and calculations of turbulent jets and dilfusion flames presented in the liter at ure it can be concluded that mechanica! and scalar scales are not equal. The measured Prandtl number in the heated round jet of Chevray and Tutu [CHE78] was not constant, neither was the Prandtl number measured by Sarh [SAR90] in a heated rectangular jet or the mechanica! to scalar time scale ratio found in in a helium-air jet

32

[PAN93-2]. Drake et al. [DRA86] argued that a constant turbulent Schmidt number may not at all be a useful concept in turbulent flames. Shabbir and Taulbee [SHA90] found a varying mechanical to scalar time scale ratio in their experiment on a buoyant axisymmetric plume. I"astly, Dibble et al. [DIB86] calculated the time scale ratio in a turbulent ditfusion flame using a second-order turbulence model and also here the ratio was not constant. Clearly the equal-scales model, in which a fixed RT is assumed is at varianee with these experiments. It is shown in the following chapters that the trends which are observed in these experiments are reproduced qualitatively by the non-equal scales model to be presented below.

A transport equation for the scalar dissipation rate must be constructed to cir­cumvent the assumption of equal-scales for scalar and mechanica! turbulence. In the literature most equations for the scalar dissipation rate are within the frarnework of second-order closures. However, Yoshizawa [YOS88] bas given a consistent derivation for this model equation within the k- e context witb his Two-Scale-Direct-Interaction­Approximation (TSDIA), see appendix C.

2.5.2 Dissipation rate equations from the TSDIA

With tbc TSDIA it is possible to derive k - e type of equations for k, e, g and e0 , as well as modeHing relations for the Reynolds-stresses Rij and the scalar ditfusion fluxes

In the present study only the TSDIA form for the scalar dissipation equation and eddy-diffusivity are used. For the sake of comparison with the standard model the TSDIA mechanica! dissipation rate equation is given as well.

Mechanical dissipation rate

The TSDIA e-equation reads [YOS87]

wbile

De ê Dk Dt = 1 k Dt

(2.46)

(2.4 7)

withï 1.7. DkisadiffusiontermDk='V·(Ckk~\lk) \l·(Ck<~\le)withCkk 1.27 and Cke 0.72, including cross-ditfusion effects [YOS87]. The solution of Eq. (2.46) can be derived to be [SAN93a]

k é = êo(-)1'

ko (2.48)

where e0 and ko are the values of k and é insome reference point in the flow field. The cross-ditfusion in Eq. (2.46) represents the ditfusion of turbulent kinetic energy

due to gradients in the dissipation rate, and vice-versa. The importance of cross-ditfusion is not clear at the moment, and it is always neglected in standard empirica! models. The standard k-Eq. (2.21) only contains the standard ditfusion D = \J. (~\lk). Tberefore the coefficient Ckk cannot be compared with the one in the standard model C ."/ CTk 0.09. However, with the solution ê êo( f;; )"~ the gradient of ê can be turned into a gradient of

k. Now the second term becomes \J. (Ck<Ï~\lk) and consequently the sum of the two

ditfusion termsis \J. ((Ckk -ïCk,)~ \lk) from which the value Ckk-ïCke 0.046 can be

33

compared with the standard value of 0.09. This also indicates the range of applicability of the constants generated with the TSDIA. In the following the cross-diffusion effects wil! be neglected because they lead to the single diffusion term if use is made of the analytica! solution of the dissipation rate equation.

The TSDIA k - E equations, without cross-diffusion, are similar to those in the standard model, including the coefficients which have values close to the empirica! ones in Table 2.1. The evaluation of the coefficients in the TSDIA involves approximations of for instanee equilibrium (inertial range) spectra of turbulent energy and they can be calculated theoretically. These numerical values of the coefficients are approximate due to the mentioned assumptions and should be modified empirically.

Scalar dissipation rate

The equation for the scalar dissipation rateis obtained from an expansion of the equation for e f' see appendix c

e1

= C g3/2E-3/2EI/2 + C g3/2E-5/2E1/2 Dg _ ft g ftg g Dt

C gsf2E-7/2EI/2DEg + C gsf2E-5/2E1/2DE. ft'• g Dt ft• g Dt (2.49)

A model equation for Eg can be obtained by demanding transferability, which means that any model based on (g, Eg) should be transferable to a model based on (g, l!g). Th is is also the case in the standard k- E model, where in principle any coup Ie from ( k, E, Ru) can be chosen. Transferability requires an algebraic relation between (g, Eg and €1 ) and therefore Eq. (2.38) must hold. From Eq. (2.49) the scalar dissipation equation can then be derived to be

DEg = E [À }:_Dg À ~DE]. Dt g 1 g Dt + 2 E Dt

This equation can be rewritten using the equations for g and E as

DEg- D Eg Eg Eg 2 EgE Dt - '• + C,.,I gPg + C,9 ,2kPk- C •• ,r-g- C,.,rk

(2.50)

(2.51)

where the diffusion term D,9

contains cross-diffusion effects from both equations of g and E. The constants C,.,1 ••. C,

9,4 follow from the values of À1 and À2 and the constants

appearing in the equations for g and E. The constants C,., 1 and C,.,3 have equal values since they are related to the equation for g. The remairring two constants are related to the E-equation and are also equal.

From Eq. (2.51) it is clear that the scalar dissipation ra te is sensitive to a scalar as wel! as a mechanica! mechanism: decaying mechanica! turbulence wil! result in decaying scalar turbulence and production of mechanica! turbulence wil! in general (if there are gradients of the mean scalar) result in production of scalar turbulence.

The solution of Eq. (2.50) can be observed to be [SAN92]

(2.52)

with 1/J a dimensional reference value

(2.53)

34

which depends on the flow considered. It is remarkable that this solution, and also the one for the mechanica! dissipation rate Eq. (2.48), apparently ha.s not been considered in the literature. A very important feature of (2.52) is the absence of the turbulent kinetic energy k, compared with the equal-scales version t:9 = 2gt: / k.

The dimensionless constants À1 and À2 were determined by Yoshizawa [YOS88], [YOS84b] using inertial range concepts, to be 1.2and 0.306 respectively. In the following chapters it will be shown t~at these constants need modifications.

The scalar fluxes -pu;' f" and are modelled (in their isotropie form) as [YOS88]

-pu:'J" (2.M)

(2.55)

with Cf 0.446; C99 = 2.51; C9 • 9 = 2.29; C9 , 0.872 [YOS88]. From the absence of k in the ditfusion terms it can be concluded that the new scalar transport model is partly uncoupled from the mechanica! turbulence, unlike the equal-scales model. The cross­ditfusion terms in the equation for the scalar varianee flux (bet ween brackets) are not considered in the present study. Although the first term within brackets would couple the first term on the right hand side, giving an ordinary ditfusion term, the second term within brackets would still represent cross-ditfusion3

• For reasous of compatibility with the equal-scales model, however, no cross-ditfusion is considered. Therefore also the coefficients Cf and C99 are taken to be equal [SAN92] which leads to

-pu:'g"

8] li-s ~x· u,

fJg Jl-s--;;­

UXi

where the eddy-ditfusivity p. is modelled according to Eq. (2.43).

2.5.3 Comparison withother models

(2.56)

(2.57)

One of the first scalar dissipation rate equations was proposed by Zeman and Lumley [ZEM76], and was remodelied by Borghi [BOR90] for a type of flow with only one dominant gradient (a turbulent jet). The equation given by Borghi is

(2.58)

with <J:e9

= 1.3 and <J i 0.83. This equation eau be rewritten into the form of Eq. (2.51) and the resulting coefficients are given in Table 2.3. Several remarks regarding this equa­tion eau be made. Firstly, the original equation by Zeman and Lumley did not contain production terms. However, in the production term associated with scalar gradients a fixed turbulent Schmidt number appears. In the present study fixed Schmidt numbers

3 Note that this is different from the diffusion term in Eq. (2.47) which was shown to represent only ordinary diffusion w hen the analytica! solution for t: was used.

35

can be avoided. Also the factor E / k in front of the squared scalar gradient is differ­ent from other models which employ Eg/ g. Secondly, the coefficients in the dissipation terms are probably not consistent. Zeman and Lumley defined the scalar dissipation rate as the dissipation associated with the destruction of !g, while the dissipation used by Borghi and in the present study is associated with the destruction of g. Therefore the dissipation termsin Eq. (2.58) should be -1.9pEg(~ + -fk). Thirdly, the diffusion term in Eq. (2.58) does not contain cross-diffusion effects. This is because all classica! mod­eHing approaches do not incorporate cross-diffusion, while it is inherent in the equations derived with TSDIA.

Another model equation is the one proposed by Newman et al. [NEW81], also used by Elghobashi and Launder [ELG83] in the absence of mean velocity gradients. It includes both dissipation terms and one production term due to mean scalar gradients. Dibble et al. [DIB86] use this equation for the calculation of an Ar/H2 flame, and consequently had to add a production term due to mean velocity gradients. Jones and Musonge [JON88] base their model on the proposal of Lumley [LUM78] and determine the constants by consiclering temperature variances in homogeneous flow behind a grid withand without mean temperature gradient.

Shih et al. [SHI90] derive an Eg-equation by applying invariance principles and invok­ing equilibrium turbulence, which is the reason for the absence of the Pk-related term, see Table 2.3. The comparison of their model with the others is difficult due to the

Author Ref. c,g,I c,g,2 c,g,3 c,g,4

Borghi [BOR88] ~_L 1.0 0.95 0.95 k <g

Newman [NEW81] 1.0 0 1.01 0.88 E+L [ELG83] 0.9 0 1.1 0.8 Dibble [DIB86] 1.0 2.45 1.0 0.9 J+M [JON88] ~_L 1.45 (= C1) 1.0 0.9

k <g

Shih [SHI90] _!7j;gSI. 0 1.0 !7/Jo -1+ 2 I g

':iJJ(1- ..&...) 2 Rr,e

Yoshizawa [YOS88] 1.2 0.52( = 0.306Î) 1.2 0.52 Present Section 3.4.1 1.5 0.25 1.5 0.25

Table 2.3: Coefficients in the scalar dissipation mte equations by various authors. Coef­ficients refer to Eq. (2.51).

functional forms of the constants which are given here for completeness. An equation for Eg proposed by Lumley is [LUM78]

DEg = D _ ~ E; 7/Jg Dt <g 2 g

(2.59)

with

(2.60)

in which

(2.61)

36

2+?b.::::J_ _&_ g _ . Rr,e Rr,e

{

( ) ( )

10

7/;1

- (2+~.~.2 )(1-0.1(1 if RT < Rr,e

) if RT > RT,e (2.62)

with 7/;0 Jt + 0.98exp ( -~~:) [l- 0.33ln (1 55I/)] ; Ret = 4k2 j9E:v; Il -!b;jbij

where bij is the anisotropy tensor b;i ""'u:uj/2k ~hij· Lastly, 1/;1 2.4 and Rr,e is the equilibruim value of R7 in the limit of isotropie turbulence, taken to be 1.6. Note that in this model, just as in the model used in the present study, a time scale ratio criterion is used, see section 3.4.1.

Most striking dilierences in Table 2.3 are seen in both production constants, where sometimes even the appearing variables are unequal (C •• ,1 ) and large numerical differ­ences occur in C,

9,2 • The condusion from this comparison is that the model used in

the present study is complete in the sense that all terms proposed by other authors using completely different arguments are present. Moreover it can readily be used in the context of the k - é model. The dilierences with the other models are the occurrence of cross-dilfusion effects, and the specific values of the coefficients.

2.5.4 Conclusions and discussion

The k-E: model has been presented with a discussion of its shortcomings and possibilities for improvement. Mass-weighted averaging (Favre-averaging) of the dependent variables leads to a compact system of equations. Although optica! measuring techniques yield conventionally averaged variables, Favre-averaging is preferabie because in experiments dilierences between both averaging methods are small. Scalar variables can be computed in terms of either method and consequently they can be adjusted to the experimental type of averaging.

For the transport of scalar variables a model has been proposed that is based on different scales for mechanica! and scalar turbulence: the non-equal scales model. The expression for the scalar dissipation rate in this model does not contain the turbulent kinetic energy in contrast with the equal-scales model and this is in line with the con­cept that scalar turbulence is governed by g,E:9 ,E: and Rt. Furthermore, the non-equal scales model does not assume a fixed turbulent Schmidt number, or equivalently a fixed mechanica! to scalar time scale ratio, which also agrees better with experimental facts.

Regarding the non-equal scales model, several aspects need attention:

1. The reference value ~in Eq. (2.53) is dimensional, and the dimension depends on the values of ,\1 and .\2 . No evident physical interpretation of~ exists.

The value of ~ should be determined by the boundary conditions of the flow. However, this is impossible if both g and ég approach zero. This happens for instanee when the nozzle exit or the radial edge of a turbulent jet are approached. Therefore there is as yet no other metbod than optimizing ~ for each individual flow configuration.

2. The expression for the eddy-diffusivity can be written as

_g2é lis""' Cjp-2

ég (2.63)

and this approaches infinity as g approaches 0 while À1 > 1. This indicates a limited range of applicability which will be discussed in the next chapter.

37

3. Similarity considerations in a round jet, where f ~ x-1 ; g ~ x- 2 and e:9 ,..., x-1, with x the distance from the nozzle, lead to the relation ,\. 1 + 2,\.2 = 2 while in a plane jet, where f ,..., x- 112

; g ,..., x- 1 and e:9 ,..., this relation would be ,\. 1 + ~). 2 The same arguments hold for the solution of Eq. (2.46) in which, for a round Î should equal 2 while in a plane jet it should equal 2.5. This problem will be addressed in the next chapter.

4. Ashurst et al. [ASH87] show in their DNS calculations of isotropie turbulence that ,..., e:0 A. This is in doser agreement with the present model with ,\. 2 R:> 1/3 tban

with the equal-scales model.

5. In the equal-scales model four constauts appear ( (J" ag, RT,C) cl,) while the non­equal scales model contains five constants ( Ch C9g, ,\.1 , ,\.2 , ,P) although it must be mentioned that the relationship between ,\.1 and ,\. 2 reduces the number of inde­pendent constauts to four.

38

Chapter 3

lsothermal Jets

In this chapter the physical charaderistics of laminar and turbulent jets are described. Numerical results obtained for these jets are compared with analytica! results for the laminar case and with experimental data for the case of turbulent flow. In this way the numerical method as well as the turbulence model are tested. In partienlar attention is given to the prediction of the spreading rate which is known to be overpredicted by the standard k - E model. Predictions of the spreading rate wi th the standard k- r:: model are compared with predictions and experimental valnes from the literature. Modifications to the k - E model are discussed and results of predictions with a useful modification are compared with experimental data. Results of the non-equal scales model for scalar transport are presented and some difficulties with the coefficients are discussed. Finally variable-density effects are investigated by reviewing experimental data and by comparison with computations.

Numerical results are obtained with the Finite Volume Method, described in ap­pendix B.l. Most computations are performed using the parabolized method while some are generated with a full Navier-Stokes method, which is called an elliptical method, for purposes of comparison. The parabolized method neglects axial diffusion and the equa­tions are solved with a marching procedure. In the elliptical method the full equations including axial diffusion are solved in the whole flow field, iteration by iteration.

3.1 Physical description

A jet consists of fiuid which issues from a nozzle (round jet) or slot (plane jet) into coflowing or stagnant surroundings. The jet can schematically be divided into two regions: the potential core, where the dependent variables are roughly equal to their nozzle values, and the similarity region where the dependent variables deercase with axial distance. In the latter region the jet spreads due to diffusion processes: diffusion of momenturn by the action of the molecular viscosity and diffusion of matter by the action of the molecular diffusivity. Turbulence enhances these diffusion processes and in turbulent jets spreading charaderistics do not depend on the Reynolds number, in contrast to laminar jets. The spreading rate of a jet can be quantified by the most commonly used halfwidth spreading rate 8 112 and by the less used global spreading rate 811100· They are defined as

dy1 12 I --,p dx

and

39

s"' _ dy111oo 1 1/HJO- dx 1> (3.1)

in which YI/2 and y1; 1oo are the radial distauces from the symmetry axis where the variabie tf is half its centreline value and 1/lOOth of its centreline value, respectively. Here, x is the streamwise coordinate frorn the jet exit. The quantity tf can be the axial velocity or a scalar variabie such as the mixture fraction. In a turbulent jet the spreading rate is independent of axial distance for typically x/ D > 10 with D the nozzle inner diameter, unless the co flow velocity is unequal to 0.

Due to the small ratio of typical radial to axiallength scales the jet is also called a thin shear layer: diffusion processes in the streamwise direction are negligible compared to diffusion processes in radial direction. In a jet into free atmosphere, with uniform pressure, the static pressure is constant to a very good approxirnation and the axial momenturn equation without buoyancy is

a a a au -(puu) + -;-(puv) = -(11-) ax ay ay ay

(3.2)

a 1 a 1 a au -(puu) + --(rpuv) = --(r-11-) ax r or r or or

(3.3)

for a plane jet and a round jet, respectively. The jets considered in the present study are a larninar round jet and turbulent plane and round jets. These jets all spread linearly,

st12 is a constant. Integration in radial direction with the boundary-conditions

v(y 0) = O;v(y --> oo) 0; ~~(y 0) 0; ~~(y--> oo) = 0 gives conservation of momenturn in axial direction foco pu2dy constant and alsof;" pu2rdr constant. In a plane (turbulent) jet the cross-section of the jet per unit length in the third dimension is proportional to y1; 2 ~ x which u ,..,. x- 1

/2

• In a round jet (both larninar and turbulent) the cross-section is proportional to YÎ/2 ,..,. x 2 and u ,..,. x-1 .

The total rnass-flux rh ~ f000 pudy or rh "" f0

00 purdr is proportional to x for the jets considered and it increases linearly with x which implies entrainment of rnass frorn the surroundings. This is illustrated by the cornputed radially integrated rnass, momenturn and energy profiles given in Fig. 3.1 for a turbulent round jet. Mass and momenturn

\ Total energy

'·; \\ I' 1\ M ,, ean en.

'\V: I, I

\\ Turb. en.

%-------0 30 60

Momenturn

90 120

XJD

40

150

Figure 3.1: Centreline vari­ation of radially integrated profiles of rnass-flux ( solid line ), momenturn (horizon­tal dashed line) and en­ergy in a turbulent round jet. The total energy (long­small dashed line) is divided into energy of the mean flow ( dashed line) and turbulent energy ( dotted line ).

profiles hebave as expected. The total time averaged mechanica! energy flux is divided into the energy associated with the mean flow and the energy associated with the tur­bulence. Clearly, the contri bution of the turbulence to the energy is largest in the near field region where velocity gradients and turbulence production are highest. In the far field this energy dies out due to ever smaller velocity gradients. The mean energy also decreases, and so does the total energy. This is due to the energy transfer from the mean flow to the large eddies and the dissipation of turbulence energy into heat in the small eddies, respectively.

3.2 Laminar round jets

If constant pressure and negligible axiaJ diffusion are assumed an analytica! solution for a laminar jet issuing from a round nozzle with diameter D with a uniform velocity profile u = u0 exists [SCH79]. Numerical results can he compared with this exact solution. The parabalie system of equations

{) 1 {) a x (pu) + -;: a/rpv) 0

{) 1 {) ~(puu) + -!>(rpuv) ux rur

(3.4)

(3.5)

is transformed into ordinary differential equations by the introduetion of a streamfunc­tion. lf the density p and the kinematic viscosity v p.f p are constant the axial velocity can he given in closed form in termsof a similarity variabie Ç as

u(x,Ç) 31" ( E_2)-2

-- 1+-4 x 4

(3.6)

Here the so-called jet invariant I u is a constant (independent of x) basedon conservation of momenturn

Iu 1

["" u2rdr = 1 (~u~D2)

Vlo V 8 (3.7)

and Ç is defined in terms of the similarity coordinate 11 = r /x as

(3.8)

Then it can be derived that the halfwidth spreading rate for the jet is

/.J2 ~ 1 5.945 16 ~--

../3Re Re (3.9)

with Re u0 D(v. Themass-flux is

(3.10)

Eq. (3.5) is a parabolic partial differential equation and can he solved by successively calculating the solution at increasing x. This marching procedure is called a parabolized method. Parabolical computations have been performed with a gridspreading rate of 0.3.

41

0.020 Figure 3.2: Relative error in computed spreading rate

0.015 . versus the stepwidth in a c: round jet with uo = 0.5 m/s ..

I

~ 0.010 c: and 50 in radial .. , direction. "' E 0.005 " c:

!d 0.000

-0.005 o.oo 0.02 0.04 0.08 0.08 0.10

Stepwldth (·)

Figure 3.3: Relative error in computed spreading rate

0.05 versus the number of grid-c: points in a round jet with ..

I 0.04 (/) 0.5 m/s and a step-::=< uo c: .. width of 0.02. (/)I 0.03

E " c, 0.02 e

0.01

-0.00 0 20 40 60 BO 100

Number of radlal grldpolnlll

The influence of the stepwidth in the marching direction and the nurnber of gridpoints in the radial direction is shown in Figs. 3.2 and 3.3. For a definition of the stepwidth and the gridspreading rate, see appendix B.l. It is rernarkable that if the stepwidth is too srnall or if the nurnber of gridpoints is t.oo large the cornputations becorne less accurate. An explanation is the effect of rounding errors in numerical differentiation that leads to an increase of the inaccuracy when the gridspacing goes to 0. At 'reasonable' valnes of about 50 gridpoints and a stepwidth of 0.02 the computation of the spreading rate is in excellent agreement with the analytica! values. It must be noted that stepwidths and less gridpoints still give accurate results with worst case relative errors of about 2 %.

In Table 3.1 computational results with 50 gridpoints and a stepwidth of 0.02 are compared with the analytica! solution. The computed spreading rates are in excellent agreement with the analytica! valnes while the mass-entrainment, defined by

1 dm

mo dx (3.11)

where m0 is the nozzle mass-flux, is predicted with reasanabie accuracy. Since turbulent flow computations are always pefformed with equations in which essentially only the

42

Re uo (m/s) D (m) 51/2 51/2 [ (m 1) [ (m 1) (anal.) (numer.) (anal) (numer.)

119.41 0.25 0.004958 0.0497 0.0487 54.04 50.50 238.65 0.50 0.004954 0.0249 0.0248 27.07 26.56 483.33 1.00 0.005017 0.0123 0.0123 13.20 13.47

Table 3.1: Analytica[ and numerical halfwidth spreading rates and mass-entrainment rates normalized by the nozzle mass-fiux in a laminar round jet with 11 1.1936610-5

kg m-1ç 1 and p = 1.15 kg m-3 •

kinematic viscosity is replaced by the effective viscosity, it may be expected that the present methad is adequate for turbulent flow computations.

3.3 Turbulent jets

The predictions on turbulent jets have been performed with the following grid charac­teristics. In the parabalical metbod a stepwidth in axial direction of 0.02 is used with 30 radial gridpoints. The gridspreading rate was 0.3. This was suftkient to obtain grid independent halfwidth spreading rates. In the full elliptical methad a rectangular grid of 80 x 50 with a length of about 50 D and a width of about 25 D was used.

3.3.1 Similarity properties

In the similarity region centreline valnes of the axial velocity and the mixture fraction decrease with powersof axial distance as derived in the previous section. In this region no external typicallength scale exists, which implies that radial profiles of velocity and mixture fraction at different axial distauces are completely similar if they are scaled in a proper manner. This was already shown for the laminar jet by Eq. (:3.6). The dependent variabie should be normalized by its centreline value and the radial distance should be normalized by the axial distance. For a turbulent jet issuing into still air the following similarity features will frequently be used:

• The spreading rate is independent of x and of Re;

• Profiles of -/kJu~ and fo)fc are independent of x and Re;

• u0 /ïic = Kv.x/ D and 1/fc = K1x/ D where I<", and KJ are constants;

• Radial profiles scale in the following way:

u 'ilcfl ( 7})

J fch('fl)

where 7} = r/x and the index c indicates 'centreline'.

43

It can he noted that the similarity assumption implies that the turbulence is completely determined by its local1 properties, regardless of its initia] conditions2 , which is the basis of the one-point dosure modelling such as the k c and second-order dosure models. lt also implies that these modeJs predict a spreading rate Sf12 and fundions !1 to fs independent of initia! conditions. Predictions using the k - c model can he compared with experiments conveniently by checking the above mentioned similarity features with experimental data.

Although it is generally belicved that jets obey this independenee of initia! conditions therc exist different experimental results regarding S112 , sec Table 3.2. George [GE089] discusses the sirnilarity behaviour of turbulent flows and its relation to initia! condi­tions. He coneindes that, although the sprcading rate must he constant, it can depend on initia! conditions, which at least in part explains the differcnccs in experimental data .

. Othcr unccrtainties are related to insufficient knowledgc or documentation of the cxper­imcntal conditions and experirnental inaccuracies. For instance, there are indications that the data of Wygnanski and Fiedler (W&F) [WYG69] are not in agreement with the concept of constant axial momenturn flux [SEI81]. However in thc present study the assumption of a universa! spreading rate and complete similarity is maintained becausc the predictions, as expected, do not show any dependenee of far field dynamics on initia! conditions.

3.3.2 Spreading rates

The main axial properties of the jet are the slope I< u of tbc inverse centreline velocity

Tïo =I< !:._ uc u D (3.12)

which is closely related to tbc spreading rate, and tbc asymptotic value of the turbulence intensity Vkc/uc. Jn this study with the standard k - t: model the latter is predicted to he 0.333 while experimental values are 0.32 [WYG69j and 0.25 [PAN93-1]. The ex­perimental value of the slope Ku is 0.165 [PAN93-l], while using the standard k t:

model present predictions yield 0.206. This implies an overpredidion of tbc halfwidth spreading rate S]12 , which from now on is called S1t2• The value for S1t2 is found to he 0.115 with tbc elliptical method and 0.12 with the parabolized method. No valnes of S 112 predicted with the elliptical rnethod have been found in tbc literature. The main difference between these methods is the neglect of streamwise diffusion in the parabolic calculations. The importance of streamwise diffusion itself is difficult to analyse since it is not possible to use the grid of tbc parabolized metbod in the elliptical metbod due to memory restrictions. To assess the importance of strearnwise gradients the streamwise gradients of tbc axial velocity in tbc production term of turbulent energy have been temporarily excluded in tbc parabalie computations leading to a value for sl/2 that was only 1% smaller. Consequently the difference of 3% between tbc clliptic and the parabolic methods might he related to differences in numerical accuracy. Tbc parab­olized metbod allows a much finer gridspacing in axial direction because there are no memory restrictions associated with tbc resolution in the marching direction. The ellip­tic method on tbc other hand has a limitation in this regard duc to the two-dimensional

1 Loc al in the sense that no correlations of the type u;( i)uj (i+ R) are considered. 2The term initia! conditions is comrnonly used in t.he literature to denote nozzle conditions.

44

grid that must be stored. It has been found that the predicted spreading rate increases with grid density in the axial direction, which is consistent with the remarksof Launder and Morse [LAU77] and this can explain the difference. lt must be noted however that the predicted spreading rates with elliptic calculations were grid independent since the final results were generated with a sufliciently fine grid.

All computed halfwidth spreading rates are well above the experimental valnes re­ported in the literature, see Table 3.2. The experimental spreading rates show significant

Table 3.2: Measured halfwidth spreading rates for round jets.

scat ter, and apparently tend to become higher with the years. Still the k- e model over­prediets the spreading rate while for a plane jet the spreading ra te is accurately predicted. This is known as the plane jet - round jet anomaly for the k e model.

In the literature predicted valnes of 51; 2 vary between 0.113 and 0.125, see the first five references in Table 3.3, where also predicted valnes in the present study are shown. All predictions we re carried out using the k- e model and parabolized methods. Some of the differences between these valnes can be attributed to smal! differences in the model coeflicients. However, with identical coeflicients there are still differences between valnes reported in the literature. The valnes of 51; 2 predicted in the present study are within the range of valnes found in the literature. These differences must be dne to differences in numerical accuracy. Regarding this matter it is worth noting that for the plane jet Ljuboja and Rodi [LUB80] found a spreading of 0.114 using a parabolized methad while Paulay et al. [PAU85] obtained a value of 0.108 (difference of 6 %) with exactly the same model but solving ordinary differential equations by introducing a similarity transformation, which they showed to be highly accurate. The same type of similarity transformation was also used by Taulbee [TAU89] for the round jet using the standard coeflicients of Table 2.1 giving 51; 2 = 0.12. This value is dosest to the present predictions with the parabolized method which again gives confidence to the numerical method used in the present study. Finally, note the difference of 20% between the two valnes of 51; 2

predicted with the same secoud-order dosure model, see the last two lines of Table 3.3, which Launder attributed to numerical problems.

The spreading rate 5~1100 predicted with the parabolized method is 0.27 while the elliptic method gives 0.21, the experimental value being about 0.22 [LAN87]. No pre­dictions of sr;wo are reported in the literature.

3.3.3 Modifications to the k - r:: model

In the literature a number of modifications to the k e model have been proposed to correct the spreading rate for the round jet, see Table 3.3, but many attempts have failed because it proved to deteriorate predictions for other types of flow. In the present study many of these modifications have been implemented and predicted spreading rates are compared with the computed results found in the literature. In addition, in the present study spreading rates have been computed with models which are extensions of the k ê

45

model, such as the multi-scale turbulence model, discussed in chapter 2, and the RNG model, which has not been discussed yet.

To improve the predictions of the round jet spreading rate the turbulent of momenturnmust be diminished, and consequently the eddy-viscosity must be decreased. Often this is done by changing the empirica! coefficients C1 and C2 in the t:-equation. A decrease of ttt can be accomplished by increasing the dissipation rates and therefore a higher value of C1 or equivalently a lower value of is needed, see Table 3.3.

Anthor I ReL Modification to 51/2 the k - é model Lit. Present

Taulbee [TAU89] Standard i 0.12 0.12 (p) (Table 2.1) 0.115(e)

Kline et al. [KLI69] Standard 0.11.5 0.12 (p) (Table 2.1) 0.115(e)

Malin [MAL89J r:Te = 1.314 0.113 0.12 (p) Pope [POP78] cl = 1.45; c2 = 1.9 0.125 0.113(p) H&L . [HAN80] cl= 1.42 0.115 0.125(p) Present C2 = 1.88 - 0.108(p) Unknown [POP78] cl= 1.6 - 0.084(p) Morse [MOR77] cl= 1.4- e 0.080 0.080(e) McGuirck [MCG77] cl = 1.14 - s.31 - -

L&S [LOC83] CJL = 0.09- 0.04Ç; C2 = 1.92- 0.0667Ç; ç = (!1!1"-(1 1-~))0.2 - 0.09l(p) 2uc

Present CJL = 0.06 0.098(p) Pope [POP78] Vortex stretching 0.079 -

Cho [CH090] Intermit tency 0.090 -

H&L [HAN80] Normal strains 0.098 Yakhot et al. [YAK92] RNG model 0.18 (p) K&C [KIM89] Multiple-time scales 0.116(p) Launder [LAU77] Second-order dosure 0.126 -

Launder [LAU90] Second-order dosure 0.105 -

Table 3.3: Halfwidth spreading rat es 51; 2 fora round jet with several modifications to the standard k-t: model. Predictions in the present study are performed with the parabolized (p) or elliptical (e) methad while all literature values are obtained with a parabolized rnethod. Abbreviations are: H&L: Hanjalic and Launder, U\fS: Loekwood and Stolakis, K&C: Kim and Chen.

The modification of increasing C1 to 1.6, reported by Pope, is in fact equivalent to the one proposed by McGuirck if y1; 2 = 0.087 · x because J-~ = -1/ x. With cl 1.6 present predictions yield sl/2 = 0.084 which is smaller tha~ the experimental values. With the modification of Morse the present elliptical method yields 51;wo 0.152 which is 32% lower than the measured value, although the halfwidth is improved. Present parabolical predictions with the modification given by Loekwood and Stolakis [LOC83] give an improved halfwidth and also a correct global spreading 51; 100 = 0.21

46

and therefore this modification is acceptable. The much simpler modification C,.. = 0.06 [JAN92] gives an improved spreading rate for the halfwidth and also an acceptable value for the global spreading S 1two = 0.23. The predicted velocity decay constant K" of Eg. (3.12) is 0.167 with C,.. = 0.06, which is in very good agreement with the value of 0.165 measured by Panchapakesan and Lumley (P&L(l)) [PAN93-1].

Pope [POP78] suggested to add a term to the e-equation basedon the idea of vortex stretching, which would only be active in the round jet. However with this term the spreading rate of a radial jet is underpredicted by 60% [RUB85]. Furthermore the resulting model does not comply with realizability which is one of the constraints which can be imposed on turbulence models. These are in short [SPE87]:

l. General coordinate and dimensional invariance.

2. Realizability, which means that both turbulent kinetic energy and its dissipation rate should be positive.

3. Material frame indifference in the limit of two-dimensional turbulence [SPE81].

Hanjalic and Launder [HAN80] added a term proportional to k'#Jb..<!l!!L88

" CijkEtmk, with u::t:J Xm

Eijk the alternating tensor, to the e-equation. However, this term violates realizabil-ity [SPE90].

It is also interesting to investigate the performance of k e type of models which are not specifically designed to improve the prediction of the spreading rate in the round jet. These models are the very recent RNG (Renormalisation Group) k- e model of Yakhot et al. [YAK92] and the multiple-time scales model of Kim and Chen [KIM89], which has been presented in chapter 2. Thus far no predictions of spreading rates in a round jet with these models are reported in the literature.

The RNG model differs from the standard k- e model in the definition of its con­stants, which are [Y AK92]

c,.. 0.085

<lk 0. 7179

C = 1.42- ((1 (/(o) 1 1 + f3(3

C7e = 0.7179.

with (0 4.38 and f3 = 0.015, while ( is the scaled strain rate given by

. sk ~ = ~

€ (3.13)

Here, Sisthestrain rate of the mean flow, defined by S in which Sij is given by Eq. (1.7). Discussion of the background of the model is out of the scope of this study. Although the coefficients resembie those in the standard model, the most important difference is the dependenee of C1 on the strain rate of the mean flow. Computational results obtained in the present study have shown that in a turbulent round jet the scaled strain rateis lower than (o which implies a smaller cl and therefore a lower dissipation rate. The higher /lt results in a much too high spreading rate (see Table 3.3).

The multiple-time scales model of Kim and Chen [KIM89] gives a halfwidth spreading similar to that of the standard model (Table 3.3). The global spreading rate, however, is somewhat large: 51/toO = 0.27. The multiple-scale, or non-equilibrium, aspects of a round turbulent jet are illustrated in Fig. 3.4 where radial profiles of the multi-scale

47

... ;

i = ï5

4

2

0 '-------~--

0 5 10 R/0

11.0 i 0.8

0.6

0.4

0.2

Figure 3.4: Radial profiles of tbe multi-scale quantities. Solid line: Transfer energy kt, smal! dasbed line: en­ergy in tbe production range kp, long dasbed line: dissi­pation in tbe transfer range Et, dotted line: dissipa.tion in tbe production range Ep,

long-small dasbes: produc­tion rate in tbe production range Pk.

quantities kp, kt, Ep, Et and Pk are plotted. It is seen that strongest non-equilibrium effects are manifest in the central part of the jet where the production and dissipation of turbulent energy differ most. The production rate attains its maximurn in the region of maximurn shear and hereequilibrium is reached in the sense that Pk, Ep and Et

are alrnost equal. Hence, the most important non-equilibrium effects occur in the region with the least impact on the turbulent exchange of mornentum, narnely in the centre of the jet.

Discussion of the modifications

The modifications can be grouped into two categories: the first category are models aimed at capturing additional physical effects, which rnay be regarcled as attempts to improve the standard k t: model. They either result in extra differential equations or in extra terrns in the existing equations. The secoud category are modifications which merely involve changing constants of the standard model.

Suggestions for the improverneut of the k-t: model that have been investigated in the present study can be cast into two forrns which have been discussed in chapter 1. These are

1. A modification to the constitutive equation for the Reynolds-stress to indude anisotropic effects. This has been discussed briefiy in chapter 2. By more accu­rately predicting normal stresses, in partienlar the predicted axial momenturn exchange in axial direction would be improved. However, the importance of the normal stresses in a thin shear flow, such as a turbulent jet, is smal!. The rnain contribution cornes from u'v' and furthermore it will be shown later that is already predicted fairly accurate with the present model.

2. Induding non-equilibrium turbulence. This has been shown to be of minor impor~ tance in the shear influenced region. Therefore this model gives the same results as the standard model for the ha.lfwidth spreading.

Consequently in the present study no model in the first category was found to be ade­quate to rernove the round jet/plane jet anornaly.

48

In the second category modifications of C1 and C2 are hazardons because these constants were determined to give good agreement with a number of elementary flows. Recent experiments, however, give indications that the value of C2 should he somewhat smaller than the standard value of 1.92, namely 1.88 [GR089]. This would lead to higher dissipation rates and consequently to a lower spreading rate. lndeed, with the parabolized metbod 51; 2 was found to he 0.108 (51; 100 being 0.25) with C2 1.88. This also shows the severe sensitivity of the predictions to the model constants.

An important reason for not using modifications which employ centreline velocity gradients is that in many flows no centreline can he identified, thereby limiting the modifications to flows with an axis of symmetry.

In conclusion, the choice of C~' 0.06 (instead of C~' = 0.09) is the simplestand ap­parently most effective modification. Since all radial profiles are similar if the similarity coordinate 11 = r /x is used, it is of minor importance which partienlar modification of the second category is employed, as long as it impraves the predicted spreading rate.

3.3.4 Comparison with experimental data

The accuracy of the model with C~' = 0.06 is now investigated in more detail using experimental data of Panchapakesan and Lumley (P&L(l)) [PAN93-l], Wygnanski and Fiedler (W&F) [WYG69], Capp, Hussein and George (CHG) [CAP90] and Hussein and George (H&G) (HUS90]. The experimental data of P&L(l), that are most recent, are used as a reference in the figures. The other data are only given if they significantly deviate from other measurements . The radial profile of axial velocity normalized by the centreline velocity Uc is in excellent agreement with experiments (Fig. 3.5). Note that the radial distance is scaled with the axial distance rather than with the local jet halfwidth, since the latter procedure would mask an incorrectly predicted spreading rate. At the edge of the jet some disagreement between the experiments exists.

The shear stress u'v' mainly determines the jet spreading. In Fig. 3.6 the normalized shear stress is seen to he in very good agreement with experiments. The predicted spreading rate of 0.098 is close to the experimental valnes of 0.094 (CHG) and 0.096 (P&L(l)). The measurements of W&F give a lower curve which is shifted to the axis. This is due to the lower spreading rate of 0.086 intheir experiment.

In Fig. 3.7 the radial profile of the axial normal stress is seen to agree very well with the experiments althongh the centreline valnes of P&L(l) are somewhat lower. The radial profile of the radial normal stress is in qnalitative agreement with the experiment of P&L(l) (sec Fig. 3.8) especially if the general weak performance of the k e: model regarding normal stresses is considered. It must he noted that there is considerable scat­ter between the individnal experiments which makes a comparison somewhat difficnlt.

The dissip at ion rate of turbulent energy e: is very difficnlt todetermine experimentally and this is reflected by the considerable scatter in the experimental data, see Fig. 3.9. Several authors have made estimates of this variabie either by direct measurements of derivatives (H&G) or by a balance ofthe tnrbnlence energy budget Eq. (2.17) (THC and P&L(l )). Particularly near the centreline the experiments disagree. Dissipation rates obtained from a turbnlence energy budget show st rong variation near the centreline with an off-axis local maximum while those obta.ined from direct measnrements assnming local isotropy or axisymmetry show a monotonons deercase with radial distance. In

49

1.00 Figure 3.5: Radial pro-file of the normalized ax-ial velocity in an air-

0.75 rur jet as a function of normalized radial distance

u, rfx. Experimental results § 0.50

of P&L(l) [PAN93-1] are W&F [WYG69] (o)

0.25 and CHG (CAP90] ( + ).

0.00 0.00 0.05 0.10 0.15 0.20 0.25

R/X

3e-02 Figure 3.6: Radial profile of the normalized shear stress

2&-02 u' v' /uc 2 in an air-air jet. See caption of Fig. 3.5 for fur-

"' 2&-02 ther details.

• " I ~ }'

1&·02 :>

0 + 5&-03 0

0.05 0.10 0.15 0.20 0.25

R/X

contrast, the predictions show a flat profile near the centreline. The dissipation ép

predicted with the multi-scale model, shown in Fig. 3.4, shows a very weak increase with increasing radial distance before falling down to 0. This would slightly favour the direct measurements of THC. For distauces from the centreline all profiles show a good agreement, which is remarkable taking into account the empiricism with which the é-equation is derived.

3.4 Scalar transport: Non-equal scales model

The transport of a scalar quantity in a round jet is addressed by testing and applying the non-equal scales model introduced in chapter 2.

In this section attention is focussed on phenomena which are density independent. This does not imply that only constant density experiments are considered because many features of the flow may be made density independent by the introduetion of the effective nozzle diameter

(3.14)

50

0.30

0.25

0.20

u I

::;) 0.15 "_

" 0.10

0.05

0.00 0.00

0.30

0.215

0.20

u I

::;) 0.15 '!;:

0.10

0.05

0.05 0.10 0.15

R/X

0.20 0.25

Figure 3. 7: Radial profile of the normalized normal stress vW /üc in an air-air jet. See caption of Fig. 3.5 for further details.

Figure 3.8: Radial profile of the normalized normal stress ..;;t;J /üc in an air-air jet. See caption of Fig. 3.5 for further details.

where w Pa/ PI with Pa the ambient air density and PI the nozzle fl.uid density. The physical idea behind this transformation is that at large x the density is equal to the ambient air density and, using conservation of momentum, the velocity decay scales in the same way as in the constant density case if D is replaced by D. (if buoyancy is neglected).

Variabie density effects resulting from isothermal mixing of two gases of different density are considered in the next section while these effects in fl.ames are discussed in chapter 5.

3.4.1 Determination of the coefficients

The coefficients À 1 and Àz

For the non-equal scales model which essentially consists of Eq.'s (2.52), (2.53) and (2.57), the constants À1 and Àz should have appropriate values. By consiclering similarity behaviour of the dependent variables in a round jet it bas been shown in chapter 2 that

51

3e-02

+

3e-02 • +

• ~

• 0

2e-02 0 • u I 0 + • "iJ

:::!. lil l> l> * z "' " @ 2e-02 " • " ;I " + • " " " . 1e-02 + a. CD

+

Se-03 + +

0 0.00 0.05 0.10 0.15 0.20

A/X

the relation between the constants should be

0.25

Figure 3.9: Radial pro­file of the normalized dis­sipation ra te ELu/'iic 3 with Lu = 0.096 x in an air­air jet. Experiments are of P&L(l) (L'l.), Taulbee Russein and Capp (THC) [TAU87] ( turbulence energy budget with SHW = Sta­tionary Hot Wire measure­ments (\7) and LDA (<>)), HG (direct measurements assuming isotropy ( +) and axisymmetry of the smallest scales ( • )).

(3.15)

Exploratory calculations have shown that the largest influence on the solution was ex­erted by À1, which led to the choice of À1 = 1.5 and consequently À2 = 0.25. These values are to be compared with the ones given by Yoshizawa: À1 = 1.2 and À2 = 0.306, which are based on the assumption that the turbulence is completely governed by the inertial range of the spectrum. The values can also be compared with those in Ta­ble 2.3 where the predictions in the present study are c.g,l c.g,3 = Àt = 1.5, C,

9,2 = À2 x C1 = 0.25 x 1.92 = 0.48 and C,

9,4 = À2 x C2 = 0.25 x 1.44 = 0.36

(see Table 2.3 and Eq. (2.51)).

The coefficient </;

The value of</; defined in Eq. (2.52), unfortunately is not dimensionless; it is dependent on the initial conditions. Therefore it has to be determined empirically for each flow. Although this is at varianee with the requirement of dimensional invariance, at this moment there is no other solution.

Each (constant density) flow can be characterized by the jet exit velocity and the initial turbulence values of k and E. The values of f and g at the nozzle are no free parameters: they are 1 and 0 at the nozzle, respectively. The resulting mixture fraction in the far field is given by 1/]c = K1x/ D where the value of KJ is closely related to the jet spreading rate s{12 . Sirree the mixture fraction field is one of the most important

quantities in a jet and a flame, K 1 and S{;2 must be predicted accurately. The scalar varianee g is also important in a flame, and therefore should be predicted accurately as well. With the non-equal scales model this must be accomplished by choosing appropri­ate values of</; and Cf. Changing Cf from its TSDIA value of 0.446 to 0.5 was found to facilitate obtaining the correct value of S{;2 and predicting g accurately with one value of <f;. It is recalled that the TSDIA values are only approximate, see chapter 2.

The value of </; in principal depends on the initial conditions. This would imply that changing the initial conditions while </;is kept constant would change the solution,

52

even in the far field. This, however, is only true in a very limited sense. Numerical experiments showed that the solution, that is S{12 for instance, does not depend on the initia! turbulence values. It only depends on the jet exit velocity. The way in which 4> depends on the velocity is discussed in section 3.4.4.

With the above constraints and with the standard k f: model used for meebani-cal turbulence, the predicted mixture fraction on the centreline agrees very well with the experiment of Birch et al. [BIR78] regarding the spreading rate S1; 2 = 0.097 with 4> 650 3

, see Fig. 3.10. The scalar varianee g is also predicted very well with this choice. Note that the scalar varianee increases immediately, without showing a poten­tial core. Although this seems to agree well with the experiment, it will later present some problems.

In Fig. 3.11 radial profiles of f and the turbulent Schmidt number are plotted. The radial profile of f predicted with Cf 0.5 is seen to agree well with the experiments. The predicted turbulent Schmidt number is compared with the measurements of Chevray

0.15

0.10

- 0.5

0.05

o.o [____ ____ ,,,,,,,,,L._ _____ _L_ ____ ____j 0.00

0 25 50 75

XID

0.5 +

0.0 L_ ____ """" __ "" ______ """,_.L.....,___J 0.0

0 2

s t: CT .

Figure 3.10: Non-equal scales (>.1 1.5; .\z = 0.25) model predictions of the centreline mixture frac­tion ( solid line) and vari­anee v'9 ( dashed line) in a methane-air jet. Triangles: experiment of Birch et al. [BIR78].

Figure 3.11: Normalized mixture fraction l/lc and turbulent Schmidt number ( dashed line) versus R/ Rt/2· Solid line: CJ = 0.5, dashed line: Cf 0.446. Crosses: Prandtl number [CHE78]. See caption of Fig. 3.10 for further details.

and Tutu [CHE78] who calculated a turbulent Prandtl number from experimental data of a heated round jet. Qualitative agreement is found in the sense that both model and experiment show a local maximum. These data can he used for a first comparison if

dimcnsion of</> is omitted.

53

the Lewis number is constant, which is a good approximation in the present case. Also recent measurements [PAN93-2] of RT, defined by Eq. (2.41) show a similar trend. The relation between S Ct and RT is

(3.16)

The time scale ratio criterion

It was demonstrated that the non-equal scales model performs well in an isothermal jet. However, in the limit as g -+ 0 the eddy-diffusivity f.ls given in Eq. (2.39) approaches infinity if Eq. (2.52) for Eg is used together with )IJ > 1: f.ls ~ -j;;g2- 2>. 1E 1- 2>.2 -+ oo. This happens for instanee near the nozzle and at the jet boundaries, where both g

and Eg are 0. In the near field region near the nozzle this large f.Ls leads to a faster decrease of Ie and to a faster increase of 9c in axial direction. In fact, the potential core region for scalar variables no longer exists. In a flame this effect would lead to an immediate temperature rise in the near field region, which is unphysical. The origin of the problem is the restricted range of validity of the equation for the scalar dissipation rate which was derived with the assumption that inertial range turbulence exists. In the flow regions where g -+ 0 the scalar turbulence is very low and the inertial range does not exist. Consequently a modification is in order in these regions. Because the classical model is empirically derived and works in all parts of the jet, a criterion is applied which distinguishes regions with high intensity scalar turbulence and regions with low intensity scalar turbulence. The criterion is chosen to be linked with RT in the following manner: if the time scale of scalar turbulence g/Eg is lower than half of the value of its mechanica! counterpart kj E, i.e., RT > 2, then the new model is applied, and otherwise the turbulent transport terms are modelled by Eq. (2.19):

{ CJrfJ-2pg2-2>.,EJ-2À2 if RT > 2

f.ls = C -k2 if RT _< 2 JP4; (3.17)

For RT :S 2, f.Ls is computed according the equal-scales model while for RT > 2 the non-equal scales model is used. In the following this working hypothesis is called the "time scale criterion". It is noted that Eg (needed in the equation for g) is always computed according to the non-equal scales expression. Furthermore, it is noted that the second equation in (3.17) implies a turbulent Schmidt numberof0.72 (at= 4CpfCJ ). Predictions with this concept for the samejet are shown in Fig. 3.12 from which it can be concluded that the agreement with the experiment still is very good while in the near field region the potential core is retrieved which is more realistic. Radial profiles of the varianeeg are shown in Fig. 3.13 together with several experimental results. The non-equal scales model shows better agreement with the experiments in the sense that the curve is broader.

3.4.2 Asymptotic behaviour, experiment al data

In this section experimental data on the far field behaviour of quantities such as the velocity and scalar decay constant, the turbulence intensity, scalar turbulence intensity ( called unmixedness) and spreading rates are discussed. In Table 3.4 results of various experimental studies are presented.

54

,----·······-·-----······-~···-----······~...., 0.15 Figure 3.12: Centreline va.l­

1.0

0.5

15 30

XID

1.5 ,-----~

1.0

0.5

• ' e" •

' @ ' " ' \<>

' ' ' ' '

60

' ' '

0.10

0.06

.!!! t:: r:r ..

0.0 L---~---'--------'---''-'--------0.0 0.1 0.2 0.3

R/X

ues of the mixture fraction and variance. Predictions with both the equal ( dashed line) and non-equal sca.les model with the time scale ratio criterion (solid line) . For further details see cap­tion of Fig. 3.10.

Figure 3.13: Radial pro­file of the normalized vari­anee V'ii!jk. Solid line: non­eq ua.l scales, da.shed line: equal-sca.les model. Sym­bols are experiments of P&L (2) (6), [BEC76] (\7), [LOC80] (o), [BIR78] ( + ), [DAH85] ( • ), [DOW90] at Re = 16000 (o) and Re 5000 ( <> ).

Scalar spreading rates seem to be independent ·of the density ratio w. This was concluded from recent experimental studies conducted to clarify the density dependenee of various turbulence quantities [GOU86, RIC93]. Some experimental data in Table 3.4 indicate a larger S{12 with larger w. However, among these experiments there are also contradictory results, compare the spreading rates reported by Birch et al. [BIR78] and P&K [PIT84]. Although the latter experiment concerned a methane jet in a slow coflow of 3% the spreading rate is larger than the one in the free methane into air jet of Birch. Furthermore, the air-air jet of Becker et al. [BEC76] gives a spreading rate equal to the one reported by P&K although it should have a smaller spreading rate because of the smaller w. This is an indication of the experimental uncertainty. Predietiens in the present study show only a very weak dependenee of the spreading rate on the initia! density ratio, both s;/2 and s{/2 decrea.se with about 4 % when the nozzle gas density decreases with a factor of 7.2. This favours the conclusions of Gouldin et al. [GOU86] and R&P [RIC93]. Especially the Jatter, very recent, experiment shows that there is no density influence on asymptotic valnes such as sft2, f{f and viiJlc with 0.64 < w < 7.2.

Unrnixedness defined by y?i)fc appears to be density independent and asymptoti­cally equal to 0.22 ~ 0.23 [PIT91, PAN93-2, RIC93], although some experiments show a

55

I Author Ref. Co ft. w Sf!z VY / fc KJ Kg K&W [KEA50) ? 7.2 0.132 0.214 Becker [BEC76] 0 1 0.106 0.22 0.186 0.84 Birch [BIR78] 0 1.8 0.097 0.29 0.25 1.16 Dy er [DYE79] 3.3 0.66 0.086 0.18 -C&R [CHE80] 0 1 0.2 L&M [LOC80] 0 1 0.21 -

Santoro [SAN81] 0 1.04 0.097 -

P&K [PIT8·1] 3 1.8 0.104 0.26 0.224 1.14 Schefer [SCH85] 17 0.65 - 0.185 D&D [DOW90] 0 1 0.23 0.196

0.211 Pitts [PIT91] 0 0.2-7.2 0.23 0.144

0.23 P&L (2) [PAN93-2) 0 7.2 0.1:38 0.21-0.22 0.271 R&P [RIC93) 0 7.2 0.113 0.227 0.212 -

1.8 0.115 0.227 0.212 0.64 0.108 0.227 0.210

•

Table 3.4: Review of experimental data on the scalar spreading, asymptotic unmixedness, and decay constanis versus the coflow {Cofl.) in% of the main jet exit velocity and of the density ratio w. Values ofl<J are including the diameter De as in Eq. {3.18) and Kg is defined analogously to Kj, but with the diameter D. Abbreviations are: C&R: Chen and Rodi; D&D: Dowling and Dimotakis; L&M: Loekwood and Moneib; P&L{2): Pan­chapakesan and Lumley; K&W: Keagy and Weller; P&l(: Pitts and Kashiwagi; R&P: Richards and Pitts.

higher unmixedness in variabie density flows. Predictions with both modelsindeed show an asymptotic value which is independent of w. Several experimental studies reported an increase of the unmixedness with axial distance [BEC76, SCH85, PIT84, EFF88] (see [GOU86)) , which D&D [DOW91] attribute to insufficient resolution.

The mixture fraction decay constant K1 defined by

1 }' x =- = iJ fc De

(3.18)

should be independent of w if the sealing procedure with the effective diameter captures all the infiuences of density ratio on entrainment. Indeed, if w is not too far from 1 then l{J ~ 0.2 in all experiments. Pitts [PIT91] gives KJ 0.2 ± 10% from a review of many experimental data4 • The most recent experimental data of R&P show K1 0.21, independent of w. Predictions in the present study show that KJ is independent of w.

4 The heliumjets seem to deviate from this trend on I\1. However, the velocity decay constant I<u was nearly the same in the air-air jet and the helium-air jet of P&L, if De was used.

56

The varianee decay constant Kg is defined by

This means that ./9cffc wKJ /Kg.

= Kgx/D without using D •.

The scalar dissipation êg is a.'lsumed to scale with x-4 in the present study. There are two experimental studies which show a much weaker decay than x-4 [EFF88, LOC80]. However, Dowling [DOW91] found the x-4 sealing law, usiug the experimental data of D&D and attributed the different finding of L&M to resolution difficulties. Predictions always give Eg ~ x-4 •

3.4.3 Predictions of constant density jets

The predictions in this section concern constant density flows and therefore consider only the effect of coflow, although measurements with varying coflow are scarce. Predictions with the equal-scales model are performed with the standard model C~' = 0.09 and rYJ = 0.9 and 0.7. In the non-equal scales model</> is chosen such that the asymptotic value without coflow of vf9cffc Ri 0.22, based on experiments. Furthermore, À1 = 1.5; ,\2 = 0.25 and Cf = 0.5. It is assumed that </>does not change with varying coflow, its value being </> = 1100, the initia! conditions correspond to fully developed pipe flow with ÏÏc = 50 mjs.

The spreading rate 5{;2 computed with the equal-scales model strongly depends

on C1, and rY 1. The standard model prediets S{;2 0.124, while changing (Y 1 to 0. 7

gives s{;2 0.14, both valnes being larger than the experimental value 0.113 measured

by R&P. The non-equal scales model prediets 5{;2 = 0.11 which is very close to the experimental value. If coflow is present the halfwidth is not a linear function of x anymore. Therefore 51; 2 depends on the axial distance and is not a useful concept in a coflowing jet.

Predictions of ,J9)fc and KJ for various coflow strengtbs are presented in Table 3.5. Without coflow the equal-scales model prediets vf9)fc 0.25, independent on values

Cofl. rYJ yf9)fc yf9)fc KJ KJ (Eq.) (N-eq.) (Eq.) (N-eq.) i

0 0.7 0.25 0.22 0.25 0.21

.~ 0.25 0.22 0.27- 0.28* 0.25- 0.29* 0.22 0.21

0.9 0.27- 0.28* 0.19 20 0.7 0.27- 0.28* 0.26 0.30* 0.18 0.17

0.9 0.27 0.28* 0.15

Table 3.5: Model predictions for asymptotic values of unmixedness and mixture fraction decay at various coflow strengths. Eq. denotes equal-scales and N-eq. denotes non-equal seales model. Unmixedness values with and without stars are the values at 25 and 50 nozzle diameters respectively. Note that the value of rY 1 is only relevant in the equal-scales model.

of C I' and rY f, which is quite remarkable. The decay constant K f predicted with the

57

equal-scales model is somewhat too large, the experirnental value being 0.21 (R&P). The non-equal scales model prediets this value very accurately.

With coflow the equal-scales model prediets V§/Jc 0.28 regardless of coflow velocity. The sarne holds for the non-equal scales model, although axial variations are sornewhat larger. It is interesting to note that the predictions for the unrnixedness show an approximate asymptotic behaviour with and without coflow, while the turbulence intensity Vkcluc only shows this behaviour without coflow and decreases with axial distance if coflow is present. However, Vkc/(uc- UcJI) would be independent of x The increase of the asyrnptotic unmixedness with increasing coflow velocity is not in agreement with the finding of Pitts [PIT91] who found a decrease of unrnixedness with increasing coflow velocity.

Although the halfwidth is not a linear function of x if coflow is present, K1 can be determined because 1/fc does depend linearly on x, even with coflow. The decay constant I<t is apparently rnuch more influenced by the coflow when the equal-scales model is used, see Table 3.5. It is hard to condude which model performs best due to the Jack of measurements for coflowing jets. At low coflow veloeities the non-equal scales model performs best. It must be noted that at higher coflow veloeities the non-equal scales model is mainly in the range R,. < 2 in most of the flow, which can he seen in Table 3.5: at high coflow veloeities predictions with the non-equal scales model become cornparable with the equal-scales model with fYJ = 0.7 because the turbulent Schmidt nurnber is approximatly 0. 72 if R,. < 2. This means that only at low coflow veloeities the mechanica! and scalar turbulence are uncoupled. Indeed, changing the turbulence coefficient CP. at low coflow velocity does not affect the predictions of the non-equal scales model (not shown in Table 3.5).

3.4.4 Discussion of the non-equal scales model

Although the non-equal scales model gives promising results there are some aspects that need further discussion.

The constant </> of Eq. (2.53) is dimensional and therefore cannot be universal. It would be best to relate it to a boundary condition but both at a radial boundary of the jet and at the nozzle g and t:9 are 0. Furtherrnore, at the radial boundary the energy dissipation varrishes as well, which makes the evaluation of </> impossible. The origin of this problem is that the theory of TSDIA is valid only for high Reynolds numbers where an inertial range for scalar fluctuations exists. At both aforementioned boundaries this is not the case. Therefore </> has to be related to an interior point in the flow.

It is interesting to analyse what happens with </> if the jet exit velocity, or Reynolds number, changes. To this end it must be noted that k, E:, g and t:9 attain their respective centreline maximum values typically at 8 or 9 diameters frorn the nozzle, depending on initia! conditions. The maximurn values of k and t: scale with the jet exit velocity u0

and the axial distance in the following manner:

(3.19)

While the profile of g is independent of the Reynolds number, the constauts Ck, c, and 9ma.x depend on initia! conditions. Now for the non-equal scales model to give reasonable

58

results, the predicted profiles should not be too far from those predicted with the equal­scales model. Therefore in the flow region considered, i.e., 8 to 9 diameters from the nozzle on the centreline, the two scalar dissipation rates must be approximately equal:

(3.20)

lnserting the expressionsof Eq. (3.19) into this equation gives

(3.21)

Although </; cannot be universa! due to its dimensionality, the sensitivity of the predict.ions to </; can be minimized by demanding that t/> be independent of u0 • This would lead to À2 = 1/3 which in turn would make À1 4/3. It is interesting to note that these values are indeed not far from both the TSDIA (À1 = 1.2; À2 = 0.306) and the modified values (À 1 1.5; À2 = 0.25). Test calculations with the set (À1 = 4/3; À2 1/3) showed good agreement with experiments if the time scale ratio criterion was used. In addition it may be noted that the same strat.egy in a plane jet would lead to (À1 5/3; À2 1/3), i.e., showing no geomet.ry dependenee of À2 .

A point needing further elaboration is the apparent geometry dependenee of À1 and À2 as discussed in chapter 2. Both constants are non-dimensional and in the spirit of the TSDIA they are geometry independent because the inertial range forms for the scalar and mechanica! energy spectrum, which they are based on, are geometry independent. The only solution which has emerged during this study is the notion that the non-equal scales model, in the form it is presented and used, has a limited range of validity with respect to the turbulence dynamics. Firstly the idea is that it is only valid for regions in the flow where an inertial range exists. This is the reason for the introduetion of the critica! time scale ratio which determines where the empirica! equal-scales model should be used and where the non-equal scales model should be used. lndeed R,. approaches 0 near the nozzle and for large radial distances. The second idea is that the model could be only valid for small scales up to the energy containing range, i.e., valid for the inertial range and below. In this case another model, which necessarily would be an empirica! model again, should be used for the energy containing scales where production of turbulence takes place. The present model for the scalar dissipation, however, contains production terms due to scalar gradients as well as velocity gradients and extends into the energy containing range. An alternative would be the redefini ti on of the equations for the varianeeg and turbulent energy k which should be associated with the transfer scales such as in Eq. (2.26). In this way production terms which act in the large scales do not appear. With respect to this matter it is interesting to note that in early papers which were publisbed on the theory of TSDIA [YOS82, YOS83] the equations were assumed to be valid up to the grid scales in the computational flow domain, which means that it was a subgrid model for LES. In later studies on the TSDIA Yoshizawa implicitly extended the theory to all scales by cernparing it with standard empirica! models such as the k ê model. Finally, in this context it can be mentioned that the solution for the TSDIA dissipation rate e, Eq. (2.48), also cannot be used in both a plane and a round jet because it implies a constant eddy-viscosity throughout the flow, which is incorrect.

A third point concerns the reliability of the model constants which Yoshizawa deter­mined analytically by using inertial range spectra for turbulence and extending these to

59

the energy containing scales, which is questionable but there is no alternative. Yoshizawa suggests to modify these constauts empirically because of the limitations inherent in the theory. Indeed in the application to the backward facing step problem [NIS86] modified constants were used.

Finally it is interesting to note that the remarkably simple solutions (2.48) and (2.52) to the Eq.'s (2.46) and (2.50), have never been reported in the literature. According to Yoshizawa [YOS93] the geometry dependenee of the constauts .\1 and .\2 is related to the plane jet-round jet anomaly. However, the sealing properties of the mean variables with axial distance are correct in both flows with the standard k - E: model, where the anomaly is present.

3.4.5 Predictions of variabie density jets

In the low-Mach-number approximation density variations are the only means by which mixing and combustion are coupled with the dynamics of the flow, which in turn is described by the Navier-Stokes equations. Variabie density effects influence the dynamics of the flow either by initia! density differences between two gases, or by density variations in a (turbulent) flame. In studies of variabie density one of the important issues is how spreading rates are influenced by density differences. These spreading rates reflect the efficiency of turbulent mixing of momenturn and of the scalar, the ratio of which is the turbulent Schmidt number Sc1 • Th is efficiency is partienlady important in a turbulent flame. Variable-density effects caused by initia! density differences will become less important with axial distance, and eventually the jet will behave as a constant density jet. Gouldin et al. [GOU86] condurled that spreading rates are insensitive to initia! density differences and Richards and Pitts [RIC93] confirmed this observation and extended it to the centreline unmixedness and the scalar decay constant.

The following ( un)certainties in the publisbed results of experimentscan be identified, see also Table 3.4.

• The dynamics of the variable-density flow is assumed to be similar to an equivalent constant density flow if the density corrected diameter De (3.14) is used.

• Gouldin et al. [GOU86] state that initia! density differences have no influence on velocity spreading rates, while most (more recent) experimental data seem to indicate a larger velocity spreading if the density of the nozzle fiuid is lower than the density of the ambient [WITSO, STR93a, PAN93-2].

• Scalar spreading rates are independent of the initia! density ratio [GOU86, RIC93]. However, there are peculiarities which have already been discussed.

• Mechanica! turbulence intensities are higher in a variable-density flow, see for instanee P&L (2) [PAN93-2] who attribute this to the near-field development of the jet, see also Ref.[KYE93].

• Most recent ex perimental investigations report an asymptotic value ·of ,.f9Jfc which is independent of the density ratio [RIC93, PAN93-2, PIT91, LOCSO].

In the following the performance of both scalar transport models is investigated regarding the above-mentioned points.

(iO

The equal-scales model does not show any significant form of variable-density ef­fects: The velocity as well as scalar spreading rates do not change at increasing w up to 7.2. The only noticeable effect on the spreading rates is a decrease by about 4%. Asymptotic values of mechanica! and scalar turbulence intensities are not affected. The only important effects which have been observed in this study occur in the near field, where density differences are largest. With decreasing nozzle gas density the peaks of turbulence quantities become higher and are located closer to the nozzle.

With the non-equal scales model, for every density ratio the constant </> must in prin­ciple he determined using a first approximation given insection 3.4.4 by Eq. (3.21), The value of </> is determined by assuming, with experimental evidence, that the asymptotic value of Jg)fc is 0.22, independent of w. Results of predictions are shown in Table 3.6, where spreading rates have not been listed because they did not change significantly. It can be seen that the asymptotic unmixedness, predicted with both models, is indeed

Cofl. w ..Jg I fc (Eq.) .fg I fc (N-eq.) KJ (Eq.) KJ (N-eq.) 4> 0 1 0.25 0.22 0.25 0.21 1100 0 1.8 0.25 0.22 0.25 0.21 1375 0 7.2 0.25 0.22- 0.23* 0.25 0.21 2275 10% 7.2 0.28- 0.29* 0.30 0.33* 0.19 0.18 2275

Table 3.6: Asymptotic values of unmixedness and mixture fraction decay at various density ratios w predicted with both models. In the equal-scales model C~< = 0.09 and O"J = 0.7. Unmixedness values with stars are the values at 25 and 50 nozzle diameters.

hardly influenced at all by density variations except in the case of coflow. Without coflow both models predicta decay constant Kj which is independent of w. This confirms the usefulness of the concept of the effective diameter De and it agrees with experimental findings of R&P. Moreover, the non-equal scales model prediets the correct value of Kj, being 0.21 (which is influenced by </>), while 4> wa.~ determined only so as to secure that Jg)fc = 0.22 at each w.

With coflow, the unmixedness is larger with both models and, remarkably, the value of Kj is almost the same predicted with both models. The latter can he explained by the fact that Rr < 2 in most parts of the flow. In that case the effective turbulent Schmidt number equals 0. 72, which approximately equals the value used in the equal­scales model.

The increase of the predicted asymptotic unrnixedness with increasing coflow velocity disagrees with an experirnental finding of Pitts' [PIT91].

3.5 Conclusions

Mechanica! and scalar turbulence in a round jet has been investigated with k e type of models.

The problem of the prediction of the round jet spreading rate has been investigated and many modifications and turbulence rnodels have been assessed. It is concluded that there is no modification on the k - e model that is both general and able to predict

61

the spreading rate correctly within the experimental uncertainty. A simple modifica­tion cl-' = 0.06 Wa.'l found to work wel!, although it is not genera!. Radial profiles of mean variables, their variances and the dissipation rate were found to be in satisfactory agreement with experiments.

Scalar transport has been investigated with a new model, which does not assume equa.l-scales for mechanica! and scalar turbulence and is basedon the two-point TSDIA theory for turbulence of Yoshizawa. This model is promising in the sense that it is able to qualitatively predict the behaviour of the turbulent Schmidt number. The geometry dependenee of some coefficients is discussed and it is tentatively suggested that the model is only valid in regions of the flow where the inertial range exists. This concept is the idea behind the time scale ratio criterion which is used to obtain realistic results in flow regions where the scalar varianee approaches 0.

In the absence of cofiow both scalar transport models predict asymptotic valnes for . the spreading rate SD2 , the mixture fraction decay constant Kf and the centreline value

of the unmixedness ..fij)fc· These variables are independent of the initial density ratio. These findings agree with the very recent experimental study on the effects of global (initial) density differences in axisymmetric jets of Richards and Pitts [RIC93]. The coefficient 4> in the non-equal scales model can be tuned to either value 8{;2 , Kf or

..j!J)fc to obtain values for the other two parameters which are in accordance with this experiment. The agreement with the experiments is better than that obtained with the equal-scales model.

62

Chapter 4

Cambustion

This chapter serves two purposes: The first is the description and discussion of the laminar counterflow (flamelet) computations and thesecondis the statistica! description of both the mixture fraction and the non-equilibrium parameter (strain rate). The statistics of these parameters is needed if the laminar flamelet data are to be used in turbulent flame computations.

In chapter 1 the choice for the laminar flamelet metbod for turbulent combustion has been motivated. In this chapter some alternative metbods are briefly discussed, wbich do not belong to the two main categories, being laminar fiamelet and pdf transport methods. The flamelet method is outlined with emphasis on the definition of the mixture fraction and on the computation of counterflow flames. Computations of the latter have been performed with an early version of tbe Cambridge code "RUNlDL" written by Rogg [ROG92]. It is based on a damped and modilied Newton iteration procedure wbich is briefly described in appendix B.3. Results of tbe computations of pure and diluted hydrogen fiames and methane fiames are presented and discussed.

In this chapter, which mainly concerns laminar fiames, the mixture fraction is called Z and the scalar dissipation rate is denoted with X· These variables are equivalent with the instantaneous values of f and €g in a turbulent situation, which gives x = cg and Z j. It is recalled that the symbols cg, g and also k and e:, which are turbulence quantities, are implicitly assumed to be Favre-averaged and tberefore the tilde is omitted.

4.1 Some alternative methods

Here some methods different from the laminar fiamelet metbod are briefly discussed and reasons are given for not using them, or for only using them occasionally. Different models have been proposed because with some methods there is a strong need to re­duce to amount of chemica! kinetic calculations to be perforrned during the turbulent combustion calculations, while this is not the case with the flamelet method. There are several ways to achieve this.

Simple Chemica! Reaction Mechanism (SCRS)

In the SCRS a one step infinitely fast irreversible reaction (equilibrium chemistry) and equal diffusivities are assurned. It can be interpreted as a (0-dimensional) perfectly stirred reactor with infinite residence time. The input for the PSR (or SCRS) is given

63

by the local value of the mixture fraction. Sirree the mixture fraction fixes the enthalpy, the temperatureis obtained through Eq. (1.8). An extension of the SCRS can be made by replacing the one step chemistry by a detailed reaction mechanism with infinite reaction rates. Although in that case many species are considered it is not essentially different.

Due to the infinite reaction ra te the model is not capable of predicting non-equilibrium effects, which preelucles predictions of localized flame quenching. Moreover, the maxi­mum temperature obtained with the SCRS is the adiabatic fiame temperature 1 . The temperature versus the mixture fraction obtained with the SCRS for a hydrogen-air flame is shown in Fig. 4.1. The cusp in the temperature at the stoichiometrie mixture fraction Z.t (Eq. ( 4.5)) is due to the irreversibility of the reaction. The shape of the profile has some similarity with the curves obtained with the fiamelet method, in which detailed chemistry induding non-equilibrium effects is accounted for. Mass fraction pro­files are shown in Fig. 4.2. No oxygen exists for Z > Z.t and no fuel exists for Z < Z.t, which is due to the assumption of au irreversible reaction.

The SCRS is often applied toa turbulent fiame using a presumed pdf for the mixture fraction. It can beseen as the limiting case of a fiamelet in equilibrium (strain rate a= 0) with equal diffusivities and a one-step irreversible reaction.

Because of its simplicity, the SCRS model can be easily interpreted. In this study it is occasionally used to support an analysis of a calculation with the laminar tlamelet model.

Eddy dissipation concept (EDC)

The EDC has been developed by Magnussen [MAG81] and it is applicable to both ditfusion and to premixed fiames. It assumes fast mixing so that reactions take place as in a perfectly stirred reactor (PSR) after molecular mixing is completed2

. Finite chemica! rate effects are incorporated by consiclering the residence time in the reactor.

The assumption that the flame locally resembles a PSR indeed leads to a rednetion of the computing time needed for the chemistry. However, in the present study it is not used because the assumption of a PSR Jacks the sound physical basis of the flamelet concept.

Reaction progress models

The reaction progress models arebasedon the concept of Janicka and Kollmann [JAN78] that only a limited number of chemica! reactions, namely the "slow" three-body reac­tions, are rate limiting, the remairring reactions being in partial equilibrium. The rate constauts of the rate limiting reactions are called progress variables and this implies that it is necessary to solve transport equations for these variables and their variances, in cl u ding a chemica! souree term. Presumed are used to solve these equations. The advantage of the method is that a rednetion of the complexity of the chemica! kinetics is achieved via a series of systematic steps. The domain of validity of these steps can be checked in laminar flame studies. For the study of non-equilibrium phenomena, which are considered in the present study, many progress variables would be needed and the computational cost would again be very high. Reaction progress models are also used in

11n some applications of the SCRS a radiative sink term is accounted for (SAL78, LOC83] in which case the maximum temperature is lower than the adiabatic flame temperature.

2 Note that with infinite residence times this concept is equivalent with the SCRS.

64

combination with the Monte Carlo joint pdf metbod mentioned in chapter 1. At present reaction progress models, with presumed pdf's, are used at the University of Twente.3

Many variabie presumed pdf metbod

The many variabie presumed pdf metbod proposed by Bockhom [BOC88] uses pdf's to model turbulent combustion with complex chemistry. In the metbod drastic assumptions on the form of the pdf are made, rather than solving the transport equation for the joint pdf. The reaction souree terms are averaged with two or three dimensional presumed pdf's. The pdf's are two or three dimensional because the souree terms depend on the temperature and on one or two mass fractions. In practice they are very often simplified to Dirac delta functions, when the chemica! reaction under consideration is not very important. A large number of transport equations for mass fractlons and their variances ( to construct the pdf's) has to be solved. The arguments not to use this model are partially the sarne as those related to the pdf transport equation method. The computational cost is high and local structures in the turbulent flame (molecular diffusion) are not modelled. At present this metbod is used at Delft University of Technology.

4.2 Flamelet metbod

The flamelet concept, which was originally proposed by Liiián and Williams, has been developed into a forma! theory by Peters [PET84, PET86].

As described in chapter 1 flarnelets are larninar, one-dimensional flarnes which are embedded within the turbulent flow field and are subject to the turbulent straining mo­tion, leading to flame stretch. This effect can be quantified by the scalar dissipation rate or strain rate. If flarne stretch is absent, and consequently chemistry is in equilibrium, the flame structure can be described solely by the mixture fraction.

4.2.1 Mixture fraction

The mixture fraction Z is a Shvab-Zel'dovich variabie [WIL85], defined as an element mass fraction of matter originating from the fuel stream. It is also called a conserved scalar because elements are conserved during cambustion and it does not influence the ftuid mechanics. The element mass fraction of element i, occurring in an N-species mixture, is

N

Z; ( 4.1)

in which aji is the number of atoms i in species j, Yj is the mass fraction of species j, M1 is the molecular mass of species j and Mr is the atomie mass of atom i. Any linear combination of Z; originating from the fuel stream is invariant under combustion.

very recent development which may be dassified into the group of reaction progress models is the 'conditional moment closure' of Bilger and Klimenko, see for instanee [BIL93] and [SMI93].

65

The mixture fraction is defined as a linear combination of Z; such that it is 0 in the oxidizer stream and 1 in the fuel stream. It can he defined as

(4.2)

with YF,I the mass fraction of fuel in the fuel stream and Yo,2 the mass fraction of oxygen in the air stream. ZF and Zo are the fuel element mass fraction coming from the fuel stream and the oxygen element mass fraction coming from the air stream, respectively. The fuel element mass fraction for instanee is equal to ZF =Ze+ ZH if the fuel consists of C and H atoms only. It must he noted that the fuel element mass fraction, and therefore also the mixture fraction, strictly can only he defined if all diffusivities are equal. Furthermore, the practical evaluation of ZF is not trivia! as wil! he seen in the next sections.

Important at this stage is the numerical value of Z near which the reaction zone wil! he located: the stoichiometrie value Zst· This value can only he defined in termsof a one-step reaction of fuel and oxidizer giving products, for instanee C02 and H20. A general global reaction is

v;..Fuel + v~Oxygen---> Products. (4.3)

A stoichiometrie balance between the mass fractions occurs if

ZF,st v;..MF Zo,st v~Mo

( 4.4)

where MF and Ma are the rnalar masses of the fuel and oxygen, respectively. With Eq. (4.4) the stoichiometrie mixture fraction can he derived to he

1 Zst = 'M y .

1 + v0 0 F,l

vf.MpYo,2

(4.5)

The stoichiometrie value thus depends on the global reaction mechanism (v~, v;..), and on the amount of pure fuel and oxygen present in the fuel and air streams (YF,h Yo, 2 ),

respectively. For metharre-air cambustion (100% CH4 and Yo2 ,2 = 0.232): Zst = 0.055 and for hydrogen-air: Zst = 0.028.

4.2.2 Flamelet theory

Results of the flamelet analysis are useful in principle only if all diffusivities are equal and if the Lewis numbers are equal to one. Furthermore, the chemistry must he relatively fast in order to have a thin reaction zone in mixture fraction space. In most applications the latter assumption is valid [PET91] while diffusivities are approximately the same in most mixtures, except in hydragen flames.

In space the flamelets are instantaneously located near the two-dimensional surface of stoichiometrie mixture (Z = Zs1 ). Perpendicular to this surface the thermadynamie variables wil! show largest variations while the tangential variations are negligible, to a good approximation. Consequently the flame structure is locally one-dimensional and can he analysed most efficiently by introducing a coordinate system that is attached to

66

the iso-surface of stoichiometrie mixture. This analysis was first given by Peters [PET84] and is not reproduced here. Only the main results are discussed.

The energy equation in terros of the temperature in the local coordinate system becomes [PET84]

ar ( a z ) 2 a2T N h, . p- = pD - -2 - L -pY; - R(T) 8t 8xk az •=1 Cp,m

(4.6)

where no convection effect is present because the coordinate system is attached to the surface (Z Z.t)· The term R(T) contains all derivatives along the surface (Z Z.t), and consequently this term can be neglected. The resulting equation is an instationary diffusion equation for T(Z, t) with a chemica! souree term localized near (Z = z.t)· The coefficient in front of the secoud derivative is equal to the half the density times the scalar dissipation rate ( called x in this chapter) defined by Eq. (2.32). A large value of x near the flame zone, for instanee at stoichiometrie conditions Xsll will result in a large heat flux across the flamelet leading to a temperature deercase near Z = z.~, which in turn will result in a lower reaction rate and eventually (at very high Xst) will quench the flame, see the S-shaped curve in Fig. 1.4. At such a situation isothermal mixing will take place, along with a large temperature gradient in the direction of the stoichiometrie surface, invalidating the assumption of small R(T) in Eq. ( 4.6). However, if it is assumed that these occurrences are short-lived, it is conjectured that the analysis is still useful.

In chapter 1 the strain rate s insteadof the scalar dissipation rate x, which emerged in the analysis above, was mentioned to be responsible for flame stretch. Indeed, it is known from the literature that in a laminar counterflow flame these variables are similar. However, in the present study it was found that, regarding quenching conditions, both variables have a different effect. In chapter 7 this is elucidated and it is shown that the strain rate is more appropriate to be used in a turbulent flame.

The analysis given so far in this chapter allows the prediction of non-equilibrium effects, manifesting themselves in for instanee a deercase of the temperature, due to flame stretch. Flame stretch in turn is quantified by a non-equilibrium parameter !1 which can be either Xst or the strain rate s. Furthermore, the flame structure at given !1 is determined by Z alone. Therefore a description of a turbulent flame can be given in terros of this analysis if the statistics of Z and !1 are known. These statistics are discussed insection 4.4. Furthermore, the quasi one-dimensionality of the flamelet opens the possibility of predicting the detailed chemica! kinetics of thermo-chemical variables in a laminar one-dimensional flame and incorporating the data thus obtained into a turbulent flame computation. The laminar flame geometry used for this purpose is the counterflow stagnation point diffusion flame, to be discussed next.

4.2.3 Counterflow diffusion flame

In this section a system of ordinary differential equations descrihing a laminar !lamelet is derived and the corresponding boundary conditions are given in section 4.3. The equations are solved for hydrogen and methane flamelets using the numerical method described in appendix B.3.

The flow considered is a plane or axisymmetric stagnation point flow, generated by two reaetarits flowing towards each other from opposite sicles (Fig. 1.3). Near the stag­nation point the flow forms a boundary layer, which can bedescribed with a boundary

67

layer sirnilarity coordinate 1J which reduces the partial differential equations to ordinary ( one-dirnensional) differential equations . The coordinate in the direction perpendicular to the stagnation plane is called y, the coordinate perpendicular to y is x. The fuel side is located at y --+ oo and the oxidizer side at y --+ -oo, see Fig. 1.3. The velocity gradient at these boundaries is called a1 and a2 respectively:

au a2 ax ly->-00 (4.7)

in which u is the velocity in the x-direction. It can be shown using conservation of

momenturn that a1 a2VP2/P1· Frorn now on the strain rate at the oxidizer side a2 is called a. where the index e denotes the edge of the boundary layer at the oxidizer side. This strain rate is the parameter that characterizes the flow field. If it is assurned that the velocity u is proportional to x in the whole flow field [BAT90a] then

(4.8)

( 4.9)

which autornatically satisfies the continuity equation. Here, and in the following deriva­tion, the prime denotes differentiation with respect to the variabie between round brack­ets. The parameter j is equal to 0 in the planar case and 1 in the axisyrnrnetric case, where x corresponds to the radial distance. Now, to transforrn the boundary layer equa­tions into ordinary differential equations in the vicinity of the stagnation point strearnline (x= 0), the dirnensionless density weighted sirnilarity coordinate 1J

(j + 1 )a. t' p(y') dy' Pe/le Jo

( 4.10)

is introduced together with the dirnensionless strearn function f/J(y)

f(y) = ..,;a;v;f/J(y). ( 4.11)

It is useful to write 1/J and f/1' in terrns of u and v:

V Pe/leae(j + 1) f/l(y)

pv ( 4.12)

and frorn pu p.xf'(y)/..Ji+I p.x';:f/J'(!J) pa.xf/1'(11) = pu.f/1'(11) it follows that

u ( 4.13)

In this derivation the identities

f'(y) and

have been used. The sirnilarity equations for continuity, rnornenturn, species and ternperature are

given by:

( 4.14)

68

1

+ .!!:_( c <!>") dr;

--=1

,---- [~w;M;h; + 4k9 uT4]

N dT -r=====-;;:;-- PCp,; Y; V; 'I) dr;

( 4.15)

( 4.16)

( 4.17)

where the prime denotes differentation with respect to r; and where it was used that x-derivatives on the stagnation point streamline are 0. The momenturn Eq. (4.15) can be considered an equation for </>' and the integration of the continuity Eq. (4.14) gives </>. In the momenturn equation C is the so-called Chapman- Rubesin parameter C = ...1!.1!.....

Pe#e

In the species Eq. ( 4.16) V;'IJ is the mass dilfusion velocity of species i in the r;-direction, and w; is the production rate of species i in kmolem-3s-1 . In the temperature equation a radiative sink term proportional to T 4 has been added for future use (chapter 6). It is not used in the present chapter. In ( 4.17) Cp,m is the mixture specific heat, À is the mixture heat conductivity and h; is the enthalpy of species i. The mixture quantities are to be calculated using weighting procedures for the individual species properties. For the evaluation of the thermodynamic and transport properties, see appendix D.

It must be noted that the axisymmetric equations (j 1) are equivalent to the planar case if p is constant and a. is replaced by ~a •.

The mass dilfusion velocity V; consists of three terms, namely concentration gradient dilfusion (ordinary diffusion) V;c, temperature gradient dilfusion (thermodiffusion) V;T and pressure gradient dilfusion [ORA81]. The latter is not important in the present (isobaric) application. The dilfusion velocity can be written as

( 4.18)

where the meaning of V;, is discussed below. Thermo dilfusion can be important for light species such as H, H2 and He. In the present study the incorporation of thermo dilfusion led to an insignificant effect on the maximum temperature and consequently it is neglected in the computations to be presented. The approximations used for V;c and V,T are listed in appendix D. The dilfusion velocity should obey the identity

N

Y;V; 0 ( 4.19)

and this is enforced by the correction velocity V;, E~1 Y;(V;c+ V?) in Eq. ( 4.18). This correction velocity, which is not thought to be extremely important [WIL85], appears because the approximations used to evaluate V;c and V,T do not aut,omatically satisfy Eq. (4.19).

The boundary conditions for the temperature and mass fractions at r; ---> -oo ( oxi­dizer) and r;---> +oo (fuel) have to be specified, and for the momenturn </>' they are:

</>'(-oo) 1; </>'(+oo) = /2fi 69

The dimensionless stream function 1> should obey Eq. (4.14) which can be satisfied only if the numerical valnes of 1} at the boundaries are large enough.

It must be mentioned that the boundary conditions used in the present study are also referred to as potential flow. The so-called plug-flow boundary conditions, which are not used in the present study, would be [CHE93]: 1>'( -oo) 0; 1>'( +oo) = 0 and both 1>( -oo) and 1>( +oo) prescri bed. The charaderistic strain rate is then determined by a (numerical) evaluation of~ in the outer oxidizer flow.

4.3 Results for counterflow ftames

Predicted temperature and mass fraction profiles of the Hamelets can be given either in terms of physical space dimensions or in terms of the mixture fraction. The latter has the advantage of being useful in the turbulent flame computations and therefore will be employed. However, the mixture fraction either has to be expressed in termsof the calculated mass fractions, for which several alternatives exist, or it can be obtained by solving a convection-diifusion equation with a mixture averaged diifusion coefficient. These possibilities are investigated based on computations carried out for the case of a hydrogen flame. Next, computations of the hydrogen and metharre fiames are presented to illustrate the possibilities of flamelet computations and to compare predicted valnes of important parameters such a.'! queuehing valnes for the strain rate with valnes from the literature.

4.3.1 Defi.nition of the mixture fraction

The definition used in the present study, which is given by Bilger [BIL88) and derived in appendix E, is based on the fact that the fuel and oxidizer eventually will form only C02 and H20. In this definition, which is valid if the fuel contains only C and H, the element mass fractions of C, H and 0 occur:

z _ 2Zc/Mc + tZH/MH + (Zo,2- Zo)/Mo

- 2Zc,dMc + ~ZH,t/MH + Zo,2/Mo (4.20)

Here Zi is the mass fraction of the element i, given in Eq. ( 4.1 ), and the indices 1 and 2 indicate that the mass fraction corresponds to the fuel stream or air stream respectively. The advantage of this method is that Zst can be computed numerically, Zst being Z at the condition (2Zc/Mc + l/2ZH/Mll Zo/Mo = 0), and the resulting value can be compared with the analytic value obtained from Eq. (4.5). This is not possible with the definitions which are discussed next.

The mixture fraction can also be defined by a transport equation

!:_(~dZ) d1] PeJLeCp d1]

( 4.21)

where the diifusion coefficient is mixture averaged. In this case the mixture fraction is solved as part of the problem, although it does not influence the other variables. This definition is used for comparison with Eq. ( 4.20).

70

Finally, two further definitions found in the literature are given. The first one is used by Dixon -Lewis et al. [DIX86]

z (4.22)

in which M is the number of elements present in the flow. This definition gave nega­tive values of the mixture fraction (due to differential diffusion effects) in [DIX86] and therefore will not be used in the present study.

The second is used by Gutheil, Balakrishnan and Williams [GUT93]. They employ the hydrogen element mass fraction ZH to define Z:

(4.23)

Also this definition will not be used in the present study. All definitions rely on the implicit assumption that all diffusivities are equal. If this

is the case, these definitions are the same, as they should.

4.3.2 Hydrogen flame

In this section pure and diluted hydrogen flames are discussed. In the case of pure flames predictions of quenching behaviour are compared with literature studies and the consequences of non-equilibrium effects, which appear at high a, are shown. The case of the diluted fiame is used to show the importance of preferential dilfusion regarding both the definition of Z and the maximum temperature in the flamelets. The reaction mechanism used is given in appendix D.3.

Pure hydrogen flame

A 100 % flame is considered first. Several studies of this flame have been reported in the literature. Most recent studies [TRE93, GUT93] indicate that rate constants for hydrogen combustion under atmospheric circumstances are now well established and that predictions of mass fractions of major species are not severely influenced by variations of the rate constants. The H 2 0 2 mechanism used in the present study is given in Table D.l. It camprises 11 species and 30 reactions (including the NO-scheme). One of the most important parameters to be predicted is the strain rate at queuehing aq which in the present study is predicted to be 14.00 x 103 s-1 for a planar fiamelet. Literature values are aq = 14260 s-1 [TRE93] with the same mechanisrn as in the present study and aq = 14400 s- 1 [GUT93] with different rate constants for sorne reactions. The agreement is satisfactory, also if the cornputed value of Dixon-Lewis et al. [DIX86] aq ( 1.3 ± 0.1) x 104 s- 1

, where an estirnate of the uncertainty is included, is taken into account. The incorporation of the correction velocity lÎ;, in the present study leads to a somewhat slower quenching behaviour: aq 15.38 x 103 s-1

• However, the structure of the flarne is not influenced. The difference between aq = 14.00 x 103 s-1 and aq = 15.38 x 103 s-1 without and with the correction velocity Vc, respectively, can be interpreted as a charaderistic uncertainty for the accuracy with which aq is determined. This difference also agrees with the uncertainty of ±1300 s-1 given by Dixon-Lewis et al.

Experimental values of aq are available only for the axisymmetric case: aç(exp) = 8250 s- 1 , referenced by Gutheil et al. [GUT93]. In the present study a value of aq =

71

8.00 x 103 s-1 is found while Gutheil et al. found aq = 8140 s-1 , both in agreement with the experiment. The planar value of aq = 14.00 x 103 s-1 in the present study can be compared with twice the axisymmetric value 2 x 8.00 x 103 s- 1 16.00 x 103 s-1 . The difference is due to the non-constant density near the oxidizer boundary, which is very close to the flame zone near Z Zst = 0.028.

Temperature profiles predicted with the present flamelet method at various strain rates and the profile obtained with the SCRS mechanism are shown in Fig. 4.1. The

2500

2000

g 1500

1-

1000

500

zat 0 0.0 0.1 0.2 0.3 0.4

z

0.25 0.10 Solld: a•100/s

~ Dashed: a•1 0000/s 0.20 ~ Dotted: SCRS 0.08

\ \ \

0.15 \ Oxygen 0.08 \ .. "_.. 0 I

>- \ 2at 0.10 \ 0.04 \ \

\

0.05 0.02

0.00 ~...=.:=..::::..;;.,=::c:___:::" ___ ~--=-====c::::l 0.00

0.00 0.02 0.04 0.08 0.08 0.10

z

0.5

.. J:

I >-

Figure 4.1: Temperature profiles ( from top to bottorn curve) obtained with the SCRS, and with the fiamelet method at fL 100 1000 s-1 , 10000 s-1 , and 13500 s-1 •

Figure 4.2: Mass fraction profiles of Hz and Oz at a 100 s-1 (solid), 10000 s-1

(dashed) and a profile ob-tained with the SCRS (dot-ted line) .

profile at a the low strain rate of a = 100 s-1 resembles the profile predicted with the SCRS. Indeed, the SCRS is equivalent to equilibrium chemistry with one irreversible reaction. The SCRS is sometimes called "mixed=burnt", which is illustrated in 4.2: no fuel penetrates at Z < Zst and no oxygen penetrates at Z > Zst· The mass fractions of fuel and oxidizer predicted in the flamelets allow coexistence. This effect is partly due to non-equilibrium as can be seen in Fig. 4.2. It can also be observed from 4.2 that CO mass fractions increase at increasing strain rate while C02 mass fractions decrease at increasing strain rate. This is to be expected: a flame near queuehing is not able to burn the fuel completely which leads to a higher CO level, while correspondingly less C02 is produced.

72

Diluted hydrogen flame

The mass fractions at the fuel side are: YH2 ,00 0.0466; YN2 ,00 0.9534, which implies Z.t = 0.384. In this diluted flame the definition of Z, which can be based on element mass fractions (from Eq. (4.20) denoted with Z.!) or on the transport equation (from Eq. (4.21) denoted with Z1.), can lead to large differences. In Fig. 4.3 the temperature in the flamelet at a 500 s-1 versus both definitions of Z is shown, together with a line representing Z = Z,1 = 0.384. The maximum temperature in the curve with Zte is seen to be located near Z ~ 0.2, while using z.1 the temperature maximum is located near Z = Zst· Therefore the latter definition of Z from the element mass fractions is preferred.

The difference between the two definitions of Z is due to the explicit assumption in the transport equation that all diffusivities are equal, which of course is not valid in an H 2-flame, due to the very high hydrogen diffusîvities. In the definition of Eq. (4.20) this assumption is more implicit and in fact it is a measure for the element mass fraction of hydrogen. Moreover, as already stated, with this definition the stoichiometrie value z.t can be calculated from the numerical solution of the flamelet while this is not possible with the transport equation for Z.

In Fig. 4.4 temperature profiles predicted for the flamelets and SCRS show that at low strain rates the maximum temperature in the flamelet exceeds the adiabatic flame temperature which is predicted with the SCRS, especially at fuellean conditions. This effect, which is surprising at first sight, is due to the preferential dilfusion of molecular hydrogen. More hydrogen is consumed by the oxygen than when the diffusivities were all equal. This leadstoa higher temperature compared with the SCRS maximum (adia­batic) temper at ure because in the SCRS equal diffusivities are explicitly assumed. In the literature this effect, which is mainly important in diluted hydrogen flames is seldomly mentioned. In pure hydrogen flames flamelet temperatures at fuel lean conditions and at fuel-rich (Z > 0.35) are somewhat higher than the SCRS temperature, which can be seen in Fig. 4.1.

Extinction behaviour

Dilution of the fuel with nitrogen will result in extinction of the flame at lower strain rates than for pure flames, see Table 4.1. This is indeed what the computations show. The diluted fiame extinguishes at a = 6150 in the present study, which is less than half the value for the pure flame. However, the scalar dissipation rates at quenching show an opposite behaviour. In the diluted flame all definitions of x at quenching are higher than in the pure hydrogen case, see Table 4.1. This would indicate that the scalar dissipation rate can not represent a parameter which characterizes non-equilibrium effects. It suggests that the scalar dissipation rate is unsuitable for purposes of turbulent cambustion calculations. This problem, which is not mentioned in the literature, is addressed further in chapter 7 about lift-offin turbulent diffusion flames. In that chapter it is shown that the sametrend is also observed in (diluted) methane flames.

4.3.3 Methane ftame

Computations for methane flames comprise, apart from the H2 - 0 2 mechanism given in Table D.l, at least the Cl mechanism (where merely species with one carbon atom

73

g 1-

g 1-

2000

' ' [~·500/S I

' 1500

1000

9L-----~----~~----~------~----~

0.0 0.2 0.4 0.8 0.8 1.0

z

2000

1500

1000

600

0 0.0 0.2 0.4 0.8 0.8 1.0

z

Figure 4.3: Tempera.ture in the diluted YH1 ,00 = 0.0466 fiame versus the mixture fraction Z obta.ined with Eq. (4.20, Ze~) (solid line) and the transport Eq. ( 4.21, Zte) ( dashed line).

Figure 4.4: Temperature profiles in the diluted hydra­gen flame obta.ined with the SCRS, and with the flamelet method for a = 100 s-1 ,

1000 .~- 1 and 6000 s-1 (from top to hottorn curve).

are considered). Most predictions have been perfomled with the Cl mechanism of Ta­bie D.2. Some of the cornputations have been carried out including the C2 mechanism of Table D.3. The latter mechanism results in a total of 26 chemica! species, for which rnass fraction equations have to be solved, and 87 chemica! reactions, which leads to long computing times. The hydrogen flarne comprised only 11 species and 30 reactions (in­cluding the NO-scheme). It should be mentioned that the computing time is determined solely by the number of species and not by the number of reactions.

The definition of the mixture fraction, which was very important in the case of the diluted hydragen flame, is much more arbitrary in the metharre flarne sirree all important diffusivities are approximately equal. Consequently, at all strain rates the maximum temperatures in the flarnelets are found to be lower than the adiabatic temperature obtained with the SCRS, see Fig. (4.5). In Fig. 4.6 it can be observed that the oxygen 'leaks' through the reaction zone (near Z 0.055) at the high strain rate near extinction while the C II4 is consumed before it can penetrate into the oxidizer boundary. The oxygen leakage due to the non-equilibrium effects is responsible for the presence of oxygen on the centreline of turbulent diffusion flames. From Fig. 4.6 it is also observed that Yco increases as expected in a flame near while Yco 2 decreases.

A compilation of extinction strain rates is given in Table 4.2. The value of aq for

74

Flame aq Xst,q (1) Xst,q (2) Xmax,q (3) Reference • (mass% H2) (s-I) (s-I) ( s-1) ( s-1)

Plane 100 14.00 x 103 1100 2900 2750 Present 100 1.3 x 104 - - [DIX86]

• 100 14260 - [TRE93] 100 14400 - [GUT93] 100 with V., 15.38 x 103 1350 3250 2100 Present 0.0446 6150 7000 3000 4100 Present Cyl. symm.

• 100 Exp. 8250 - - In [GUT93]1 100 8.00 x 103 590 i 1470 1300 Present

1100 8140 [GUT93]

Table 4.1: Quenching values of the strain rate and several definitions of the scalar dissi­pation rate for diluted and pure hydrogen fiames. The scalar dissipation rates are defined according to Eq. {4.20) (1); Eq. (4.21) (2) and maximum temperature conditions (3), evaluated with Eq. (4.21).

a plane 100% CHrair flame predicted in the present study is found todependon the mechanism used and on the preserree of the correction velocity. Without V., and with only the Cl mechanisrn aq 505 s-1

• Again this value is sornewhat lower than twice the value of ag in the axisymrnetric case aq = 270 s-1 , just as in the hydrogen flame. If V., is included in the planar case, aq is found to be higher (ag = 556 s-1 ) and induding the C2 rnechanism of Table D.3 leads to a queuehing value of ag = 595 s-1 .

The queuehing value found by Peters and Kee [PET87] of aq = 450 s-1 in the planar flame with V" is significantly lower than the value in the present study of aq 505 s-1 •

According to Ref. [CHE93] this is probably due to the neglect of the species CH and CH2 in the study of Peters and Kee.

A recent numerical study about metharre counterflow Harnelets by Chelliah, Seshadri and Law [CHE93] concerns an axisymrnetric flame computed with the same Cl mecha­nism as in Table D.2, except that they used Lindernann forrns for some reactions as in Ref. [PET93]. They found an extinction strain rate of aq 259 s-1 which is close to the value in the present study of aq 270 s-1

•

From the scatter in aq predicted by Chelliah et al. [CHE90] the influence of different chemica! mechanisms is seen to be large. The experimental value of ag = 190 s-1 given in Ref. [CHE90] is lower than in most numerical studies, induding the present study. This is probably due to the experimental configuration in which the outer flow was closer to plug flow than to the potential flow used in the present study. Indeed, the plug flow boundary conditions yield lower valnes of aq with the same mechanisrn [CHE93].

Given the considerable scatter in the publisbed predicted values of aq it rnay be concluded that the flarnelet computations presented in this chapter are in satisfactory agreement with those found in the literature. The scatter is rnainly due to different chernical mechanisrns used. However, it was shown in the present study that the cor­rection ditfusion velocity V" may have an influence of 10 % on predicted values of ag

75

2500

2000

1500 g 1-

1000

i

500

0 0.0 0.1

0.20

o.1s L

~ 0.10

0.05-

0.00 0.0 0.1

(Table 4.2).

4.4 Statistics

0.2

z

0.2

z

0.3

0.3

Figure 4.5: Temperature profiles for a methane flame obtalned with the SCRS, a.nd with pla.nar flamelets without Vc for a = 100 s-1 ,

400 s-1 and 500 çi (from top to bottorn curve).

Figure 4.6: Mass frac­tion profiles of CJI4, 02; co a.nd co2 for methane flanlelets at strain rates of a = 100 s-1 and 500 s-1 •

See a.lso caption of Fig. 4.5.

In a turbulent flame the flamelet method can be used if the joint pdf P(H, Z) is known, where n is the parameter which is responsible for the non-equilibrium which would formally be X· Fortunately it appears that Zand x are statistically independent in a mixing layer type of flow, except in the near field region [ASH85]. Therefore to a good approximation P(H, Z) can be written as the product P(Z)P(H). Formally n should be conditioned near the flame zone but it will be assumed that the statistics of n conditioned on stoichiometrie conditions, are identical to the unconditioned statistics. The scalar variables such as mass fractions and temperature are then given as

(4.24)

in which if>(f; !!) indicates that </> is a function of f with n as a parameter. In the following sections the presumed pdf's for f and for n are introduced.

76

! Flame aq Xst,q (1) Xst,q (2) J;_ax,q (3) Reference Type (s-1) ( s-1) ( s-1) -1)

Plane

No Vc; Cl 505 29 43 42 Present With \/";Cl 556 33 48 41 Present With \/"; C2 595 36 52 43 Present With Vc; Cl 450 - [PET87] -, ..... Cyl. symm. No V:,; Cl 270 14 21 20 Present No Vc; Cl 259 - 17 - [CHE93] Diff. mech.; Cl 177-272 1 - 15 - [CHE90]

I ~~~g flow; Cl 194 - 17 - [CHE93] 190 i - 19.9 [CHE90]

Table 4.2: Quenching values of the strain rate and several definitions of the scalar dis­sipation rate for methane-air fiames. The scalar dissipation rates are defined according to Eq. (4.20} (1); Eq. (4.21} (2) and maximum temperafure conditions, evaluated with Eq. (4.21} (3).

4.4.1 Mixture fraction

The mixture fraction obeys the {:1-function dis tribution toa good approximation [GIR91], while it is well established that the exact form of the pdf does not very much influence the predicted mass fractions of the major species, temperature and density. The Favre­averaged {:1-function pdf is completely determined by its first two moments j and g:

_ r-1 (1 nb-1 P(f) = f:1(a, b)

where a ]c, b = (1 ])c with c ](1- ])jg- 1

f:1(a,b) l r-1(1- f)b-ldf.

The conventional pdf P(f) is related to the Favre-pdf F(f) as

F(J) = p(f)P(J). p

(4.25)

( 4.26)

(4.27)

Since F(J) can only be constructed if the Favre-averaged J and g are predicted, which is the case in this study, also the conventional averages of scalar variables can be obtained with the help of Eq. (4.27):

(4.28)

(4.29)

77

In particular the mean density is obtained as

fJ P(f) df. Jo p(f)

( 4.30)

An illustration of the pdf's in a turbulent jet flame is given in Fig. 4. 7. The pdf's near the centreline are approximately symmetrie around the mean mixture fraction, while the pdf's at larger radial distance show a sharp peak near f = 0, which reflects intermittent behaviour. Intermittency is the feature of turbulentflowsof alternatively being turbulent and laminar in time, sec Fig. 4.8. In a jet contiguration the period of laminar flow is

f

l

15

: 1- 0.01 '/

10

f- 0.11 f = 0.25

- / .... -.... ( / (centra)

5 :I ' j ',

{; ' I : ' I --

1 I

I OL------d~-------L~-------L~~--~~~--~

0.0 0.1 0.2 0.3 0.4 0.5

~A, l r .. y--~~"~ " ' t, I ,,, - ,, fE-----?-1 ~"--"'

t T

Figure 4.7: Pdf's of the mixture fraction at three radial distauces from the centreline of an ax:isymmet­rîc turbulent diffusion flame. Solid line: j = 0.25 ( centre­line), dasbed line: j 0.11, dotted line: j 0.01.

Figure 4.8: Schematic pic­ture of a signal which is alternatively turbulent and laminar in time. The in­termittency factor is 7 = (T- t1 t 2) /T where t1 and t 2 are intervals in time where the fluld is laminar.

associated with the preserree of free air which is represented by f = 0. Therefore the intermittent behaviour can be modelled by placing a so-called intermittency spike at f = 0 in the pdf for the mixture fraction. In order to quantify the intermittency an indicator function I is introduced which is equal to 1 when the flow at the instant under consideration is turbulent, while it is 0 when it is laminar. The mean of this function is called the intermittency factor i. The tot a] ( unconditioned) pdf then becomes

P(f) (1 - :Y)ó(f) + :yPt(f) (4.31)

78

where Pt(J) is the ( conditioned) pdf, representing the turbulent signa!, which is equiv­alent to Eq. (4.25) with a and b replaced by at and b1• These are in turn given by the previous expressions fora and b evaluated with ft and 9t insteadof J and g. The inter-

. . ry-0.32; f-0.01 ~~

20 i.

1600

g 1000 1-

600

0 0 5 10

R/D

0.26

0.20

Without

0.10

-----10.05

20 25

Figure 4.9: Thrbulent nal Pt(f) of Eq. (4.31). The intermittency spike at f 0 (with strength 1 - t) is not shown. See caption of Fig. 4.7 for further informa­tion.

Figure 4.10: Radial profiles of temperature and mass fractions predicted for a nat­ural gas flame with (open symbols) and without (filled symbols) intermittency.

mittency factor 7 can be obtained by solving its transport equation [DOP77, CHE87], but also by using an empirica! correlation given by Bilger [KEN77]

(/{+1)/(~ +1) J2

( 4.32)

with an empirica! constant K ~ 0.3, which implies 9t J{]?. The relationship between the turbulent and unconditioned variables is given by

j (g+ }2)

ilt 7(9t + ]?)

( 4.33)

( 4.34)

In Fig. 4.9 the turbulent signa} P1(f) of the pdf's are presented at the same locations as in Fig. 4.7. The radial profiles of temperature and some mass fractions, computed with and without intermittency spike in the pdf are shown in Fig. 4.10. The profiles

79

are not significantly different. Moreover, the empirica! Eq. ( 4.32) leading to g1 = /{ R is incorrect if i = 1. It would only hold approximately in the far field on the symmetry axis of an isothermal jet. Indeed in a flame problems occur near the nozzle. Furthermore, if ;y 1 then in fact g should equal g1 , which is not the case if g1 [{ ff.

4.4.2 Non-equilibrium parameter

The non-equilibrium parameter can be either the scalar dissipation rate orthestrain ra te. From the literature it is known that the first one obeys a log-normal distribution and the second a quasi Gaussian distribution [ABD88]. The scalar dissipation distribution is not used in this study and the strain rate distribution is given in chapter 5.

4.5 Conclusions

A short review on other turbulent cambustion models than the flamelet methad has been given induding the reasons for not using them in the present study. The concept of the mixture fraction as a conserved scalar in turbulent flames and the theory of the flamelet method have been presented. The ordinary differential equations descrihing the counterflow geometry of the laminar flamelets, that is used to obtain the thermo­chemical data for turbulent flame computations, have been solved for hydragen and metharre flamelets. It was found from computations on diluted hydrogen flames that the partienlar way in which the mixture fraction is defined is not arbitrary when differential dilfusion effects are present. In the case of diluted hydragen flames the definition based on the element mass fraction gives most realistic results, while in all other cases the definitions give similar resnlts.

The extinction behaviour of the 100 % hydragen-air and the methane-air flamelets in terms of the queuehing value aq is in agreement with valnes found in the literature, although the valnes show considerable scatter. This may be due, however, to different chemica! mechanisms used. With the same mechanism, valnes predicted in the present study agree with those reported in the literature. Indusions of thermo-diffusion effects have been found to he negligible while the indusion of a correction diffusion velocity was found to have a rather large influence of about 10 %.

The statistics of the mixture fraction have been given by a presumed pdf in termsof a iJ-function, for the case of Favre-averaged variables. The way to obtain conventionally averaged scalar variables has been ontlined. Intermittency effects, which can be captured by introducing an intermittency spike in the pdf, have been shown to have a very small effect on cornputed mean variables in a turbulent flame.

80

Chapter 5

Turbulent hydrogen flames

Predictions of hydrogen ftames are compared with experiment al data from the liter at ure. This comparison usually is complicated by the existence of turbulence model deficiencies as well as experimental uncertainties which both can he quite large. The additive effect of these uncertainties sometimes makes a detailed quantitative comparison less useful. In that case it can he very profitable to investigate trends or sealing behaviour that can he observed in computations and experiments. For instance, the behaviour of various quantities with varying coftow, the inftuence of combustion on the flow, the sealing properties of the NO concentration with varying coftow and with the Reynolds number etc. are investigated in the present chapter. Also detailed quantitative comparisons of predictions with experimental data from the literature are made.

The modeHing of the thenno-chemical state with the ftamelet metbod is described in section 5.1. Some physical charaderistics of ftames are compared with those of isother­mal jets in section 5.2. The influence of coflow on various physical quantities in a 100 % H2 flame is investigated in section 5.3 and it is compared with the experimental data of Kent and Bilger (K&B) [KEN73].

The prediction of thermal NO is given special attention in section 5.4. In particular its dependenee on the strain rate, which is usually neglected in NO predictions in the literature, is investigated. The sealing behaviour of the NO concentration with the jet exit Reynolds number in a turbulent hydrogen ftame is investigated.

In section 5.5 the non-equal scales model that was introduced in chapter 2 is applied to investigate its performance in the flame measured by K&B and Kennedy and Kent (K&K) [KEN81].

Differential dilfusion effects are investigated in section 5.6 with emphasis on the applicability of the flamelet method to diluted turbulent hydrogen flames. Finally, the conclusions are summarized in section 5. 7.

5.1 Laminar flamelet modeHing

The principal idea in the present application of the ftamelet method is to model the mean thermo-chemical state in the following way. Given the local turbulent strain rate the probahility of burning Pb, which is associated with local flame quenching, of the flamelets can he defined. Consequently, 1 -Pb is the probability of isothermal mixing at that position. The burning flamelets experience a strain rate, the pdf of which (together with the pdf of the mixture fraction) determines the average thermo-chemical state in

81

the burning part. This pdf of the strain rate is approximated by a delta function pdf. For a given mean strain rate appropriate laminar flamelets, characterized by their respective strain rates, are chosen and the mean thermodynamic variables can be computed.

A veraged scalars

The Favre-averaged value ~of a scalar rjJ(f,O) is given by Eq. (4.24). For the non­equilibrium parameter n now the strain rate s is substituted, and this choice will be motivated further in chapter 7. (4.24) can be modified into

J> l dfF(f) fo""" F(s)r/>(f;s)ds

l df [fo'" F(f)F( s )<P(f; s )ds + 1~ F(f)F( s )<Po(f)ds]

f1

df f'• F(f)P(s)<t>(f;s)ds +<Po(]) r'"' P(s)ds (5.1) 1o 1o 1 ••

where <Po(!) r/>(f; s > sq) equals the variabie <Pat isothermal mixing. The last step in which J0

1 F(f)<Po(f)df is replaced by <Po(j) is justified because in the isothermal mixing situation the average J>o may be written as <Po(]) sirree all rPo(f) profiles are linear. Eq. (5.1) is simplified by the introduetion of the probability of burning Pb

and (1 H) = t"" F(s)ds. 1 •• (5.2)

Now the mean value :/> can be written as

~ (1 H)<Po(j) + {• dsF(s) l F(f)<P(f;s)df. (5.3)

The integral over f is renamed as

~(s) t F(f)<P(f; s)df (5.4)

which gives

(5.5)

Here, the first term is the isothermal mixing term and the second one wil! be called the burning part.

The strain rate obeys the quasi-Gaussian distribution [ABDSS]

P(s) 2

where ers is the variance. It is coupled to the mean s by

s [!er. This leads to the probability of burning

aq ) H erf(_ 112

. S1f

82

(5.6)

(5.7)

(5.8)

Here erf denotes the error-function and aq is the strain rate at quenching (obtained from the laminar flamelet computations in chapter 4). It is recalled that strain rates associated with laminar flamelet computations are denoted with a, while those associated with turbulence are denoted by s. In order for Eq. (5.8) to be useful, the average turbulent strain rate s must be estimated. This is a controversial matter since a choice has to be made about the size of the eddies with which the strain rate is associated. The strain rate of the large ( energy containing) eddies is proportional to e: / k while the strain rate

of the small eddies is proportional to ~· The constants of proportionality of both strain rates are unknown at the moment. However, in chapter 7 approximate valnes for these constants for the case of lifted diffusion flames are proposed. They are ~ 6 for the large eddies and ~ 0.3 for the small eddies. Although in that chapter these values are found for a natura! gas flame the order of magnitude is assumed to be the same for hydrogen flames.

The term in Eq. (5.5) representing the burning part is approximated in two ways, depending on the objective of the specific computation. In the case that variables which are to be predicted are sensitive tothestrain ra te (such as N 0 concentration for instanee) the burning part will be simulated with M (1'.1 > 1) flamelets. In the case that only the mean temperature and mean density are required it suffices to approximate the burning part with only one flamelet. This is a reasonable approximation because these variables (in contrast to NO concentrations) do not vary severely when the strain rate varies between 0 and the quenching value ag.

The simulation of the burning part in Eq. (5.5) with M flamelets with strain rates ai (i running from 1 to M) is practically performed by neglecting fluctuations so that the pdf of sis a single delta function at s = s, namely F(s) ó(s- s). Thesetof M flamelets is called a 'flamelet library'.

If it is assumed that the burning part of Eq. (5.5) can be approximated by only one flamelet tPb, characterized by ab, then

(1 Pb)q;o(i) +Pb t F(J)Ij;b(f; ab)df

(1 Pb)~j;o(}) + H~b(ab)

(5.9)

(5.10)

which is the simplest approximation. It is noted that the mean density pis obtained by integration over each flamelet and using Eq. (4.30):

(5.11)

Similarly, the mean temperature is obtained as

(5.12)

The computation of the mean species mass fractions is analogous to that of the mean temperature. In the first term in the above equations the isothermal value is evaluated at the Favre-mean mixture fraction ], instead of the conventional mean ]. This ap­proximation is allowed because the actual difference between a Favre-averaged and a conventionally averaged mixture fraction is small (according to predictions performed in the present study).

83

In the case that the burning part is described with more flamelets, the mean ~(s) with a; < .5 < ai+ I is obtained by linear interpolation between ij>(!; a;) and ij>(!; ai+ I). In order for this procedure to be accurate the flamelet library must be large enough. A smalllibrary suffices for variables with only a weak dependenee on the strain rate. When a variabie depends approximately linearly on 1 I a then a linear interpolation based on the reciprocal strain rates 1la;, llai+I and 113 should be used. The NO concentratien will beseen to be linear with lla (especially at low strain rates) and for this variabie it is essential to use this type of interpolation.

Finally a remark on the geometry of the laminar flamelets (plane or axisymmetric) is in order. The choice of the configuration influences the value of aq, but not the structure of the laminar flamelet. Since aq is different for both geometries the response of the flamelet to an applied strain rate is different for the plane and the axisymmetric case. However, the coefficient in the evaluation of the turbulent strain rate s is far from well established. Therefore the uncertainty on the choice of the geometry of the laminar flamelets is transferred to the uncertainty of the strain rate coefficient.

5.2 Flame-isothermal jet characteristics

The flow field is modelled with the k ê model and the equal-scales scalar transport model. The choice for C~< = 0.06 has been motivated in chapter 3 by consiclering the spreading rate of isothermal jets. The temperature and density are computed with one burning flamelet at a 100 s-1

, assuming also Pb f'::l 1 in the whole flow field. Computations are performed with the parabolized method, using 30 grid nocles in the radial direction and a step-size of 0.02. A computation with the elliptic method has been used to produce some iso-scalar contours. These contours are intended to visualize the jet and the flame in a manner different from the profiles that are used in the rest of the thesis.

Predietiens of a vertical hydrogen jet (exit velocity 151 mis with coflow of 1 rnls) and corresponding flame are now discussed and are compared with experimental data found in the literature.

In a flame all variables decay more slowly than in a conesponding isothermal jet due to the expansion effect (Figs. 5.1, 5.2 and .5.3). This is an evident feature of flarrtes and it is confirmed by all experiments. An illustration of this expansion effect in terms of iso-mixture fraction contours in a hydrogen jet and in the flame, predicted with the elliptic method, is given in Figs. 5.4 and 5.5.

Experimental data suggest that the turbulence intensity is usually smaller in a flame than in the conesponding isothermal jet [WITSO, KEN73]. However, due to the inherent assurnption in the turbulence model of u' "' v'k ~ ü it can be expected that the predicted turbulence intensity is not altered by the flame: veloeities are higher ( expansion) and the turbulent kinetic energy is accordingly also higher, leading to same turbulence intensity in the far field (Fig. 5.1). However, in the near field the turbulence intensity in the flame is lower because both k and u attain their asymptotic behaviour at x I D, due to the expansion. An illustration of the temperature field in terrus of iso-temperature contours is shown in Fig. 5.6.

The smaller turbulence intensity observed in experiments can be attributed to the increase of molecular viscosity (because of the higher temperature), which will deercase the turbulence Reynolds number (Re 1 ~ P Iw) and consequently deercase the turbu-

84

.. .. .. c ... ~ E c

"'

1.0:n \--~····-~~-···-~---···---;::=====::==-~;=::;--l

. \ Normalized r== lsc)fh.l o.s \ velocity . =-= Flamel

l

0.6 ' ' I l I \ I

\ Turb. lntensity 0.4

,, ~~-~------------~~~~----= 0.0

0

1.0 ll ~ 11 \Mixture

o.6 \ fraction l

0.6

0.4

I I I

' I \ \ \

\ \

50 100

X/D

1--=-

150

lsoth. Flame

Unmixedness -------------------

X/D

Figure 5.1: Normalized ve­lodty Uc/ Uc( r = 0) and turbulence intensity A/fic in an isothermal hydrogen jet (solid line) and flame ( dashed line).

Figure 5.2: Mixture frac­tion ie and unmixedness ..;Fc/ ie in an isothermal hy­drogen jet (solid line) and flame (dashed line).

lence intensity as well, see section 1.5.2. This effect cannot be captured with the high Reynolds number k e; model which does not take into account laminarization due to changes in the viscosity. It must be noted that the very weak decreasein the turbulence intensity in Fig. 5.1 is due to the slow coHow of 1 m/s. In the complete absence of coHow the curves would be horizontaL

The trend of increasing values of the unmixedness with axial distance in a Hame (see Fig. 5.2) agrees with experiments (KEN78, KEN81].

Most experimental studies report smaller spreading rates in Harnes than in the cor­responding isothermal jets [WITSO, STR93a]. This is reproduced by the predictions in Fig. 5.7. However, experiments indicate a larger difference in spreading rates (20-25%) [DRA86].

Some aspects of the simulation of Harnes concern intermittency and the difference between Favre and conventional averages. The intermittency has been modelled using the empirica! intermittency function given in chapter 4. No significant effects on tem­pcrature or mass fractions were found. Therefore this effect is neglected in the present study. Differences between Favre and conventionally averaged quantities are also found

85

0.02

0.01

0.00 L=~'---'--_:.:.::.::::ï.:::::::::=~""""'""'"-.;;;.;:.;;;.;:.=.;;:=.;;;;1 0 10 20 30 40 50

X/0

Figure 5.3: Turbulent ki­netic energy ( solid line) and scalar varianee g ( dashed line) in an isotherm al jet and fiame.

to be insignificant in this study. I<or instance, Favre-averaged unmixedness valnes on the symmetry axis are computed to be about 6 % higher than the conventionally averaged values at x/ D = 180, the difference being less at smaller axial distances. In their ex­periment, K&K observed a larger Favre-averaged unmixedness although Favre-averaged values at xj D = 120 already exceed the conventionally averaged unmixedness by a fac­tor of almost 2. However, as indicated by Drake et aL [DRA82] the very large valnes of g > 0.4 reported by K&K, obtained with Mie-scattering are probably due to exper­imental error. Furthermore, the difference between Favre and conventionally averaged mass fractions obtained with Raman spectroscopy, reported by Masri et al. [MAS88], are usually less than 10%.

Finally some remarks on buoyancy effects are in order. In a flame the hot gases will experience an upward force that leads to an increasing momenturn flux in vertical direction. Indeed, predictions showed that veloeities in the axial direction are somewhat higher in a vertical flame than in a horizontal flame. Turbulence intensities, however, are not affected due to the already mentioned proportionality between velocity fluctuations and mean velocity. Predictions of unmixedness and velocity and scalar halfwidths do not show significant changes with or without gravity.

The main results of the predictions are summarized below.

• In a flame all profiles show a slower axial decay due to the expanswn effect (Figs. 5.1, 5.2 and 5.3).

• The centreline turbulence intensity Vkc/tLc in the far field is not altered by the flame (Fig. 5.1), while it is somewhat lower in the near field of the flame.

• The centreline unmixedness v'fk/ Je increases with axial distance rather than at­taining an asymptotic value1 . The result of the equal-scales model is shown in Fig. 5.2.

• The turbulent kinetic energy k and scalar varianeeg are higher in the far field of a flame and the centreline ma.xima are located at larger axial distauces (Fig. 5.3).

1This holds for both scalar transport models.

86

5

Coflow

y

X/0

X/0

5

y /0 3

2

87

Figure 5.4: Iso-mixture fraction contours for an isothermal hydragen jet, predicted with the elliptic method. The jet and coflow entrance are in­dicated.

Figure 5.5: Iso-mixture fraction contours for a hy­drogen flame.

Figure 5.6: lso· temperature contours for a hydragen flame.

Q :i: jl

~ :1:

12 24 Figure 5.7: Velocity and lsoth. mixture fraction halfwidths

Flame in an isothermal hydrogen 9 ..., 18 jet ( solid line) and flame

"' , ...

( dashed line ). /

... <é--- Velocity _"/

8 / / 12

// /

/ ---/ ---/

3 /.

8 /.

0 0 0 50 100 150

X/0

In the near field x/ D < 10, however, the values of k and g in a flame are lower than their isothermal counterparts.

• The spreading rates are not very much affected by the flame, except if cofiow is present. Without coflow the spreading rate in the flame is somewhat lower than in the isothermal jet case (Fig. 5. 7).

• lntermittency is insignificant regarding the predicted mean quantities.

• Favre and conventionally averaged variables are approximately the same.

5.3 Influence of coflow

The influence of coflow is investigated using the experimental data of the 100 % H2

flame of K&B. The horizontal hydrogen jet issues from a nozzle with D 0.00762 m into coflowing air with various coflow to fuel velocity ratios UcJI/fio = 1/10; 1/5 and 1/2. These ratios are now called 'coflow strengths'. The experimental details are given in Table 5.1. According to K&B buoyancy effects are negligible for the present case.

uo (m/s) 151.1 107.1 48.81 Uma:c (m/s) 178.3 126.5 63.3 Ucfl (m/s) 15.1 21.4 24.4: n (-) 8 8 5 UcJduo 1/10 1/5 1/2

Table 5.1: Mean fuel exit veloeities fio and co flow veloeities Ucfl· The mean exit velocity

is calculated with fio = J0D/2 u(r )rdr / J~12 rdr. The maxim']Lm velocity Umax is the on­

axis velocity u(r = 0). The integer n indicates the power in the powerlaw descrihing the initia! velocity profile u(r) fimax(l- 21) D) 11" {BIR60}.

88

The coefficients C~' = 0.06 and a1 ag = 0.7 are used together with the equal-sca.les model for scalar transport. One fiamelet at a = 100 s-1 characterizes the thermo­chemical field. The following nozzle boundary conditions are used. The velocity profile is given in the caption of Table 5.1. The turbulent kinetic energy is k = O.OOlu!,"x and the dissipation is E = Ci'k312 j(0.03D). It is recalled that these boundary conditions have no influence on the flow downstream.

In Fig .. ').8 the centreline temperature predicted with the current value of Cl' 0.06 and with C~' 0.09 are compared with the measurements of K&B. With C~' == 0.09

2500

2000

8 c 1500 .. ~ > g 1000 I-

500

1.0

0.8

~ 0.6 I

::> er I 0.4 ::>

0.2 l.

0.0 0

Cmu•O.O~<::::-:c"'-~····· -t:.- .. :.:.:.::~ . .:.:~ ~---

/ /

/ I

I I

I.

"

--. ,. .· """" . /

/ /

/ / 6

/

50

50

Cmu•0.06 No coflow

Varianee

100 150

X/D

100 150

X/D

/'.

Figure 5.8: Centreline tem­perature of a 100 % H2 flame with Ucf/ = 15 mjs. Dashed line:· C~-' 0.09; solid line: cl-' 0.06; dot­ted line: C ~-' = 0.06 without coflow. Also shown is ..JTI2. Symbols: measurements of K&B.

Figure 5.9: N ormalized ve­locity profiles Uma:r:/ûc with coflow strengths of 1/10 (solid line, L'l); 1/5 (dashed line, +) and 1/2 ( dotted line, o ). Symbols are mea­surements of K&B.

the temperature peak is too close to the nozzle. This is because the model with C~' 0.09 prediets a larger spreading, a faster decay of mixture fraction and consequently a temperature peak closer to the nozzle than predicted with the modified model (cl' = 0.06) and found in the experiments2

. In the far field the temperature deercases faster with cl-'= 0.09 because of stronger mixing, leading toa faster mixture fraction decay.

Also shown in Fig. 5.8 is the centreline temperature predicted with C~-' 0.06 but without coflow. Near the nozzle the temperatures predicted with both C~-' 0.06 sit-

can also be observed in the temperature contour plots of Bockhom [BOC90) who also used the k ·- é model, although with c2 "' 1.85.

89

uations (with and without cofiow) are identical since in that region cofiow is not yet important. For large distances, however, the predictions with cofiow using Cjl- = 0.06 and cjl- 0.09 are almast the same, indicating that the situation without cofiow gives more eflicient mixing. This means that without cofiow the computed mixture fraction decays faster than if cofiow is present. Indeed, predictions showed (not shown) that without coflow the computed decay of the mixture fraction in the flame is not a lin­ear function of x: the mixture fraction decay rate increases with axial distance, i.e., 8//ox is notconstant but increases with x. However, in cofiowing flames (and also in isothermal jetswithor without coflow), of fox is constant, giving 1// K1xf D with K1 a constant. As aresult of the difference in mixture fraction decay between cofiowing (constant Kj) and non-coflowing (non-constant Kj) fiames, the centreline temper at ure at large x is different.

Detailed experimental data on the decay of the mixture fraction in flames with vary­ing coflow have not been found in the literature. However, Drakeet al. [DRA86] exper­imentally found an initia! decay of the Favre-averaged mixture fraction and the excess velocity ( velocity on the axis minus the cofiow velocity) like x-213 and a decay of x-3 for x/ D > 150. It is of interest to note that Glass and Bilger [GLA 78] found the same initial 50 < x/ D < 150 excess velocity decay, but for x/ D > 150 they found a decay of x-312

insteadof x-3, which may be an indication of the experimental uncertainty. It must be

mentioned that these experiments were conducted with coflow. Since predictions only show a decay of 1/ x if cofiow is present, these rneasurements cannot be reproduced.

The variation of the axial velocity of ftc/ümax with Ucjl/fio = 1/10; 1/5 and 1/2 are shown in Fig. 5.9. It is seen that the velocity is slightly underpredicted for coflows of 1/10 and 1/5, while for a cofiow of 1/2 the velocity is somewhat overpredicted. The agreement between predictions and measurements is very satisfactory.

In chapter 3 the asymptotic values of the unmixedness V§) Je have been shown to be fairly independent of the initia! density ratio. Centreline unmixedness values were shown to be not infiuenced by the coflow and remained constant. In the present hydragen into air jets this is also the case, irrespective of the cofiow velocity. In the turbulent fiame, where density differences are sustained throughout the flow, the centreline values of the unmixedness in Fig. 5.10 are seen to increase with axial distance, the increase being more pronounced at higher cofiow strengths. This behaviour must be attributed to the variabie densîty effects in a fiame since they are not present in isothermal jets. Additionally it must be noted that without cofiow (Fig. 5.2) the unmixedness increases faster with x/ D than at the current coflow strengths of 1/10 and 1/5. The reason for this peculiarity may be related to the already mentioned behaviour of the mixture fraction decay coeflicient I<j. The value of ]{1 without cofiow is not a constant, while it is a constant when coflow is present.

The mechanica! turbulence intensity vfkc/fic in a flame deercases with axial dis­tanee (Fig. 5.10). The effect heemnes more important at higher coflow velocities. How­ever, when the turbulence intensity is defined in termsof the 'excess' velocity, namely vfkcf(uc Ucjt), the effect is reversed (not shown in the figure) in the sense that the turbulence intensities resembie the curves for the unmixedness. Since in isothermal jets (with and without coflow) the turbulence intensity, defined with the excess velocity, is constant this effect is also a typical variabie density effect due to the flame. The centre­line temperatures, predicted with one burning flamelet, show a slight shift to the right of the temperature maximum with increasing cofiow strength. The maximum temper-

90

0.6

2500

2000

1500 g 1-

1000

500

0 0 50

.. ·· .··

Unmixedness

X/D

100 150

X/0

I Figure 5.10: Axial vada­tion of the un­mixedness ygj Îc and tur­bulence intensity ...fkc/ûc in flames with coflow strengtbs of 1/10 (solid lines), 1/5 (dashed lines) and 1/2 (dot­ted lines ).

Figure 5.11: Mean tem­perature T and varianee ...JTI2 in flames with coflow strength of 1/10 ( solid line ); 1/5 (dashed line) and 1/2 ( dotted line ).

ature at cofiow strength 1/2 is clearly lower, which is due to the higher values of the unmixedness (Fig. 5.10). This also explains why the predicted temperature varianee is higher with strong cofiow. It should be noted that if more than one fiamelet is used in the cofiow 1/2 case, the maximum mean temperature wiJl decrease. This is because the mean strain rate .5 at this position is smaller than the current laminar flamelet strain rate a 100 s-1 (Fig. 5.20).

Figs. 5.12 and 5.13 show that the halfwidths yr12 and yft2 in the flames with cofiow are much larger than those of the corresponding isothermal jets, i.e., a trend opposite from the case without coflow (Fig. 5. 7). Here, the velocity halfwidth is defined by yr12 r(u = 0.5(iic + Ucfl)). This effect deercases with increasing cofiow strength (Fig. 5.12 and 5.13). lt is unknown how this agrees with experirnental results because no ex perimental data are available. It is recalled frorn Fig. 5. 7 that without coflow the predicted halfwidths of the jet and the flarne are approxirnately equal.

91

!coflow 1/1 o I 4

OL_----~~----~-----=~==~ 0 50 100 150

X/D

X/D

5.4 Predictions of NO

Figure 5.12: Predicted half­widths yr12 and y{12 of a hy­drogen jet ( solid lines) and flame ( dashed Iine) with a coftow strength of 1/10.

5.13: Predicted half­widths y~12 and y{12 of a hy­drogen jet ( solid lines) and ftame ( dashed line) with a coftow strengthof 1/5.

In this section it is shown that NO concentrations are extremely sensitive tothestrain rate, or equivalently flow time scales. The flamelet method is able to take into account these effects, and it is applied to predict NO concentrations in the flames of (K&B) and Bilgerand Beek [BIL74]. However, the formation of NO is generally considered to be a relatively slow process that is rate limited, rather than mixing limited. This may lead to the condusion that the fiamelet method, which is based on the assumption of fast chemistry, is not appropriate for predictions of NO concentrations. Nevertheless, in the litera.ture NO predictions with the SCRS, which is basedon the assumption of infinitely fast rea.ctions, ha.ve been reported [CHE87]. Moreover, flow time scales are important in chemica! reactions that are rate limited, and while these time scales are neglected in the SCRS, they are represented in the ftamelet method by the strain effects. Therefore it is argued tha.t it is useful to investigate the capability of the ftamelet method to predict NO concentrations and to investigate the effect of flow time scales, represented by the strain rate.

The computations reported in this sectionare performed with cl' 0.06, O'J O'g

92

0.7. When the small eddy strain rateis used s = C.,1 ~ with C.,1 = 0.4 while for the large eddy strain ra te s = Cs, 2ë./ k the coefficient is C.,2 = 6. The probability of burning Pb is taken to be equal to 1 throughout the flow field. The latter approximation is exact for x/ D > 10 since for large axial distauces the strain rate has diminished sufficiently. The number of fiamelets constituting the fiamelet library in the present chapter is either one (at a= 100 s-1 ) or nine (at a 100; 200; 400; 800; 2000; 4000; 8000 and 12000 s-1 ).

5.4.1 Effects of strain

In the laminar fiamelet computations the thermal N 0 production is modelled with reactions 25-30 in Table D.3 (appendix D.3), in addition to the other reactions in that table. Since the NO formation is effectively a process with a very large adivation energy, the production is limited to the high temperature regions in the fiame. This implies that the NO concentration is very sensitive to temperature variations and therefore also very sensitive to strain rate variations. The latter effect of strain rate variations on NO concentration is demonstrated in Fig. 5.14 where the maximum NO concentration (in ppm) 3 drops from about 1900 ppm at a low strain rate of a= 10 s-1 (Tmax = 2643 K) toabout 5 · 10-5 ppm at a 14000 s-1 (near quenching, Tmax = 1390 K). In Fig. 5.15 the NO concentration and the NO mass fraction are presented versus the mixture fraction. The NO mass fraction is approximately constant along a large fraction of the mixture fraction valnes at the rich side, while the concentration decreases fast for Z > Z(T Tmax)·

The strong dependenee of X,'Vo on the strain rate leads to several important and interesting results of the prediction of the NO concentration in a turbulent flame. Pre­dictions using only one bnrning fiamelet (at one fixed strain rate) are compared with predictions with nine flamelets. In the case of only one fiamelet the axial varlation of XNo with axial distance x/D can be expected to follow the mean temperature closely: if T increases, XNo will increase and vice versa. The behaviour of the temperature in turn, is completely determined by the axial varlation of the mean mixture fraction4

•

In the case of many flamelets descrihing the burning part, the NO concentration is determined by both the mean mixture fraction, which determines the mean temperature, and by the mean turbulent strain rate s. The latter decays as 1/x2 in an isothermal jet and shows approximately the same behaviour in a flame. Moreover, the strain rate s mainly influences while the temperature is less influenced (Fig. 5.14). In the turbulent flame at axial distauces x/ D smaller than the position where J RJ fat and T >:::! T max both effects (mixture fraction variation and strain rate varlation) increase the NO concentration. In the far field however, where xjD is larger than the position of maximum temperature, both effects compete with each other: the mixture fraction decays below stoichiometrie J < !st and consequently the NO concentration is expected to decrease as well (Fig. 5.15). The decaying strain rate, however, will increase the NO concentration (Fig. 5.14 and 5.17). The computational results of the NO concentra­tion predicted VI'Îth one and nine burning fiamelets are shown in Fig. 5.16. The NO concentration in Fig. 5.16 predicted with one flamelet shows a maximum, and so do

3 ppm, parts per mi!lion, is X No x 106 . 4 Although T is computed with a ,8-func.tion pdf its main features can in deed be discussed by con­

siclering only the mean mixture fraction.

93

g

= E 1-

.. c: 0

~ . . .. E ... c • lii ö .!§. 0 z

1!1100

1!1!00

11100

1800

1300

1000 0

1&-03

88-04

8e·04

48-04,

28-04

0 0.0

1000

100

10

0.1

1 0.01

1 0.001

0.0001

0.00001 8000 11000 11!000 15000

a

2500

2000

1500

1000

500

--'----'--- -'-----===...:11 0 0.2 0.4 0.8 0.8 1.0

z

ê Q. e = E 0 z ~

Figure 5.14: Maximum temperature and maximum NO concentratien (in ppm) versus the strain rate in a laminar 100 % hydro­gen counterflow :flame. Note that the N 0 con centration scaJe is logarithmic.

Figure 5.15: NO con­centration (molar fraction) (solid line), mass fraction ( dashed line) and temper· at ure ( dotted line) versus the mixture fraction in a laminar 100 % hydrogen counterflow :flame at a

500 s-1 •

the experiments. However, in the near field the nine-fiamelet predictions look superior; this is due to the high strain rate in the near field, which suppresses NO formation. In the far field the NO concentration predicted with nine-fiamelets still increases while the measured values decrease. This is due to the already mentioned competition between mixture fraction and strain rate effects. Both quantit,ies decay for larger x, but the NO concentration increases due toa decreasing strain rate and it decreases due toa decreas­ing mixture fraction. This effect seen to be dominated by the strain rate since the NO con centration still increases at x I D > x I D Î=f". The fact that the predicted N 0 profiles with nine hurning fiamelets agree less wel! with experimental data than those with only one fiamelet does not mean that the one-fiamelet model is superior. It does mean that the mean temperature should he predicted more accurately for large x/ D, i.e., that the mixture fraction should decay faster. This can turn around the competitive effect of decreasing strain rate and decreasing mixture fraction such that the N 0 concentrat ion decays, instead of increases, at large x/ D. In deed, it has already been discussed that there is experimental evidence that at large x I D the mean mixture fraction decays faster than the computed mixture fraction that scales with "" 1/ x if cofiow is present. lt must also he stressed, however, that the experimental results, obtained with a sample probe, are regarcled to he only indicative due to large experimental errors, as stated by Chen

94

2500

2000

1500 g 1-

1000

500

200

150

100

50

I 0 z

0 ~~----~··~····~····~~~~····~··-~·--~·--···~·····~Jo

0 50 100 150 200

X/D

600

500

ê 400 Q, e --)(

300 .. e 0 z >< 200

100 -------ó:o1--

-------------0 0.000 0.005 0.010 0.015 0.020

1/a

[CHE87].

Figure 5.16: Centreline N 0 concentra.tion a.nd tem­pera.ture. Predictions with one burning fia.melet a = 100 s-1 (da.shed line) a.nd nine burning Hamelets ( solid line). Experiments of K&B [KEN73] ( /::,) a.nd Bilger a.nd Beek [BIL74] (+).

Figure 5.17: NO con-centration at fixed mixture fra.ction valnes Z 0.01 ( da.shed line ), Z = 0.02 (sma.ll da.shes), Z = 0.04 ( dotted line) and Z = Zst 0.028 (solid line) versus the inverse strain rate 1/a in a la.minar counterflow hydro­gen fia.me.

Another explanation for the increa.sing NO concentration is that the strain rate that the Hamelets experience at the downstream positions under consideration, can be much higher than the loca.l strain rate. If the history of a Hamelet is taken into account then the Hamelets experience a strain rate that is associated with much smaller axial distances, and consequently is much higher tha.n the local strain rate.

It can be concluded that the prediction of the NO concentration depends strongly on the strain rate s, which makes modelling of this parameter important. Furthermore, NO predictions where only one burning flamelet is considered, a.nd where consequently no account is taken of flow time scales, must be regarcled with ca.ution, although they decep­tively give better agreement with experirnents. Model predictions with many Hamelets show a large discrepancy with the experiments but it wil! be shown in the next sections that sealing properties of the NO concentration with initia! conditions can only be pre­dicted if strain rate variations are accounted for, i.e., many Hamelets are considered. In addition it is mentioned that the phenornenon of a more elaborate model giving less satisfactory agreement with experimental data than a simpte model is also sometimes encountered in turbulence modelling. Also in those cases, it may be just a coincidence

95

of the simpler models giving good agreement with experiments.

5.4.2 The choice of the small eddy strain rate

The strain rate a has to be identified with either the large eddy or the small eddy turbulent strain rate. In this section this ehoiee will be basedon experimentally observed sealing properties of NO eoncentration with the jet Reynolds number.

The axial sealing properties of the strain rates are 5

(5.13)

for the small eddy strain rate and

(5.14)

for the large eddy strain rate. Experimentally the maximum NO eoneentration oeeurs at maximum temperature

eonditions. Maximum temperaJure eonditions in turn, for a given fuel, oeeur at a fixed value of x/ D approximately. Therefore, using the strain rate sealing properties of Eq. (5.13) and Eq. (5.14) and the fact that maximum NO eoncentrations scale with 1/a (espeeially fora < 1000 s-1 ), the maximum NO concentration seales with ïï0 , D and v as 6

ïïo -3/2 (Dv)1/2

Dïïo -1 Re-1/2

for the small eddy strain rate (with Re ïï0 Dfv) and

XNO,max Dïïo-1

D2 v-1 Re-1

(5.15)

( 5.16)

for the large eddy strain rate. These two sealing properties are completely different as far astheir Re dependenee is concerned. Bilgerand Beek [BIL74) find a relationship between the measured XNo,max and the Reynolds number which is described by Eq. (5.15) rather than by Eq. (5.16), thereby giving confidenee to the choice of the small eddy strain rate for flamelet computations. This result implies that the Reynolds number should be kept as large as possible while mini mi zing the overall residence time D /ïïo.

In addition it is noted that these sealing properties cannot be reproduced by the SCRS sirree this model does not take strain rate variations into account.

5.4.3 Infl.uence of cofl.ow on NO predictions

The experimentsof K&B indicate that the maximum NO concentration increa.ses with increasing coflow strength. Predictions with nine flarnelets using the small eddy strain rate indeed show a similar trend (Fig. 5.18) while, in contrast, predictions with the large eddy strain rate show the opposite trend (Fig. 5.19). The very high values of NO

5 Based on sealing properties of free jets into still air. 6 Note that these sealing relations have a dimeosion of s. In order to non-dimensionalize them they

could be divided by a reference strain rate.

96

200 Figure 5.18: Centreline

~mali ~ëddyl NO concent ration (ppm) rain rate 1/2 for various co flow using

150 ----~---------

the small eddy stra.in rate

Ê 0 (lines ). Experiments (sym-

Q. 0 bols) of K&B [KEN73]: 6 e 100 ~ + + 0 1/2 +

0 + = coftow 1/10; + co flow z ö ö +1/5 1/5; o = coftow 1/2.

50 ö ö ö

1/10

50 100 150 200

X/D

200~ . Figure 5.19: Centreline

[lirgeeddy strain rat9j NO con centration (ppm) for various coflows using

150. the large eddy stra.in ra te

Ê 0 0 1/10 (lines ). Experiments (sym-

Q. 0 bols) of K&B [KEN73]: 6 e 100 0

+ g.. .......... 0 ! -<~~ . = coftow 1/10; + coftow z ... --~.~-; ... ······· + 1/5; o = coftow 1/2 .

"' ;-:.-::····" ö 1/5 .......-"'<i/2 b. b.

-"'".,.. 0

0 50 100 150 200

X/D

concentration predicted with the srnall eddy strain rate at coflow 1/2 are unsatisfactory. They are caused by the srnall values of the smal! eddy strain rate (Fig. 5.20). Sirree there are no measurernents available on strain rates in this flarne it is hard to conclude whether this is a (turbulence) model deficîency.

The large differences between rneasured and predicted N 0 concentratîons are not surprising for several reasons. Firstly, the NO concentration has been shown to be extremely sensitive to the strain rate for which only rather crude roodels exist. Secondly, the mixture fraction in the far field is overpredicted, and thirdly there are experimental uncertainties as indicated by Bilger and Beek [BIL74], see also Fig. 5.16. This means that the focus should be on trends that can be observed in experiments and predictions, rather than quantitative results.

It can he noted in Figs. 5.18 and 5.19 that forsmaller xj D the NO predictions with the smal! eddy strain rate are satisfactory, while the predictions with the one-flarnelet model (Fig. 5.16) overpredict the NO concentration in that region. The reason for this difference is that for smaller x j D the rather large val u es of s prevent a fast increase of the N 0 concentration. This effect cannot be simulated with only one :!lamel et with a fixed strain rate. In the far field the situation is reversed. The predictions with one flarnelet show a decrease because the mixture fraction decreases with x/ D. With nine flamelets,

97

1000

800

~ 800

~ c:

J 400

200

0 50 100

1000

800

:ë: c 800

~ c: 'iii 400 IA

200

0 50 100

I Small ~ddy strain rate I

-----150

XJD

eddy straln rate

150

XJD

200

Figure 5.20: Centreline small eddy strain rate 8 0.4Jë72il in the :flames with co:flow strengths of 1/10 (u0 = 151 m/s); 1/5 (ïïo = 107 m/s) and 1/2 (u0 = 48 m/s).

Figure 5.21: Centreline large eddy strain rate 8 = 6e/k in the flames with co­flow strengths of 1/10 (uo = 151 m/s); 1/5 (u0 = 107 m/s) and 1/2 (u0 = 48 m/s).

however, the strain rate deercases as wellleading to the increasing NO concentrations with xfD.

The experimental evidence that the fiames with the stronger cofiow have the high­est NO concentrations can be explained by noting that the fiames with higher cofiow strength had lower fuel exit veloeities (Table 5.1 ). As aresult of the u0 dependenee in the sealing laws (with fixed D), Eq. (5.15) and Eq. (.'\.16) suggest that NO coneentrations at lower exit velocity (strongest coflow) are higher. Therefore the higher NO eoneentration at higher cofiow is aresult of the lower exit velocity and not a eonsequence of the eofiow itself. The experiments and the predictions with the small eddy strain rate agree quali­tatively with this explanation. The predictions with the large eddy strain rate, however, do notchange (Fig. 5.19) although the sealing behaviour Eq. (5.16) suggests an inerease with lower exit veloei ties. Apparently the coflow has sueh an influenee on the large eddy strain rate that the expected trend of deereasing large eddy strain rate with decreasing exit velocity is not reprodueed. This is illustrated in Fig. 5.20 and Fig. 5.21 where the smali eddy strain rate and the large eddy strain rate respeetively are plotted for the three coflow strengths.

While the small eddy strain rate deercases with inereasing cofiow ( decreasing exit

98

velocity), by contrast the large eddy strain rate increases with increasing cofiow leading toa wrong trend in the predicted NO concentration. Consequently this presents another argument in favour of using the small eddy stra.in ra.te in fia.melet computa.tions.

5.5 Scalar transport

The non-equa.l scales scalar transport model is applied to the flame of K&B. This ap­plication is based on the modified turbulence model with CJ.t. 0.06, also used in the previous section.

Befare the results are discussed some details of the non-equal scales model are reca­pitulated.

5.5.1 The time scale ratio

The key parameters of the model are the eddy-diffusivity lts and the scalar dissipation rate êg· These variables are computed according to

and (5.17)

where e;e is the non-equal scales scalar dissipation rate eg = </>g>-'e>-2 and e;q the equal­scales scalar dissipation ra te e~ Rege / k. The mecha.nical to scalar time scale ratio Rr is defined as

Rr = kje gje~e

(5.18)

and usually Re is equal to 2 (standard equal-scales model). It is noted that Rr = Ree~• je~.

From these expressions it can be derived that the effective turbulent Schmidt number, for regions with R-r <Re, is O"t ~ R~CJ.t.fCt. Consequently in these regions the value of the mechanica! turbulence coefficient CJ.t. is important. In chapter 3 the value of CJ.I. was not found to he important in the application of the non-equal scales model, but this was because in the most important regions of the flow Rr > Re. This will turn out not to be the case in a flame.

In the previous section CJ.t. = 0.06 was chosen and this value is also used in the present section. To keep the effective turbulent Schmidt number O"t = ptf lts ~ 0. 7 the value of Cf must deercase to a bout 0.33.

These values of C~" and C1 are different from those used in our paper presented at the "Joint Meeting of the British and German Sections of The Cambustion Institute" in Cambridge 1993 [SAN93a] where C~" 0.09 and C1 0.5 were used, also giving fft ~ 0. 72. The results are not significantly different.

Finally the coefficient </> is chosen by noting that in the far field of the turbulent flame the mixture fraction scales as 1/ fe I<,xj D, according to equal-scales predictions. The value of the slope K f is demanded to be the same as predicted with the equal-scales model, i.e. 0.29. This value is reproduced with </> ~ 1350.

It is noteworthy that Janicka in [JAN79] expressed the time scale ratio R-r according

(5.19)

99

with 7 1 RJ 0.4 and 72 RJ 4. He used the experiment of Warhaft and Lumley [WAR78] to deduce the logarithmic dependenee of the scalar varianee and used the experiment of Kennedy and Kent [KEN78] to determine the coefficients 71 and 72 . He used the standard k- e model with the eddy-diffusivity 1-l• modelled by the equal-scales expression J.l 8 J.td O't with O't = 0. 7, but with the scalar dissipation rate given by

ê êg R,.gk. (5.20)

A peculiarity of this method is that Rr becOines negative for larger axial distances than shown in Fig. 5.22.

I\ IJ..----1 \

2 I \ I \ I ', I ' I ' I '',, I "-,

1 _

11 ,------ ________ -'.::-::-.. ........ -___ ~chmidt --1 --1 R -Janick'à-------

T

0~---------L----------~--------~----~

0 50 100 150

X/D

Figure 5.22: Centreline val­ues of the time scale ratio 14 2E;• / f:~q ( solid line) and R7 according .Janicka's model [.JAN79] ( dashed line) in the flame of K&B. Also shownis the effective turbu­lent Schmidt number (dot­ted line).

The values of .&, predicted by the present method in Fig. 5.22 show a decrease with axial distance, which is in qualitative agreement with Janicka's model. The decrease of R,. with axial distance is attributed both to the variabie density effects of the flame and to the preserree of coflow. The decrease of R,. with axial distance in a flame is most pronounced if ·coflow is present. In the absence of coflow R7 decreases as well, but not that fast. The latter is also the case in isothermal jets.

In Fig. 5.22 the full non-equal scales model is used in the region 9 < xj D < 73, since R7 > 2. For x/ D < 9 and x/ D > 73 only e9 is evaluated as e~· while 1-ls is evaluated according to Eq. (5.17) with e9 = Rcgt:/k. Furthermore, in the region where the full non-equal scales model is active, i.e., between 9 and 73 diameters, the Schmidt number is about 1. This agrees with the rneasurernents of Glass and Bilger [GLA78] who measured turbulent Schrnidt nurnbers of 1 for x/ D < 80 and values of 0. 7 for x/ D > 160.

Predicted radial profiles of R,. at x/ D = 40 and x/ D = 80 are shown in Fig. 5.23 along with the constant equal-scales value Rr = 2 and the predictions of Dibble et al. [DIB86] using a more elaborate second-order turbulence model applied to a diluted H2 / Ar flame. Although the fiarnes in the present study and that of Dibble et al. are different, sorne qualitative sirnilarities are observed. The profiles of Rr at smaller x/ D show an off-axis maximum and they flatten with increasing axial distance. The model of Janicka, (not shown in Fig. 5.23) shows an off-axis local maximurn of R7 at every axial distance since g has always alocal maximurn in radial direction. The quantitative values predicted by the present method and the model of Dibble et aL are different, but unfortunately there are no measurernents of Rr available in flarnes.

100

1 .. E i=

X/0=40

----....... , ' ' + + + \

".+. +~ "' \

\+ X/0•80 '

----___". \ +

Equal-scales

+ \

' ..... 0 L---~~-~-~J-~- --~-,..__ _ __:""_....__~· __._ ___ __J

0.00 0.05 0.10 0.15 0.20 0.25

RJX

5.5.2 Comparison with experiments

Figure 5.23: Radial profiles of the time scale ratio at x/D 40 (solid line) and x/ D = 80 ( dashed line) pre­dicted with the non-equal scales model. Dotted line: Equal-scales model, second order predictions of Dibble et al. [DIB86] at x/ D = 30: 6. and at x/D = 50: +.

Measured normalized radial mixture fraction profiles obey a Gaussian fittoa very good approximation [KEN73], and this is reproduced by both the equaJ and the non-equal scales model (Fig. 5.24). However, the non-equal scales model at x/ D 40 is somewhat closer to the experiment than at xl D 80. The reason for the difference is that at x I D 40 Rr > 2 at the centreline. The equal-scales model predictions are the same at both axial stations. The mixture fraction halfwidths (Fig. 5.25) show significant ex­perimental scatter. Very large differences between rneasured Favre and conventionally averaged halfwidths are reported with Mie-scattering (K&K) and the probe rneasure­ments (indeterminate regarding the type of averaging) of K&B are in between. The large differences between measured Favre and conventionally averaged scalars are prob­ably overestimated [DRA82].

The non-equal scales predictions agree better with the ( conventionally averaged) experiments. Here, it is mentioned again that the numerical difference between Favre and conventionally averaged variables is very small.

Fig. 5.26 shows that the predicted centreline temperatures agree reasonably well with experimental data, except at large axial distances x I D > 150. The rnain reason for this deviation is that the predicted mean mixture fraction is too large in that region. It should decrease faster, and thus more efficient turbulent mixing is required. Principally this could be accomplished with the non-equal scales model if the time scale criterion was not used there. A low value of Rr (see Fig. 5.22) implies a srnall value of the effective Schrnidt nurnber since Se1 = p,tf p,., ( C"IC f )R~. A srnall Set in turn implies intense scalar mixing. Although this seerns an ad-hoc solution, there is experimental evidence that the effective turbulent Schrnidt nurnber decreases with axial distance in turbulent hydrogen fiarnes: Glass and Bilger [GLA 78] measured it to be about 1 for xl D < 80 and a bout 0. 7 for x I D > 160. This experimental finding of lower Schrnidt nurnbers farther downstrearn agrees with the already rnentioned experimental evidence of faster mixture fraction decay rates at large axial distauces (xiD > 150), namely j""' x-3 according to Drake et al. [DRA86]. It has already been mentioned that the computed mixture fraction decays as j ""' x- 1

• The predicted centreline values of the unrnixedness $cl j

lOl

1.0

0.8

0.8 ,:, ;:,

0.4

0.2

o.o 0.0 0.6 1.0 1.6

R/R_1/2

7

Non-equal[ 8 Equal 1-6

- 4 • ~ ~, ... 3

2

0 0 60 100

X/D

b.

X/0•40

X/0•80 Gauss

"""tt> 2.0

160

2.6

Figure 5.24: Radial mix­ture fraction profiles f! Je ver-sus R/y{12 predicted with the equal-scales and with the non-equal scales model (solid lines) at x/D = 40 and 80. Equal-scales predie­tions at x/ D 40 ( da.shed line) annd x/ D = 80 ( dotted line). Symbols are a Gaus­sian fit through experimen­tal data of K&B.

Figure 5.25: Halfwidths of the mixture fraction pre-dicted with the equal-scales ( da.shed line) and with the non-equal se ales model ( solid line ). Symbols are probe mea.surements of K&B (L':.) and Mie-scattering mea.sure-ments of K&K ( o, + ).

show an increase with x/ D, particularly the non-equal sc.ales predictions (Fig. 5.27). The Mie-scattering mea.surements of K&K show a very sharp increase, especially the Favre-averaged values. It is recalled that according to Drakeet al. [DRA82] these Mie­scattering measurements of K&K for scalar variance, and also the very large difference between Favre-averaged and conventionally averaged values, are erroneous. It eau also be noted that the unmixedness is not 0 at the nozzle, although theoretically it should be 0. The probe rneasurements in Fig. 5.27 reported by Drake et al. are in excellent agreement with the predictions in the far field.

Radial profiles of scalar varianee predicted with both models agree reasonably with the conventionally measured data of K&K while the Favre-averaged results are quite different (Fig. 5.28).

Due to the normalization of the varianee absolute differences between the models are obsc~red, although in Fig. 5.29 the values predicted with the non-equal sc.ales model are seen to be rnuch higher (at x/ D = 120). This is because the scalar dissipation rate ê~e is smaller than its equal-scales counterpart (Rr < Re)·

102

g 1-

2000

1500

1.0 ···~·

0.8

• • 0.8 • c ... Jj E

0.4 c :::1

0.2

' ' ,

_;:;

Equal --.. ,•' .. "'"" D.

50

____ !:. ____ El

Non-equal

100 150

)(JO

~----·--··-·-----··--·--··

0

Favre

0 /Conv.

0 0 +

+ Non-equal

+ ______ tl ___

Equal

150

XJD

Figure 5.26: Ax:ial vada­tion of tbe temperature pre­dicted witb tbe equal-scales ( dasbed line) and non -equal se ales ( solid line) model. Symbols are experimental data of K&B.

Figure 5.27: Centreline variation of unmixedness predicted witb tbe equal­scales ( dasbed line) and non-equal se ales ( solid line) model. Mie-scattering mea­surements of K&K ( o and +) and Probe measurements of Drakeet al. [DRA82J (6.).

In Figs. 5.30 and 5.31 radial temperature profiles predicted with the equal-scales and with the non-equal scales model are seen to agree best with the experiment at x/ D 40. The non-equal scales values agree slightly better with measurements in the sense that the profile is narrower. The deviations at larger axial distance can he explained by the overpredicted mixture fraction.

Axial values of the molar fractions of the stabie species H 2 , 0 2 , H20 and N2 show reasonable agreement with the experiments, see Fig. 5.32 and Fig. 5.33. In the near field the non-equal scales model again performs somewhat better, while both models at xfD > 150 predict concentrations of H2 that are too high and Oz concentrations that are too low, again explaining the overpredicted temperature in that region.

The radial concentration profiles, given in Fig. 5.34 and 5.35 show a better agreement of the non-equal scales predictions with the experimental data, specifically for H 2 and N2.

103

2.0 .-----------------------~····--------,

1.5

1.0

0.5

Favre ",.o

0

+ + 0

Equal ........... ,/ 0

/ .... , Non-eq + ',~

I ~ Conv. --;W

0.0 '----------'-------···------L-----="""'---' 0.00

l!e-04

,. 1e-04

5e-05

0.05

"".---.... .-" ' - ' ' \

\

/\ \

Equal \ \ \ \

0.10 0.15

RJX

' o~' ______ L_ ___ ,~~----~-----~

0.00 0.05 0.10 0.15

R/X

5.6 Differential ditfusion effects

Figure 5.28: Normalized radial scalar varianee pro­files vfijfi; versus R/ D at x I D 55 predicted with the equal-scales ( dashed line) and non-equal scales model (solid line). Sym­bols are experimenta.l data of K&K.

Figure 5.29: Radial scalar varianee across the jet at x I D 120 pre­dicted with the equa.l-sca.les ( dashed line) and non-equa.l scales model ( solid line ).

The :llarnelet metbod is basedon the mixture fraction concept, which can only be defined if equal diffusivities of species are assumed. Since laminar fiamelet data are generated us­ing full elementary transport properties in which differential diffusion can be important, the application of such data may be problematic. Small hydragen atoms and molecules have molecular diffusion coefticients that are much larger than those of larger species, and therefore differential diffusion may be important in laminar hydrogen fiames. In turbulent fiames, however, differential diffusion effects may be insignificant if the large scale mixing processes proceed fast enough. In that case the application of laminar flamelet data is problematic.

In chapter 4 temperature versus mixture fraction plots have been presented (Fig. 4.1 and 4.4) for pure as wellas diluted hydragen larninar fiamelets. At low strain rates the diluted hydrogen flamelets (induding detailed transport) were found to have maximum temperatures much higher than the maximum fiame temperature that can be calculated with equal diffusivities (using the SCRS, for instance). However, experiments on tur­bulent diffusion fiames showed that differential diffusion effects become less important

104

40

g ....

RJD

40

2500 .-----------------------------~

[tion-equal I 80

2000 120

1500 180

g .... 40

1000

• 80

500 + 120

oL-----•~~---~····~~ .. ~~--~----~ 0 2 4 8 8 10 • 180

RJD

Figure 5.30: Radial tem­perature profiles at x I D 40; 80; 120 and 160. Predie­tions with the equal-scales model (tin es) and experi~

ments of K&B ( symbols ).

Figure 5.31: Radial tem­perature profiles at x I D = 40; 80; 120 and 160. Pre­dictions with the non-equal scales model (lines) and ex­periments of K&B (sym­bols).

at increasing Reynolds numbers. Drakeet aL [DRA86] experimentally found that the effect is negligible at Re~ 8500. Therefore it may be concluded that laminar flamelet data (of the counterflow configuration) are not applicable to diluted hydrogen ftames if computed mean temperatures are much higher than measured mean temperatures. Whether this is the case is investigated in the following.

With respect to chemica! composition boundary conditions the diluted free jet tur­bulent H 2-fiame of Takagi et al. [TAK81] is analogous to the diluted laminar counterflow ditfusion ftame discussed insection 4.3.2 and thus it is a convenient test case. In that sec­tion the laminar fiamelet computations at small rates of strain were found to have higher maximum temperatures than the maximum equilibrium temperature obtained with the SCRS. For instanee at a 10 s-1

; Tmax 2229 Kandat a= 30 s- 1; Tmax = 2093 K

while the SCRS gives Tmax 1889 !{. In Fig. 4.4 it could be observed that especially at fuel-lean conditions Z < Zst fiamelet temperatures are higher than the equilibrium temperature. Only at very high strain rates of a ~ 6000 s-1 (near extinction) fiamelet temperatures are lower than equilibrium temperatures. Large values of turbulent strain

105

-, )(

-, )(

1.0

0.8

0.8

0.4

I Equal ___ t ... J ~ ... --- + +

---- + + ,-~,-. ++

0.2

0.0 0 50 100 150

X/D

1.0

0.8

0.8

0.4

0.2

0.0 .._.,. _ _. • ...____..,._ _ _,__ •• __ .~--.=

0 50 150

X/D

Figure 5.32: Axial pro­files of molar fra.ctions pre­dicted with the equal-scales model. Experimental data (symbols) are of K&B.

Figure 5.33: Axial profiles of molar fractions predided with the non-equal scales model. Experimental data (symbols) are of K&B.

rates only occur in the region close to the nozzle where the mixture is mainly fuel-rich {near the axis ). This will result in only small differences between flamelet-based and SCRS-based mean temperatures. However, in the far fieldthestrain rateis relatively low while the mixture is fuel-lean (at all radial distances). This willlead to flamelet-based mean temperatures that are much higher than the SCRS-based mean temperatures.

Predietiens for the flame of Takagi et ai. have been performed with both the SCRS and the flamelet method. The latter consists of eight burning flamelets with a 30; 100;

200; 400; 1000; 2000; 4000 and 6000 s-1 using the smal! eddy strain rate s Cs,! Jt:/(2v) with C,,1 = 0.4, which is the same value used in the previous sections. The probability of burning Pb has been set equal to one. The k- é model with C" = 0.06 and the equal­scales scalar transport model with t:TJ = t:T9 = 0.7 are used. The experiment of Takagi et al. consists of a coflowing (ucfl = 5.1 m/s) vertical hydrogen flame at Re ~ 11000. The mean fuel exit velocity is u0 ~ 55 m/s with u,.,,ax ~ 70 mjs. The flame length is relatively short because of the dilution of the hydrogen with nitrogen. It was shown in chapter 4 that this increases Z.t from 0.028 for pure hydrogen to 0.384 for the diluted fuel. Therefore the axial position where f ~ Z,1 will he reached at smaller x/D.

106

~

~

1.0

IEquall

0.8

0.8 À

0.4

0.2

0.0 0 2 3 4 5 a 7

R/D

1.0 -~--··-----

0.8

0.8 H2

0.4

0.2

0.0 0 2 a 4 s a 1

R/D

Figure 5.34: Radlal pro­files of molar fractions pre­dicted with the equal-scales model. Experimental data (symbols) areofK&B.

Figure 5.35: Radial pro­files of molar fractions pre­dicted with the non-equal scales model. Experimental data (symbols) are of K&B.

The velocity at the nozzle (D = 0.0049 m) is modelled numerically with a powerlaw (n = 6), see caption of Table 5.1. The nozzle boundary conditions for k and e; are k 0.00lu2 and e; C~"k312 /(0.03D). In Fig. 5.36 the mean centreline temperatures predicted with the SCRS and with the flamelet method in deed show the expected trends. In the near field the SCRS-based temperature is higher than the experimental data because non-equilibrium effects are negiected and because the mean mixture fraction (Fig. 5.37) is underpredicted. The flamelet-based temperature agrees better with the experiment in that region. Here it must be mentioned that the experimental mixture fraction has been calculated from measured stabie species concentrations of H2 , 0 2 , H20 and N 2 • The molar mass is then computed and used to obtain the mass fraction of N2 from which the mixturefraction is calcuiated by YN, = YN,(air)+ ](YN,(fuel)- YN2 (air)).

In the far field the flamelet-based temperatures are too high but the SCRS-based temperatures agree very well with the experiment. This can also be observed in Fig. 5.38 where the radial profile of the SCRS-based temperature at x/ D 33 is in excellent agreement with the experiment, while the flamelet-based temperatures are too high. The radial profiles of the concentrations of the stabie components 0 2 , H20 and N 2 in

107

g ....

~o.-----------------~-------------------,

SCRS

1600

1000

500

20 40

XID

1.0 .....,,....,_ ________ _

.;,

0.8

0.8

0.4

0.2

\ .,. \ \ + \ \

' + Flame \ \~

\

', + ' ', +

~~· .......... +--_

80 80

--:po---.±___ ' -+---;

0.0 '----------'-------'--------'-----___;

0 20 40 80 80

XID

Figure 5.36: Centreline temperature predicted with 8 burning Hamelets (solid line) and with the SCRS (dashed line). Also shown

temperature varianee predicted with the

SCRS. Symbols are experi~ mental data of Takagi et al.

Figure 5.37: Centreline mixture fraction in the isothermal jet ( solid line) and fiame (dashed line). Symbols are experimental data of Thkagi et al.

Fig. 5.39 predicted with the SCRS agree wel! with the experiments. The predicted flamelet-based hydragen concentration is much higher than the SCRS­

based valnes because of the preferential diffusion in the hydragen fiamelets. Especially at fuel-lean conditions (large r / D) the flamelet-based hydragen concentrations are over­predicted. Since the mixture fraction in the far field is very well predicted (Fig. 5.37), this is a dear confirmation of the already anticipated result that preferential diffusion effects in turbulent diffusion flames (at sufficiently high Reynolds numbers) are negligi­ble. lt also indicates a limited applicability of f!amelet data that are generated with the full elementary set of transport coefficients, in which automatically differential diffusion can be important.

For completeness, predictions of the centreline variation of velocity and turbulent ki~ netic energy for the f!ame and for the corresponding isothermal jet are shown in Figs. 5.40 and 5.41, respectively. Whether the density is basedon the SCRS or the flamelet method is not very significant for these results.

Although the isothermal jet is predicted very accurately, see also the mixture fraction in Fig. 5.37, the results for the flame do not agree very wel! with the experimeuts. The predictions of both velocity and mixture fraction in the flame in the region between

108

2000

1500

g 1000

1-

500

Varianee 0

0 2

AID

...... __

3 4

AID

i X/0•33/

5 8

Figure 5.38: Radial pro­files of the tempera.ture a.t x I D 33 predicted with 8 burning fia.melets ( solid line) a.nd with the SCRS ( da.shed line ). The va.ria.nce is also predicted with the SCRS. Symbols a.re experi­mental data. of Ta.ka.gi et al.

Figure 5.39: Mola.r fra.c­tions a.cross the jet a.t x I D = 33 predicted with 8 burn­ing fia.rnelets ( solid line) a.nd with the SCRS ( da.shed line). Symbols are experi­mental data. of Ta.ka.gi et al.

x I D 10 a.nd 35 show a fa.ster decay than the measurements. This is most probably the result of relaminarization in the experimental situation a.s is also suggested by the very pronounced differences in the mea.sured centreline variation of the turbulent kinetic energy in Fig. 5.41. The experimental data in the flame are qualitatively different from both the computations and the experimental data in the isothermal jet. This type of low-Reynolds number effect cannot be simulated with the present model.

5.7 Conclusions

Predictions of mechanica! turbulence in turbulent hydrogen flarnes using the k -E model (C~" = 0.06 instead of0.09) and the flamelet metbod showed areasonably good agreement with experimental data. The most important features of the flarnes such as spreading rates, velocity and turbulence intensities are predicted satisfactorily. The experimental values of the Favre averaged mixture fraction fluctuations are much too large compared with conventionally averaged values and predictions. Moreover, the predictions do not show a significant difference between Favre-averaged and conventionally averaged vari­ables. A number of interesting computational results on the influence of coflow on

109

70 Figure 5.40: Variation of

" centreline velocity in the

eo \+ isothermal jet (solid line) \

50 \ +

\ and fiame (dashed line). \ + Flame Symbols are experiment al \

i: 40 \ /. data of Takagi et al. ' .5. ' ' :::1 30 ' ' + ' ' ' 20

.... ..,.. -... ---+--10 '

0 0 20 40 eo 80

X/D

100 Figure 5.41: Centreline tur-bulent kinetic energy in the

80 ,-, isothermal jet (solid line) I \ and fiame (dashed line ). I \

I \ Symbols are data of Takagi \ ": 80 \

\ Flame et al. .. \ :s ' ',/ ... 40 + + +',+

' ' 6 6

........ .... 20 -6 :-------6 6

ó 0

0 20 40 80 80

X/0

spreading rates, turbulence intensities and strain rates of small and large eddies, have been found. Experimental data on these are scarce, however.

The predicted mechanica! to scalar time scale ratio, and therefore also the turbulent Schmidt number, predicted with the non-equal scales scalar transport modelagree with trends observed in experiments and second-order turbulence models.

Predictions of thermal N 0 concentrations with the flamelet method have been shown to be extremely strain rate dependent, casting doubt on models that do not account for variations inthestrain rate. However, comparisons with experimental data showed that the present results are not satisfa.ctory which may be due to an inaccurate prediction of the mixture fraction and of the strain rate in the far field.

Experimental observations of the sealing properties of NO with Reynolds number and global residence time, reported in the literature, can be simulated with the flamelet concept only if the smal! eddy strain rate is employed as stretch descrihing parameter. In the derivation of the NO sealing properties, the inversely linear dependenee of the maximum NO concentration on the strain rate, that has been observed in the present chapter, must be used. This gives confidence that the flamelet method is useful in research on NO formation in turbulent flames. However, due to the strong dependenee

110

of NO formation on the strain rate and on the mixture fraction, the accurate prediction of these variables in a turbulent flame is important.

Finally, preferential diffusion effects have been shown to he important in flamelet computations of diluted laminar hydrogen counterflow flames (chapter 4), but to he less important in turbulent diluted hydrogen flames (present chapter). The flamelet method leads to an overprediction of the mean temper at ure in the fuel-lean zone of the turbulent flame. The SCRS model, in which equal diffusivities are explicitly assumed, does give satisfactory agreement with measurernents in this area.

111

112

Chapter 6

Stabilized Natura} Gas Flames

In turbulent natura! gas fiames local flame extinction due to high rates of strain is important even at fairly low fuel exit velocities. This local extinction can lead to lift­off of the flarne. To maintain a stabie flarne which is attached to the nozzle rim, in experirnental studies a stabilization gas (usually oxygen or hydrogen), is often added through a ring-shaped opening around the nozzle ( annulus ).

In the literature, numerical studies on gas-stabilized flames always neglect the pres­enee of the stahilizing gas, as fa.r as the author knows, and this is also clone in the present study.

The focus in the present chapter is on the effect of ra.diation in hydroca.rbon, i.e. natura! gas, flarnes.

The two most important sourees of radiation in natura! gas fla.rnes are the thermal radiation of H 2 0, C02 and soot particles. The latter, if soot is present, is by fa.r the most important. In a turbulent natura! gas flarne differences in maximum mean temperature between adiabatic and non-a.diabatic predictions (without and with the effect of soot radiation, respectively) can be of the order of 400 K (Lockwood and Stolakis [LOC83]) at maximurn temperature on the syrnmetry-axis.

Section 6.2 gives background information on radiation in flames and indicates the way in which the effect of ra.diation is accounted for in the present study. In section 6.3 results of cornputations are compared with rneasurements on the natura! gas (mainly methane) flame of Streb [STR93a]. Finally, in section 6.4 a summary of conclusions is presented.

6.1 Stabilization

Two important ways of flame stabilization are: the addition of oxygen through an annu­lus (WIT80]; and the addition of both oxygen through an inner annulus and hydrogen through an outer annulus [STR93a]. In the first metbod the amount of oxygen that has to be added is so large that the flame is influenced by the oxygen strea.rn. In the second method the influence of the stahilizing streams is srnall [STR93a].

With the flarnelet concept the stahilizing rnechanism can be explained by consiclering the queuehing value of the strain rate of the flamelets within the turbulent flarne. In the case where only oxygen is added, the fuel issues into an environment of pure oxygen. The queuehing strain rate, aq, increases when the oxygen rnass fraction at the oxidizer boundary is increased. This makes the flarnelets more resistant to strain and therefore

113

explains the stahilizing effect. In the case where hydrogen is added through an outer annulus and oxygen is added

through the inner annulus, the stabilization occurs due to two effects. The natural gas issues into an oxygen stream, which stahilizes the flame, but in addition the oxygen from the inner annulus tagether with the hydrogen farms a very hot hydragen flame surrounding the natural gas. Therefore, effectively the natural gas jet issues into an oxygen-rich hot mixture. Since increasing the temperature of the oxidizer makes laminar Hamelets more resistant to strain, there are now two stahilizing effects: an oxygen-rich environment and a high oxidizer temperature. Due to the second effect less oxygen has to be added and the infiuence of the stabilization gas is reduced [STR93a].

When the stahilizing gas does not influence the fiame it is a good approximation in computational studies to completely neglect the stahilizing gas. To simulate a stabie flame with the flamelet method in a first approximation the effect of the stahilizing gas is accounted for by taking the probability of burning (Pb) equal to one. In the literature, where often the SCRS is used to model the thermo-chemical field, this is automatically asumed since local extinction cannot occur within the SCRS.

The neglect of the stahilizing gases, which implies that boundary conditions of the flamelets remain those of the free air and pure fuel, only affects the reliability of the computations in the near field. The thermo-chemical field far downstream can still be modelled using fiamelets which have chemical composition boundary conditions equiva­lent to the fuel and air stream.

The difficulty in explicitly modeHing stahilizing streams is that the flow system is no longera two-stream problem. The oxidizer boundary condition has to be specified in a different way. Under some restricting conditions this can be accomplished by introducing a variabie that denotes the amount of an inert component, for instanee nitrogen, to discriminate the concentrations of the several streams. For instance, when oxygen is added through an annulus, the local amount of nitrogen together with the mixture fraction is an indication for the oxygen mass fraciion at the oxidizer boundary condition [SAL78]. When both hydragen and oxygen are added, the problem is even more complex. However, it has been discussed that in this case the influence of the stahilizing gases is minimal. Therefore the neglect of the stahilizing streams is warrantable.

6.2 Radiation

A principal difficulty in modelling the effect of radiation within the fiamelet concept is that radiation is a geometry-dependent phenomenon, while in the flamelet concept all thermadynamie quantities are related to the geometry-independent mixture fraction. This problem, however, is less severe if it is assumed that the free jet flame only emits energy and does not absorb energy from the environment. Furthermore, most of the fiame radiation is assumed to originate from soot; contributions from other sourees such as H 20, C02 , and also CO and CH4 molecules are much smallerand it is assumed that their contribution may be neglected. This can be inferred from the results in chapter 4.3.2. Here, adiabatically predicted mean temperatures in the hydragen flames were in reasanabie agreement with the experiment.

In the sections below two methods are presented to account for the effect of radiation within the flamelet concept.

114

6.2.1 Empirica! metbod

Crauford et al. (CRA85] artificially red u eed the adiabatic flamelet temper at ure Taa( Z), with Z the mixture fraction, by the very crude assumption

(6.1)

where is a 'fitting' temperature related to any experiment (with radiation) and Xr is an empirica! constant. Crauford et al. used Xr 0.1 fora large scale natura! gas flame and Fairweather et al. [FAI91] used Xr 0.15 fortheir natura! gas flame in a cross-flow. The instantaneous density p(Z) is to be adjusted using the equation of state.

This empirica! method is used in the present study for the turbulent natura! gas flame with fuel composition of the flame of Streb (STR93a], namely 98.084% CH4 , 0.05 %CO, 0.1 % C02 , 0.22% H20 and 1.546% N2 by volume. In order to obtain a satisfac­tory maximum mean temperature in the turbulent flame in section 6.3, a temperature difference due to radiation of about 350 K in the laminar flamelets is required. For this reasou the valnes of Texp = 2050 K [CRA85] and Xr = 0.15 [I<'AI91] are used.

A temperature versus mixture fraction profile of a flamelet at a = 100 s-1 is shown in Fig. 6.1. The maximum adiabatic flamelet temperature at this strain rate is about 2100 K, while the non-adiabatic temperature is about 1750 K. This difference of 350 K can be compared with the temperature difference that was obtained by Fairweather et al., namely 390 K.

A disadvantage of this methad is that species concentrations, which may be sensitive to the temperature, are not adjusted. Furthermore, laminar flamelet temperatures in the fuel-rich region are effectively not modified, leading to a decrease in the temperature only near the stoichiometrie mixture fraction (Fig. 6.1 ).

This problem does not occur when the temperature equation is extended with a radiati ve sink term, a method discussed in the next section.

6.2.2 Radiation modelling based on soot volume fractions

Theory

A radiative energy sink term 4rad/Cv,m has been added to the temperature Eq. (4.17). The volumetrie radiative energy IJ.rad (W m-3 ) is obtained by consiclering the emitted and absorbed energy within a volume element dV. Equilibrium between emitted (4rad · dV) and absorbed energy (47rk9 h · dV) in the volume is assumed (HOT67J. Here k9

is the absorption coefficient (m- 1) and Ib is the black-body intensity of a single ray

(W m- 2 steradian-1). Noting that 1rfo = O'T4 with 0' the constant of Stefan-Boltzmann

(5.67 x 10-s W rn- 2 K-4 ) then gives

(6.2)

The absorption coefficient is related to the emissivity c by the grey-gas relationship

(6.3)

which is based on the assumption of equality of absorptivity and emissivity (Kirchhoff's law). The mean beam length L appears because radiation emitted frorn a volumeelement

115

in the gas is attenuated by k9 over a mean length L. The length L should be related either to a dirneusion of the turbulent fiame or to the (much smaller) dirneusion of a laminar fiamelet. The latter is chosen since the soot- and radiation model is to be implemented in the laminar flamelets.

The emissivity of the soot, ~:, in a ftame can be approximated using the soot volurne fraction J", the rnean beam length L, the temperature (T), and a non-dirnensional number ,.. [SAR86]:

(6.4)

where c2 is Planck's second constant 1.439 x 10-2 m K. The number ,.., which is a function of optica! soot constants, is in the case of oil or coal flames between 4 and 10 [RIS78, SAR86], dependent on the ratio of C to H atoms. The assumptions in the

· derivatiou of Eq. (6.4) are [SAR86] that the soot absorption coefficient is independent of partiele size, and that the monochromatic emissive power E;., (in Wm-3 , with À the wavelength, obeys Wiens' law rv À -s exp( -c2/ ( ÀT) ).

From Eq.'s (6.3) and (6.4) it is seen that

kg= 4 In c + Kj~LT/cJ. (6.5)

Choice of the constauts

The rnean beam length L is taken to be 1 mrn, which represents a typical dirneusion of a flamelet. However, the spatial flame-thickness of a dilfusion flame depends on the strain rate, or mixture fraction gradient, in contrast to a laminar premixed flame where the flame-thickness is relatively constant. For instance, at a = 100 s-1 the physical distance between the leftand right boundary 1 in the flarnelet is about 5 mm while near queuehing of a radiating flamelet (a 380 this difference is only 2.5 mm. Nevertheless, a length of 1 rnm reasonably represents the dirneusion of a laminar flamelet.

The soot volurne fraction fv must be deterrnined as a fundion of the mixture fraction in the laminar fiamelets. In the present study only a very crude model for fv is chosen. Since the chemica! soot chain begins at C2H 2 , which in turn is produced by collisions in volving C H molecules, the soot volurne fraction is taken to be X eH. The X eH profiles qualitatively resembie the soot volume fraction profiles obtained by Moss et al. using a two-equation soot model [MOS88]. In Fig. 6.2 a charaderistic profile of XcH in a 98 % CH4 -air flamelet at a 100 s-1 shows a peak near the stoichiometrie value ( Zst 0.0566), which agrees qualitatively with the soot volume profiles obtained by Moss et al. Quantitatively, the XcH profiles are not expeded to represent the true soot volume fraction very accurately, but the main goal of the present metbod is to obtain the already rnentioned temperature difference of 350 K (section 6.2.1) with a reasonable model. Therefore the final coefficient to he determined, K, is chosen to be 50. This value is five times as high as the maximum value in the literature which is designed for coal or oil flarnes. The value of 50 then is somewhat high since K is expected to be lower for natura! gas flames. However, the important factor in Eq. (6.4) is Kj11L which implies that an uncertainty in K can be compensated by one of the other variables. The major issue is the approxirnate correct variation of Kj11 L with the mixture fraction.

values are the boundaries used in the numerical computation of a fiamelet.

116

g ...

:x:

~·

c-;-~~~~--~~~-----------------

2000 ~ ;j · .. ~iabatic --< Empirica!

~::---.. ~. 1500

1000

500

o'-"----~--~--L------'----~~---------~

0.0 0.2 0.4 0.6 0.8 1.0

z

1e-05 .--~-~~-----~-----~,---~--~~~~--,

ra;;,100/s 88-08

i With heatloss I

88-08

4e-08

2e·08

oL-----~----~---L-~-~----~~~~

0.00 0.02 0.04 0.08 0.08 0.10

z

Figure 6.1: Predicted fta.melet tem­pera.tures for a == 100 s-1

without ra.dia.tive heat loss ( dotted line), with heat loss a.ccounted for with the em­pirica.l method ( dashed line) a.nd with the ra.dia.tive sink term (Eq. (6.2) a.dded in Eq. (4.17) (solid line).

Figure 6.2: Predicted C H mola.r fra.ction versus the mixture fra.ction Z in a. la.minar tla.melet a.t a = 100 s-1 (ra.dia.tive heat loss included) for XcH, 0.98.

A typical temperature profile obtained with this model for a = 100 s-1 is shown in Fig. 6.1. The inclusion of the radiative heat lossesindeed shows that the temperature at the fuel rich side is lower than the empirically modified temperature. This will be seen to agree better with the measured temperatures in the turbulent fla.me. Also in the fuel lean side the temperature is somewhat lower than predicted with the empirical model.

Quenching behaviour of flamelets with heat loss

Radiating methane (98% )-air flamelets ( Zst 0.0566), as described previously with the soot-radiation model, quench at aq 380 s-1 • As expected, this value is lower tha.n that obtained in chapter 4 for adiabatic rnethane-air flamelets2, i.e., aq = 505 s-1

•

A peculiarity concerns the maximum temperature that occurs in the flamelets. Adi­abatic maximurn flamelet temperatures are always between 1840 K and about 2300 K. The former is the maximum temperature in an adiabatic flamelet near quenching (a = 505 s- 1

) while the latter is the equilibrium adiabatic maximum fla.me temperature (predicted with the SCRS).

In Table 6.1 typical flamelet temperatures and extinction strain rates are presented.

2% difference in CH4 volume fraction at the fuel boundary does not inlluence aq.

117

Flame type aq (s 1) Tmax (K) at a= 100 S 1 I T max (K) at a = aq i

Adiabatic 505 2200 I 1840

I Radiating 380 1767 i 1727

Table 6.1: Characteristic flamelet data for adiabatic and radiating methane (98%}-air flamelets.

In non-adiabatic (radiating) Hamelets the maximum temperature at a 100 s-1 is 1767 K, which is already below the temperature at which an adiabatic flamelet would have been quenched. Near quenching of the radiating flamelet ( aq = 380 s-1 ) its maximum temperature is only 1727 K, which is more than 100 K below the adiabatic quenching

· temperature of 1840 K. This is clearly an indication that queuehing behaviour is governed by strain (residence

time) effects, rather than temperature effects.

6.3 A stabilized methane flame

The vertical flame of Streb [STR93a] consists of fuel at 323 K, with concentrations of 98.084% CH4 , 0.05% CO, 0.1 % C02, 0.22% H20 and 1.546% N2 by volume. The fuel issues from a nozzle with D =10 mm and with an average speed of 60.1 m/s into still air. This relatively large flame (2 m long and a power of 0.15 MW) is stabilized with two separate H2 and 0 2 streams which have minimal effect on the downstream dynamics since their total momenturn flux is only 7 x 10-4 N, while the fuel jet issues withaflux of 0.1744 N.

Adiabatic predictions for this flame are performed with 6 burning flamelets with a =50; 100; 200; 300; 400 and 500 s-1 using the Cl-mechanism of Table D.3. The quenching value aq is 505 s-1 , which is the same as the value given in Table 4.3.3 for

100% metharre flamelets. The small eddy strain rate s = C,,l\je;f(2v) with C.,1 0.3 and P6 1 are used. Predictions including radiative heat loss are performed with one flamelet at a = 100 s-1 (see the previous section).

The turbulence model is the k- é model with Gil 0.06 and the equal-scales scalar transport model with CTJ u9 0.7. The fuel exit velocity, which had a fairly flat distribution experimentally, is modelled with a powerlaw (n=lO, see caption of Ta­bie 5.1) and Umax =69.3 mjs. Turbulence quantities are taken as k = 0.001u2 and é Cll-P12 j(0.03D). Again it is mentioned that these turbulence quantities have no influence on the downstream dynamics.

In Fig. 6.3 the predièted centreline temperature variation (6 flamelets), without ac­count for radiation loss, is seen to be about 300 K higher than found experimentally. The order of magnitude of this difference agrees with the difference of 400 K reported by Loekwood and Stolakis [LOC83]. Moreover, according to Streb [STR93b], the flame radiated and sooted intensely.

The predicted temperature using the empirica! correction shows the right values of maximum temperature, while the temperature using the Hamelets based on the soot­radiation model shows a still better agreement. The overpredicted temperature in the near field is due to laminarization in the experiment. The effect is a faster decay of the

118

1500

1000

500

0~.~~-----~-~~---~--L-~---~---L---------~

0

80

80

40

0

'" I I I \ r+j I I I I

I I

"' \ \

50

ö.\ ~' ' '

50

100 150 200

X/0

150 200

X/0

Figure 6.3: Predicted cen­treline ternperature vada­tion with 6 flarnelets with­out heat loss, with ernpirical heat loss, and with heat loss based on the ternperature equation sink term. Exper­irnental ternperature data ( syrnbols) are frorn Streb [STR93a].

Figure 6.4: Predicted cen­treline variation of veloc­ity (solid line) and turbu­lent kinetic energy { dashed line ). Experirnental data ( syrnbols) are frorn Streb [STR93a]: Velocity ( .6.) and turbulent kinetic energy (+).

predicted mixture fraction, leading to a faster increase of the temperature. In Fig. 6.4 this is illustrated again by the faster decay of the predicted velocity. The turbulent kinet-ie energy is seen to he predicted very wel!. Another example of the fast mixing in the predictions is presented by the predicted and measured halfwidths for the veloc­ity and the mixture fraction (Fig. 6.5). The predicted halfwidths are larger than the experimental value in the near field which indicates that the predictions show too fast m1xmg. In the far field the measurements, especially the mixture fraction halfwidth, are a little inconclusive, although the predicted halfwidths seem to be somewhat large. It must he noted that the experimental values of the mixture fraction halfwidth have been calculated from the measured molar fractions of the species using J = Ze /Zc,1 •

Here is the carbon element mass fraction calculated with Eq. ( 4.1) and Zc, 1 is its value at the nozzle. The expression of Eq. ( 4.20) for the mixture fraction cannot be used for this experiment because of the preserree of the stahilizing gases.

In the far field the predicted velocity (Fig. 6.4), is in excellent agreement with the experiment, but the temperature is overpredicted. This can have several reasons: ra­diation losses, underpredicted scalar varianee and overpredicted mixture fraction. The first explanation has already been discussed, and the predicted maximurn temperatures agree with the rneasured ternperatures if radiative heat loss in included. The second

119

e. = i • :1:

;;::•

111 Figure 6.5: Predicted + halfwidths for the velocity

(solid line) and the mix-/

x ; ture fraction ( dashed line ). 10 ;""

LJ. ,..-""' Symbols are measurements ---+-/1:-~ + [STR93a] ( velocity !::::. and

_"",.- + mixture fraction ( + )) . /- A

5 : Mixture fraction

100 150 200

XJD

0.25 .---c-------------------. 1.0 Figure 6.6: Centreline vari­

0.20

0.111

0.10

0.05

..,_..H 0 0

2 0

0 0

0.8

0

0.4

0.2 0

L-8~~===-===:':':_::::::;:::::::i::::::==oe.-.,.,.......---,;,.--..J 0.0 0.00 <«

0 50 100 150 200

XJD

ation of mass fractions of CH4 (solid line and 6), Oz ( dashed line and 0) and H20 (dotted line and 0) predicted with radia­tive heat loss. Symbols are experimental data from [STR93a].

explanation is unlikely to be completely responsible for this difference in the far field, although the temperature varianee is indeed somewhat underpredicted (Fig. 6.3). The last explanation of an overpredicted mixture fraction in the far field is the most likely explanation, which also agrees with the observations made in turbulent hydrogen fiames (see chapter 5). Indeed, an overpredicted mixture fraction in the far field is also able to account for the difference in predicted and measured 0 2 mass fraction, (Fig. 6.6).

In the maximum temperature region, where the temperature is predicted most accu­rately, Yo2 agrees with experiments (Fig. 6.6), while in the near field the measured valnes show a small hump which is not reproduced by the predictions. This small hump is due to the neglect of the stahilizing H2/02 streams. The somewhat underpredicted water vapour mass fraction (Fig. 6.6) is also due to this effect, while the overall agreement is satisfactory. The underprediction of YcH4 for 10 < xf D < 75 is due to the overpredicted mixing in this region.

It is worth mentioning that generally CO mass fractions measured in turbulent ditfu­sion fiames are relatively high, compared with mass fractions obtained in steady laminar fiamelets. This is also the case in the present fiame (not shown in a figure). The max­imum centreline value of Yco in the experiment was 0.08 while the Hamelets show a maximum value of 0.03. According to ManBet al. [MAU90] and also to Barlow and

120

2000 Figure 6.7: Mean temper-

Without ature across the fiame pre-

/ dicted with heat loss based 1500 on soot radiation ( solid line)

/ With + and adiabatic ( dotted line ).

S2" + Temper at ure varianee T'2 ;:: 1000 + ( dashed line). Symbols are

data from Streb [STR93a].

500

5 10 15 20 25

RIO

10 Figure 6.8: Prediction of axial velocity and turbulent

a + + + + kinetic energy across the jet

~ at x/D 120. Sym-+ .. bols are experimental data (;;" 6 .

!. i+ [STR93a]. ... + ~ 4 t + :§: Turbulent energyl ;:)

2

0 0 5 10 15 20 25

R/0

Chen [BAR92], transient effects are the reason for the discrepancy in the CO mass frac­tions. A sudden decrease of the strain rateis foliowed by an oversboot of Yco, and this type of effects do occur in turbulent ditfusion fiames. Since CO is a fuel, the difference between predicted and measured valnes of Yco can explain part of the temperature dif­ference between the measurements and the adiabatic predictions. It canrwt, however, explain the complete difference.

Predicted radial profiles of temperature and temperature variance, at x/ D = 120, agree reasonably with the experiments when radiative heat loss is taken into account (Fig. 6.7). Finally, in Fig. 6.8 the radial profile of the axial velocity at x/ D = 120 is seen to be in good agreement with the experiment, while the predicted turbulent kinetic energy shows a good qualitative correspondence with the experimental data. The reason for the apparent vertical shift between predictions and measurements is unknown.

6.4 Conclusions

In this chapter radiative heat transfer, which is usually neglected in the literature, has been shown to be important in a free jet natura! gas fiame. A relatively simple model

121

for soot-related radiation, has been implemented in the flamelet method. In the model a number of parameters, such as the mean beam length and an optica! soot constant, appear. The values for these parameters are unknown and difficult to interpret within the flamelet method. Furthermore, the soot volume fraction is estimated with a rather crude model. Therefore, these parameters have been chosen so as to provide adequate agreement between measured and predicted maximum mean temperatures in the turbu­lent methane flame of Streb [STR93a]. The difference in maximum mean temperature obtained with adiabatic predictions and predictions including radiative heat loss is more than 300 K.

An interesting aspect concerning flamelets with radiative heat loss is that the queueh­ing behaviour has been found to be mainly a residence time effect, rather than a tem­perature effect. This was concluded from the predicted maximum temperatures in the

_ flamelets near quenching. Without radiative heat loss the maximum temperature at queuehing is much higher than with radiative heat loss. The queuehing value for the strain rate in the flamelet with radiative heat loss is lower than for adiabatic flamelets, as expected.

Predictions show a mixing rate in the near field which is too large compared with experiments. This results in underpredicted velocity and mixture fraction values and overpredicted temperatures in this region. The cause for the overpredicted mixing rate are low Reynolds number effects, which were also observed in the experiment of Takagi et al. [TAK81] in chapter 5. In the far field the temperatures are also overpredicted, which is attributed to an overprediction of the mixture fraction, which was also observed in chapter 5.

Measured CO mass fractions in turbulent flames are higher than those predicted with the laminar flamelet metbod using steady flamelets. Transient effects are thought to be responsible for the highervalues in experiments [MAU90, BAR92].

The overall condusion is that a reasonable agreement between predictions and ex­periments has been obtained.

122

Chapter 7

Modeling and Calculation of Turbulent Lifted Diffusion Flames

The present chapter is an artiele which is accepted for pubikation in Gombustion and Flame, see Ref. [SAN94]. The notation in the artiele is self-contained but some minor deviations with the notation in the other chapter in this thesis are the following.

All turbulence quantities are denoted induding the tilde, which indicates .Favre­averaging. The scalar dissipation is denoted with x instead of r::9 for reasons already stated in chapter 4.

123

COivfBUSTlON AND FLA:\t!E 00: 000-000 (1993) 000

Modeling and Calculation of Turbulent Lifted Diffusion Flames

J. P. H. SA..'IDERS and A. P. G. G. LA.c'viERS EindhOfJen Uniuersity o{TecM.ology, Faculty o[Mecho.nica/ P.ngineen'ng (WOC), PD. Box 513, 56()0 .\1B

Eindho.oen, Tne Netherlands

Liftoff height.s of turbulent diffusion flames have been modeled using the laminar diffusion. fl~elet concept of Peters and Williams [A!AA J., 21:423-429 (1983)]. The strain rate of the smallest edd1es 1s used ~ the stretch descnbing parameter, instead of the more common scalac dissipation tate. The_ h(U) curve, w?1ch IS

the mean liftoff height as a function of fuel exit velocity can be accurately pred1cted, wh!le th':' was impossible with the scalar dissipation rate. Liftoff calculations performed m the flames as well as m the equivalent isothermal jets, using a standard k-• turbulence model yield approximat~ly the sa~e ":'rrect slope for the h(U) curve wbile the offset bas to be reproduced by chosing an appropr1ate coe~1ent l!l the strain rate model. For the flame calcu!ations a model for the pdf of the fluctuatmg flame base IS proposed. The results are insensitive to its width. The temporature field is qualitative!y different from the field calculated by Bradley et al. (Twenty-Third Symposûun on Combustion, 1990, pp. 685-692) who used a premixed damelet model for diffusion dames.

INTRODUCTION

Turbulent jet ditfusion flames constitute an important class of fiames, in theoretica! and experimental studies as well as in industrial applications, such as gas turbine combustors and fiare bumers.

For most fuels liftoff of the flame occurs at fairly low fuel exit velocities, and consequently liftoff is an important phenomenon. A lifted turbulent diffusion flame consists of a cold flow near field region and a nonequilibrium flame region, separated by the fluctuating flame base. The distance from the nozzle to the flame base is called the liftoff height. The physical mechanism that is responsible for the lifting of the flame is the competition between flow and chemica! effects, characterized by their respective time scales, the ratio of which is the Damköhler ii.umber. If typical chemica! time scales are everywhere smaller than flow residence times, the fiame will be rim stabi­lized. lf the fuel exit velocity increases, then the residence time, which is the inverse of a strain rate, decreases and so does the Damköhler number. At a critica! exit velocity the fiame can no Jonger be stabie at the nozzle rim and liftoff occurs. From this point increas­ing the fuel velocity further leads to an in-

Copyright ~ 1993 by The Cambustion lnstitute Publisbed by Elsevier Science Publishing Co., Inc.

crease of the liftoff height proportional to the exit velocity [1]. This regime is the subject of the present paper.

Current predietien methods for rim stabi­lized turbulent diffusion flames are reasonably accurate [2, 3]. For flames with liftoff the main difficulty is the accurate predietien of the liftoff height for which there exist two major classica! proposals. The oldest considers the base of the flame as a premixture of fuel and oxidizer. The buming velocity of this m..ixture equals the flow veloei tv at the flame base [ 4]. Peters and Willi~ [5], however, have argued that no suffi.cient premixing of the reactants can take place for this concept to be valid, in aceer­dance with experimental results of Pitts [6]. Therefore they proposed a liftoff theory based on laminar diffusion flamelet quenching. Lami­nar counterflowing diffusion flames, called flamelets, are extinguished if the stretch expe· rienced by the flame exceeds a critica! quench­ing value (7]. If a turbulent diffusion fiame may be considered as an ensemble of laminar dif· fusion flamelets with all different stretch rates [8], then liftoff of a turbulent flame can be described with this concept [5]. The liftoff con­dition is obtained by rea!izing that liftoff is a threshold phenomenon. The variabie that is responsible for this behavior is the fraction of

0010.2180/93jS6.00

124

. I

2 J. P. H. SA.NDERS AL'<D A. P. G. G. LAL'v!ERS

bumabie fiamelets, or the probability of burn­ing [5]. The probability of buming is defined in terms of a variabie descrihing stretch effects, and usualJy the scalar dissipation rate is cho­sen for this purpose [5). lf this quantity is used in the liftoff condition, predicted liftoff heights are not in very good agreement with experi­ments.

A third concept for predicting liftoff heights of turbulent diffusion Harnes was proposed by Bradley et al. [9] with an altemative, so called mixedness-reactedness, Hamelet model. Here the cambustion is considered to take place in a premixed mode in the diffusion Hame. At each given mixture fraction within the fiammability limits, a premixed fiame is established. Be­cause the most obvious independent parameter for a premixed fiame is the temperature, in­stead of the mixture fraction, all scalar profiles are given in terros of the temperature. In this way, the heat release in temperature space is determined, and the mean heat release is cal­culated by integrating over independently cho­sen probability density functions for mixture fraction and temperature. The liftoff height is determined by the axial distance where the mean heat release begins. The obtained liftoff heights as a function of fuel exit velocity h(U) correspond wel! with the experimental data of Kalghatgi [1]. Despite the success of this method, it is not completely clear whether the assumption of premixing is justified. Further­more apparently no threshold behavier is contalned in the model, while this is a very characteristic feature of a lifting fiame.

Therefore in the present artiele attention wil! be focused on the liftoff mechanism based on laminar diffusion fiamelet quenching of Pe­ters and Williams ~5, 10] mentioned earlier. Predietiens of ·the liftoff height based on this concept have been made by consictering the equivalent isothermal jet flow [5, 11], but the slopes of the h(U) curves were not in agreement with experiments.

Very recently Peters [12] modeled the liftoff height using the strain rate of the large eddies instead of the scalar dissipation rate in the liftoff condition. The calculated liftoff heights, based on isothermal jets are in good agree­ment with experiments.

In the present work the liftoff height will be

predicted based on the flow fields of the isothermal jet as well as of the Hame. The strain rate will be used in the liftoff condition instead of the scalar dissipation rate. The rea­sans for choosing this variabie are discussed. In the Hame calculations a probability density function (pdf) for the liftoff height, which takes care of the fiuctuating fiame base, is proposed and the infiuence of the shape of this function on the predicted temperature profiles is. inves­tigated. The infiuence of variabie density ef­fects on the prediered liftoff height is deter­mined by camparing predicted liftoff heights in the fiame with liftoff heights obtained from isothermal jet calculations. ·

A..NALYSIS

Two main probieros characterize the calcula­tion of a lifted turbulent diffusion fiame. Firstly the mean liftoff height has to be determined. Secondly the f!uctuating behavier of the fiame base must be taken into account.

Modeling of the Liftofr Height

If lift off occurs through laminar diffusion fiamelet quenching, then the topology of the lifted fiame is a disconnected surface of instan­taneous stoichiometry. By the disconnected­ness local quenching of the fiame is meant This is caused by local values of the stretch or nonequillbrium parameter, higher than the quenching value. At these points no fiame can exist, and these positions are called holes [10]. The fraction of burnable fiame!ets, which is equal to one minus the probability that a hole is present at a certain position, can be esti­mated by integrating the pdf of the relevant stretch parameter up to the quenching value of a laminar diffusion fiame. This fraction of bumabie Hamelets is also called the probability of burning

(1)

in which n is the stretch parameter, which can be the scalar dissipation rate x as in refs. 5 and 11 or the strain ra te s. The quenching value nq is determined from laminar diffusion

125

LIFTED DIFFUSION FI...A..\1ES

ftamelet calculations [13-15] or from experi­ments [10]. Tbe form of the pdfs P(n) are a lognormal distribution for the scalar dissipa­tien [5] and a quasi Gaussian pdf for the strain rate [16]. Tbe latter is wrinen as

(2)

Here o; denotes the variance. Tbe mean and varianee are coupled by

(3)

If too many holes are present, or the proba­bility of burning is too low, on the contour of stoichiometrie mixture where the fiame is lo­cated instantaneously, the fiame cannot be sta­bilized at the burner rim and liftoff occurs. The fraction of holes which is allowed until the fiame lifts off can be roughly estimated with percolatien theory. Tbis theory can be used to descrlbe the threshold behavior of the liftoff of the fiame by means of the probability of burn­ing [10]. Tbis concept leads to a liftoff condi­tion involving both a percoladon threshold P, and a probability of burning Pb [5]

(4)

at a radial position where the mean mixture fraction i is stoichiometrie i= f ... If Eq. 4 is used at the mentioned radial position, with the appropriate pdf P( n) inserted into Eq. 1 for Pb, then a relationship is obtained between fl. and the mean Ö, with the varianee of fl. as ~ parameter. Tbe mean and varianee of fl. are supposed to completely determine the pdf P(fl.). Tbe above procedure leads to the fol­lowing liftoff condition if the strain rate is used for fl.

P =erf 9 ( s )

c s( r = r .. )1TI/l • {5)

where r is the radial distance. from the symme­try a-ris. If the value of P. is about 0.63 (5], then this equation can be simplified to approxi-

3

mately

• Sq s(r = r,.) = -vz

P,1T (6)

If the scalar dissipation ra te is used, then the simple liftoff condition

X(r = r,.) = Xq (7)

is obtained with P. 0.63 and ux = 0.5 [5]. In the following several arguments are given

to support the choice of the strain ra te for fl., and the practical justification of this choice will be given by the results of this artiele in which the slope of the h(U) curve is the main impor­tant parameter.

Four arguments are presented in favor of the strain rate as the stretch parameter.

Firstly laminar counterflow diffusion fiames are considered. They are the comerstone for laminar fiamelet modeis in turbulent oombus­tion (8]. In these counterflow diffusion fl.ames the strain rate imposed on the fiame gives rise to stretch effects, leading to nonequilibrium chemistry. The sca.lar dissipation rate is just the consequence of this strain rate. Addition­ally, the sca.lar dissipation rate is zero at both cold flow boundaries, which necessitates the introduetion of a particular location in the fiame, where the scalar dissipation rate should be evaluated. Often the scalar dissipation rate at stoichiometrie conditions x,1 is chosen (8], but conditioning of the scalar dissipation rate on a specific value of the mixture fraction leads to a fundamental problem. The fiamelet con­cept leads to the following general expression for the mean of a scalar

~ = t df{ dOP(!, O),P{f,O), (8) 0 0

in which P(f, 0) is the joint probability den­sity function for the mixture fraction and the stretch parameter n and ,P(f, n) is a scalar and a function of the mixture fraction with n as a parameter. If fl. is chosen to be the scalar dissipation rate conditioned on stoichiometry (fl. == x<!= f .. )) then P(f, Xst) has no mean­ing other than P( x .. ). Tbis is not realistic because no account would be taken of the mixture fraction fluctuations. In practical cal­~lations th~ inconsistency has been solved by Sliilply stating that there is no satisfactory

126

4 J. P. H. SANDERS AND A P. G. G. LAMERS

model for the conditioned scalar dissipation rate, and so the unconditioned model x Ugjk was used, with g the scalar varianee and k the turbulent kinetic energy. In that case, assuming statistica! independence, P(f, n) = P(f)P(fl.), the integral in Eq. 8 can be done. The strain rate, however, need not be conditioned on any particular mixture fraction value because it is a boundary condition in a counterflow ditfusion iiamelet, so this problem does not arise.

Secondly a more serieus problem is the in­crease of the quenching value of the scalar dissipation rate at queuehing of a laminar counterflow ditfusion fiamelet if the fuel mass-fraction at the fuel boundary is de­creased. This is contrary to the value of the strain rate at quenching, which decreases with decreasing fuel mass-fraction, as it should. These observations can be interred from Fig. 10 in Ref. 8, and from fiamelet calculations perforrned in the present work (see Fig. 1), where the calculated quench values aq and Xq are plotted versus the methane volume frac­tion in diluted methane-air counterflow dilfu­sion fiames. Consequently, if the scalar dissipa­tien rate is used as the stretch parameter, then the application of the liftoff condition as de­scnöed above, would yield shorter liftoff heights if the fuel stream is diluted, which is unphysical. The experiments of Miake-Lye and Hammer [17] indeed indicate an increase of the liftoff height if the fuel is diluted, even though air was used for dilution instead of an inert gas.

Thirdly, in both Refs. 5 and 11 a liftoff

... -- • •• 0

A

••• •• .... 0

a" .. ~ ,.. 0

0 ... ...

... .. 0.4 a.a u 1.0

XCH,

Fig. 1. Quenching values "• (triangles) and x. (circles) venus tbe methane volume fra<:tion XCH, in diluted methanc:-air countc:dlow diffusion llames.

127

condition similar to Eq. 7 was used. The slopes of the predicted h(U) curves were not in agreement with the experiments. The discrep­ancies might be explained by the different seal­ing properties of the scalar dissipation ra te and the strain rate. The scalar dissipation rate on the axis of an isotherrnal axisymmetric jet scales with x-• while the strain rate scales with x- 2

•

Although the points where the meao mixture fraction is stoichiometrie are not located on the axis, it is fair to say that there is a large difference between these two variables regard­ing their sealing properties. Due to the slower decrease of the strain rate in axial direction, the liftoff condition, Eq. 6 wi1l be met at higher x-values leading generally to higher liftoff heights in better agreement with experiments.

Lastly, if the scalar dissipation rate is re­tained as the descriptor of stretch effeets, it still essentially is a strain rate [18]. This is due to the fact that the fiame thickness in m..ixture fraction space is constant, independent of the stretch imposed on the fiamelet. Eventually the scalar varianee g is replaced by the Hame thickness, leading to a sca1ar dissipation rate at the position of the fiame XF- ijk, which es­sentially is the strain rate of the large eddies. The details of this analysis are given by Peters [12].

Modeling of the Fluctuating Flame Base

The fl.uctuating fl.ame base wi1l be handled by introducing a pdf for the location of the fl.ame base P(xL) where xL denotes the axial dis­tanee from the noz:zle to the fl.ame base. The mean scalar quantities such as density and temperature can now be written as

x {'dsP(f,s)c!J(f,s)

+(1- (P(x>xddxL)

x fa1P(f)cb(f) df. (9)

In this equation the last term is the isothermal

LIFfED DIFFUSION FLA.\.IES

value of the scalar, and the factor in front of it is the probability that the flame base is located downstream of x. The probability that the in­stantaneous flame base is upstream of x is denoted by P(x > xL)- The first term in Eq. 9 is valid in the burning part of the jet, and is due to both burning and nonburning flamelets, where the average must be calculated over the mixture fraction and the strain rate and their appropriate pdfs. Statistica) independenee be­tween the mixture fraction and strain rate must be assumed if no transport equations for pdfs are solved, so P(f, s) = P(f)P(s). In this work presumed shapes for P(f) and P(s) are taken, namely a beta tunetion for the fust and a quasi-Gaussian for the Jatter. The pdf for the tlame base is unknown. To calculate it, two point pdfs are needed, which is computation­ally intraetabie for this situation. Therefore a proposition for this pdf is made. The farm is assumed to be a triangle pdf around the mean liftelf height h determined from the liftalf condition, and the width is taken to be 5 diameters.

TURBULENCE AND COMBUSTION MODELING

Tnrbulence Model

The turbulent reacting flow is modeled using Favre-averaged quantities [19-21] tagether with the k-e turbulence model. The Favre~ averaged velocity vector is denoted by Ü and its Favre tluctuation by il'. The Favre­averaged turbulent kinetic energy, its dissipa­don rate, the mixture fraction flucuations and the scalar dissipation rate are denoted by k, ê, g, and x. respectivdy. The pressure and den­sity are conventionally time averaged and denoted by p and p, respectively.

The equations consist of the continuity equation

(10)

and the momenturn equation

(11)

5

No buoyancy terms are included because the flame under consideration is completely mo­mentum driven in the region of interest. The Reynolds stress tensor is modeled as

- 2 ( - .::.) = - 3 .s,i pk + J.L;r:; • u

(12)

where .Sii is the Kronecker delta. Th! turbu­lent viscosity is modeled as J.L, = C"_pk 2jê.

The turbulence model consisrs of equations for the turbulent kinetic energy k and dissipa­tion ê. The kinetic energy equation is

( .=..) ( J.L, -) V· pUk =V· o-k Vk + P•- pê, (13)

with P• the production term of turbulent en­_:_aü

ergy -pfi;u'j -'. The dissipation equation is axj

(14)

Furthermore the equations for the mixture fraction i and scalar varianee g f 2 are

V· (iiÖi) =V· (~vj), (15)

V· ( pÓg) =V· ( :; Vg) + P1 - PX. (16)

Here Pg is the production term of scalar fluc­

tuations P1

= - C1

f;u";f" a j. The scalar dis-a x;

sipation rate x is modeled according to stan­dard practice by assuming equality of veloci;y and scalar integrallength scales: x= cx.t êgjk, where the coefficient cx,l is empirica! and usually taken to be equal to 2. The assumption of equal length scales leads to fixed turbulent

128

6 J. P. H. SA.'>:DERS Ac"<D A P. G. G. LMfERS

Schrnidt numbers u1 and u, transport model.

- af - ---;;--!" p.' pU· =-, ui ax:,.

in the scalar

(17)

The constants in above equations are given in Table 1. A scalar transport model that does not assume equallength scales and fixed turbu­lent Schrnidt numbers is given by Sanders and Lamers [22]. The expression for the scalar dissipation rate obtained with this model is

(18)

with. A1 1.5 and A2 = 0.25 determined from isothermal jet calculations and C 2 a coeffi­cient (not dimensionless) to be ~djusted to experimental data.

To be able to use the liftoff condition (Eq. 6) the straln rate must be modeled. If the strain ra te s is interprered to be the strain ra te of the smallest eddies then

s= (19)

which is based on the concept of dissipation of turbulent energy by strain rate fl.uctuations on the smallest scales. This explains the molecular viscosity ~ in. Eq. 19. The coefficient C,,1 in thJS equation lS unknown, but ît wil! be deter­mined ln the course of the work If s is inter­prered as the straln rate of the large eddies, then

Ê s=c, • .."., .• k (20)

and also here, the coef:ficient C,,2 has to be determined later on.

A short discussion on the sealing properties of bath strain rates with ax:ial disrance and fuel

TABLE 1

Turbulcnce OmstaniS Used in thc S tandard k-• Model

c". u, u, c 1 c1 c, c,.1 u1 u,

0.09 1.0 1.3 1.44 1.92 2.8 2 0.9 0.9

exit velocity is in order because bath strain rates vary as x- 2 on the ax:is of an axisymmet­ric jet that rnight give the impression that they are completely similar. In the first place it must be mentioned that the strain rates are used in the liftoff condition only along the line of stoichiometrie mixture, which makes an asymptotic sealing analysis somewhat difficult to perfarm since this line of stoichiometrie mixture farms a closed contour and conse­quently does not exist at very large x:. Never­theless different sealing properties on the axis wil! also have their inlluence on the line of stoichiometrie mixture.

The axi~ scalin_g pr~erties .9f €, k, and U" are: ê- U,/x-\ k- U" and U"- UDx- 1

• In­serting these expressions into the equations for the strain rates gives a sealing behavier of s - U 312D 31'Lv- 112x-2 for the smal! eddy strain rate and s - unx-• for the large eddy strain ra te. Consequently both strain rates have the same x-sealing behavier but scale differ­ently regarding U. Additionally it may be re­marked that along the llne of stoichiometrie mixture both strain rates vary approximately as x-t, although at small x the large eddy strain rate falls off more rapidly.

Combustion Model

Cambustion is modeled by introducing the coi?b~tion invariant mixture fraction f [23], wh1ch lS only convected and diffused through the flow. Considerlng the mixture fraction as an independent variable, all scalars except the pressure are coupled to the mixture fraction by invoking the . fl.amelet concept [8, 24]. Each fiamelet is characterized by the value of the strain rate experienced by the flamelet. So at each strain rate a table of scalars versus mix­ture fraction exists. The total of tables is called a fiamelet library. In the present work main attention is focused on fiuid mechanical pro­cesses in the lifted fiame. From numerical cal­culations and from lirerature it is known that a fiamelet hörary consisring of two fiamelets one burning and the ether isothermal, is aCC:U.ate enough for purposes of determining the mean density and temperature. Consequently the

129

LIFTED DlFFUSION FLA..'-fES

integral in Eq. 9 can be written as follows

<Ï>(x) = P4 P.fP(f).Pb(f) df 0

+((1 -P.)Pd +P .. )

X fP(/)c/>0 (!) df. 0

(21)

In tlus equation Pb is given by Eq. 1 with the appropriate pdf, the subscript b denotes the burning fiamelet, and the subscript o denotes the isothermal value. Pd is the abbreviation for io P(x >x~.) dxL• the probability that the loca­tion x is downstteam of the mean lift off position and P.. is the complementary proba­bility. The beta tunetion pdf for the mixture fraction has the form

P(f) =/"- 1(1- f)b- 1/t3(a,b) (22)

with a= f<f(l - Ï)fg- 1) and b = a(1 - f)j g and

t3(a,b) = fr- 1(1 n•- 1 df (23) 0

In the flow calculations the conventional time averages for the density and temperature are needed, and therefore for these variables Eq. 21 is rewritten as [25]

- pd.pb p(x) = f6P(f)/p

0(f) df

and

(1 - Pb)Pd + P. + -

f6P(f)IPaU) df

T(x) = - P~~.P" Pb/JP(f)T"(f)/ p11(f) df

(1 - P11 )Pd + P ..

COMPUTATIONAL SETIJl>

Turbulent Flow Calculations

(24)

The experimental results used to oompare with the numerical predictions are those of the free

7

vertical natural gas jets and lifted flames mea­sured by Wittmer [26]. The fuel exit veloeities in the jet and fiame are between 40 and 71 mjs with a slow coflowing air stream of 0.4 m/s. The nozzle diameter D is 8 mm.

Due to the fluctuations of the flame base, there might be a back coupling from down­stream to upstream positions in the ftame [9]. This necessitates the use of an elliptical com­putercode. The finite volume computercode Teach [27] was chosen as a framework for this study.

Most calculations were performed on a rect­angular grid consisting of 80 x 50 nodes in a.'lial and radial direction, respectively.

Boundary Conditions

Variables at the radial boundary are zero, ex­cept the a."<ial velocity, which is equal to the free stream velocity, and the radial velocity that was determined by the continuity equa­tion. At the outlet zero gradients are applied. At the inlet a flat velocity profile was imposed ~d the turbulent kinetic energy was taken as k =:. U 2j1000 and the dissipation as € = C,,_kl12j0.03D. The mixture fraction at the inlet is equal to 1 and its fluctuation is 0.

F1amelet Calculations

Prior to solving the elliptical equations given in the previous section the scalar profiles as a function of mixture fraction with the strain rate as a parameter in a laminar counterfiow­ing diEfusion flame are needed. These profiles are obtained by numerical integration of the boundary layer similarity equations descnbing two counterflowing streams [28]. The equations solved are the continuity, momentum, temper­ature, and species equations. The numerical code essentially is a damped and modified Newton methad [29]. This flamelet code is only used as a tool and not descnbed in this article. The program is a precursor of the Cambridge fiamelet code RUN !DL [30].

The profiles that are used in the present work are the density and temperature as a function of mixture fraction.

130

8 1. P. H. SANDERS AND A P. G. G. LAlvfERS

RESULTS

Isothennal Jets

The isothermal flow is used to get an indica­tion of the usefulness of the several liftoff models. The upstream part of the flow in a lifted flame is mainly isothermal, but in the vicinity of the flame base variadons in temper­ature and density are steep. Therefore the isothermal flow is only a fust approximation. To get an idea of the accuracy of the calcula­tions in the isothermal flow, the curve of radial position where the mean mixture fraction is stoichiometrie as a function of axial distance is compared with measurements of Horch [31] (Fig. 2).

The agreement between measurements and calculations is very good for xjD < 20. The deviation of 13% farther downstteam could be caused by both an experimental uncertainty and a model deficiency.

Liftoff heights are calculated with four dif­ferent liftoff conditions. The fust and second are based on two different modeling assump­tions for the strain rate, see Eqs. 19 and 20, while the third and fourth are based on the scalar dissipation rate as the relevant stretch parameter. For liftoff calculations a!one the scalar dissipation rate was modeled acco;dina to x= C tÊg-/k and x·= C g·Lsio.2s Al~ x, x.2 • though the strain rate already was chosen to be the appropriate stretch parameter, the results obtained with the scalar dissipation rate serve

r--------~·-·~-----~

10 20 30 40 !0

XJC

Fig. 2. Radial position where the mean mixture fraction equals the stoichiometrie value as a function of axial disu.nce. The d.rawn line corresponds to calculations and the triangles are experimental data of Horch [Jl].

as an indication for the supereority of using the strain rate.

The quenching values for the strain rate and scalar dissipation ra te are sq = 565 ç 1 and Xq = 48 s·t, which are obtained from laminar diffusion flamelet calculations of a planar flamelet.

The coefficients c,,l and c,,2 in the strain ~ate roodels and_ the coefficient cx,l and c(,2

m the scalar disstpation model are to be deter­mined. To this end the liftoff height of the flame with exit velocity of 71 m/s is taken as a reference. The coefficients are adapted so as to reproduce this liftoff height. Using the ob­tained value in the cases with the lower exit veloeities as well, the h(U) curve given in Fig. 3 is obtained.

It must be noted that in the turbulent flow equation for the mixture fraction f!uctuations g the standard model x = cx.l ig;k with c;r,l = 2 was used and only in the lift off condition the c;>efficients c;r,l and c;r,l are adapted.

It IS clearly seen in Fig. 3 that the curve with the strain rate of the smallest eddies with c,,l = 0.116 gives the best results. The curve obtained if the strain rate of the large eddies is used wi th C •.2 == 6. 4 is aJso sa tisfactory.

In order to give an indication of the sensitiv­ity of the h{U) curve with respect to, for in­stance, V OC C,,lt the VÎSCOSÏty bas been varled by factors of 2. As this is equivalent to varying C,,1 at constant v, a vertical shift of the h(U) curve is to be expected. A lower strain rate (higher v) wil! shift the curve downwards, while a higher strain rate will shift it upwards. This is illustrated in Fig. 4, where it is also observed that the slope of the curves is hardly influ­enced, which indicates that it is determined mainly by the dynamics of the stretch descrih­ing parameter.

The liftoff heights obtained with the scalar d~ipation rate with Cx.1 = 423 do not agree Wlth the experimental results. Furthermore this large value of Cx,t is at varianee with the normally a~epted value of Cx.t = 2. The rea­son for this unacceptable discrepancy is the fact that in this work the coefficient is adapted, so as to reproduce the experimental liftoff height at a fuel exit velocity of 71 m;s. If the

. st~ndard model of the scalar dissipation rate Wlth C x.t = 2 was used the liftoff heights would

131

LIITED DIFFUSION FlA\.!ES

~r---~------------------------ï

10

o------~----~----------~----~ 30 .. 50 ao 70 ••

U (m/.tl

Fig. 3. Uftoff beight h as a tunetion of fuel exit velocity U obtained with the isothormal jet simwation with several different variables in the lift off condition. Drawn lino: 'Strain rate of the small eddies. Dasbed lino: Strain rate of the large eddies.. Broken line: Scalar dissipation rate with equal length scales. Dottod lino: Sc:alar dissipation rato with non equallength scales. E.xperiments of Wittmer [26!.

be very severely underpredicted. The slope of the h(U) curve obtained with the second model for the scalar dissipation rate, with Cx,z = 1.8 105 is underpredicted as well. Also in thls case the value of Cx.z is much larger than the value wbich was used in a previous study [22], namely cx.Z = 730. However, even with the higher values for cxl and cx.2 the slopes of the curves are unde;predicted. An underpre· dicted slope was a1so found by Peters and Williarns (5] (Fig. 5), who used the liftoff data of Horch [31] of a natura! gas jet flame. This

..

.. . 30 ..

2 20

10

• 30 .. 70 eo

u (01/•l

Fig. 4. Uftoff heigilt h as a function of Cue! exit velocity U obtalned with varying molecular visoosity in the st:rain rato of the smal! cddies. Drawn lino: Same as in Fig. 3 with • - "•· Dasbed linc: • - O.Sv0 • Dottod lino: • - 2 •o· For further information sec caption of Fig. 3.

9

••r-----------------------------, .. lO

20

10

oL-~----~--~--~-----------0 10 2:0 30 40 so ao 10 ac 90

!J{m/t)

Fig. S. Uftoff beight h as a tunetion of fuel exit velocity U. drawn line are data adapted from [5]. Triangles are experi­mental data of Hordt [31].

indicates that the scalar dissipation rate is not the correct variabie to use.

The fact that the liftoff heights in Fig. 5 are not severely underpredicted is partly due to the lower queuehing value of the scalar dissi­pation, namely xt = 5 s-I, instead of the cur­rent Xq = 48 s- . There is significant uncer­tainty about the exact value of Xq• and because this value mainly determines the base of the h(U) curve, most attention should be paid to the slope of the curve, and this is predicted well with the strain rate of the smallest eddies.

From the above it may be conc!uded that the strain rate of the small eddies is the relevant stretch parameter. However, because these cal­culations correspond to an isothermal flow without strong density variations the first model of the strain rate will be tested in a real flame .

Lifted Flames

The flame calculations are performed with the liftoff condition described by Eqs. 6 and 19 . The pdf of the liftoff height is used to calculate the scalar variables such as density and tem­perature, according to Eq. 9. The form of thls pdf is chosen to be a trlangle with a width of 5 diameters. This value is taken from Ref. 32, but its exact value only marginally influences the calculations.

The cambustion is modeled using the fiamelet concept, with two fiamelets. One is

-burning and experlences a strain rate of 100 s -l and the other is isothermal.

132

10 J. P. H. SA:...:DERS AND A. P. G. G. LA.VfERS

In Fig. 6 the liftoff height as a funcrion of exit velocity is given. The calculations corre­spond extremely well with the experimenrs. However, the coefficient C, 1 0.27 is larger than in the isothermal jet c<iJculations. This is due to the increase of molecular viscositv with temperature. 1}lereby the strain rate in Éq. 19 evaluated at f = [,, decreases sharply in the flame zone. To match the liftoff condition, the coefficient C,,1 has to be increased. This coef­ficient mainly determines the actual liftoff height but does not infiuence the slope of the curve and therefore this most important feature of the flame can be accurately pre­diered both with isothermal and non isothermal calculations.

It is noticable that Bradley et al. [9] with their mixedness-reactedness fiamelet model used a value for C,,1 of the same order of magnitude, namely C,,1 = 0.081. Given the fact that there is no universally accepted value for this rnadeling constant and that Bradley et al. [9l used a different fiamelet model, this correspondence is remarkable.

The prediedons of the a:tial velocity in the fiame are given in Fig. 7. The sudden onset of combustion, accompanied by the temperature rise, leads to an expansion effect that gener­ates the spike in the calculated velocity profile. This is not measured, but possibly the a.'tial distances between the measuring stations are too large to allow a detection of this son. The expansion-induced diEferenee between the isothermal and fiame calculation is visible downstream of the mean liftoff height.

~~----------------------------~

~ 20

10

.3~.-----~------~ •• ------~----~,.----~~ U (m/a)

Fig. 6. üftoff heigilt k as a function of fuel exit veloei tv U in a t!ame. The coefficient in Eq. 19 is C, - 0.27. Exp~ri­ments of Wittmer [26].

"r-----------------------------~

X/0

Fig. 7. Axial velocity in the f!ame éalculated (drawn line) and mea.sured by Wittmer [26]. Also given is the calculated wal velocity in the îsothennal jet (dashed line).

The temperature contours in the fiame with exit velocity 71 m;s are presented in Fig. 8. It can be seen. that the contours of maximum temperature show behavior, for instanee the ~ollow center of the fiame base [17, 33], which IS to be expected in a lifted flame. This figure can be ooropared with Fig. 4 given in Ref. 9. The main diEferenee is the axia1 position where the mean temperature starts to increase on the ~etry axis. In Fig. 8 this position is approx­JIDately equal to the mean liftoff height while ~ Fig. 4 in Ref. 9 this position is roughly 4

, omes the liftoff height. Because there are no experiments available, it is difficu!t to make a judgment about this discrepancy. How­~ver, due to streng turbulence mixing, it seems likely that heat generared at the mean flame position diffuses towards the axis more rapidly such as in Fig. 8. On the other hand if ~ velocity spike such as present in Fig. 7 is ~ot in agreement with experiments, a less streng tem­perature rise on the symmetry axis is to be expected, which could be in favor of the results presented in Ref. 9.

CONCLUSIONS AND DISCUSSION

Turbulent lifted natura! gas diffusion flames have successfully been modeled and calculated using a k-€ turbulence model and the laminar diffusion fiamelet concept. The most difficu!t aspect of the fiame, namely the liftoff height, has been calculated in the flame as well as in the equivalent isothermal jet. From these cal-

133

LIFTED DIFFUSION FLA..\.1ES

culations it can be inferred that the strain rate instead of the scalar dissipation rate is the relevant parameter to descnbe stretch effects in turbulent flames. The resulting curves of the liftoff height as a tunetion of fuel exit velocity h(U) are in good agreement with experiment if the strain rate of the smallest eddies is used, while the strain rate of the large eddies gives satisfactory results. The correct slope in both the isothermal jet as well as in the fiame is obtained, while the only coefficient in the model for the strain rate that has to be fitted fixes the absolute liftoff height at one fuel exit velocity. This coefficient has no generally ac­cepted value. In this study it also serves to take account of possible uncertainties in the numer­ical values of sq and of the percolation thresh­old P •. lts value is about 2 times larger in the fiame than in the isothermal case, which is due to temperature and density effects in the flame. The close agreement between isothermal and flame based h(U) curves, supports the condu­sion of Pitts [34] that isothermal mixing processes can be used to descnbe fiame stabilization mechanisms.

The calculation of the turbulent fiame cam­prises a model for the fiuctuating flame base. The scalar variables, such as densiry and tem­perature, are calculated using a probability density function for the liftoff height. The exact farm of this pdf being unknown a tri­angle pdf is proposed. Numerical variations of the width of this pdf showed no significant infiuence on the results.

The axial velocity profile in the fiame ex­hibits a spike as a consequence of expansion effects in the viciniry of the liftoff position. Experimental data do not show this spike, but it is unlmown whether it is just not resolved,

V/0 :I X.t)

Fig. 8. Mean temperarure oontoun for tbc 71-m/s flame.

11

or is a flaw in the proposed model for the fluctuating flame base.

Calculated temperature profiles show an earlier increase of the temperature on the symmetry a.'CÎ.S than do the temperature profiles of Bradley et al. [9], obtained with a flamelet model based on premi"<ed combustion. No judgment can be made because no tempera­ture measurements are available.

The major condusion of this work is that ie is possible to accurately calculate the liftoff height of a turbulent diffusion fiame with the laminar diffusion fiamelet concept if the strain rate instead of the normally used scalar dissi­pation rate is used. Furthermore, isothermal jet calculations do allow a correct slope of the h(U) curve to be determined, but the coeffi­cient in the strain rate then has a slightly different value compared v.ith the one in the fiame.

The awhor would like to thank Prof. N. Peters of the RWTH Aachen Germo.ny for kindly supply­ing the program with which the /aminar flamelet calculatiorzs were performed and fora stimulating discussion.

REFERENCES

l. Ka.lghatgi, G. T., Combust. Sû. TeelmoL 41:17-29 {!984).

2. Loclcwood, F. C., and Stolakis, P., in Turbuknt Shear FT(}Wf 0.. J. S. Bradbury, F. Durst, B. E. Launder, F. W. S<:hmidt, a.nd J. H. Whltelaw, Eds.), Springer, Ber!in, 1983, Vol. 4, p. 328-344.

3. Chen, J. Y., Combust. Flame 69:1-36 (1987). 4. Vanquickenborne, L., and van Tiggelen, A., Combust.

Flam11 10:.59-66 {1966). 5. Peters, N., and Williams, F. A., AIAA J. 24:23-429

{1983).

6. Pins, W. M., Twemy.Second Symposium (lntema· tiona/) 011 Combu:stion, The Cambustion Institute, Pinsburgh, 1988, pp. 809-816.

7. Liiián, A., Acta.Astro1!1:1Ul. 1:1007-1039 {1974). 8. Peten;, N., Prog. Eng. Combust. Sc< 10:.319-339 {1984). 9. Bradley, D., GaskeU, P. H., and Lau, A. K. C.,

Twenty-Third Symposium (lnterrwliona/) 011 Combu.r­lion, The Cambustion lnstitute, Pittsburgh, 1990, pp. 685-692.

10. Peten;, N., Combust. Sci. TechnoL 30:1-17 (1983). 11. Janicka, J., and Peters, N., Ni=Uenih Symposium

(lnumo.tiona/) on Combustion, The Cambustion Insti· tute, Pittsburgh, 1982, pp. 367-374.

12. Peters, N., Le<::ture notes of E.'Wftac summer school

134

J. P. H. SA.t'IDERS At'ID A. P. G. G. LA.t'v!ERS

on laminar and turbuient combustion~ Aachen, 199'2, pp. 156-166.

13. Peters, N., and Kee, R. J., Combusr. Fla.me 68:17-29 (1987).

1-'. Oixon-Lewis, G., David, T., Gaskell, P. H., Fukutani, S., Jinno, H., Millet, J. A, Koe, R. J., Smooke, !'.(. D., Peters, N., Effe!sberg, E., Wa:rnatz, I., and Bohrendt, F., Twenlieth Symposium. (JnLemationa/) on Combus­tion, The Cernbustion Institute, PittSburgh, 198-l, pp. 1893-1904.

15. Peters, N., and Rogg, B. (Eds.), Redu.ced Kin.etic Mechanisms for App/ications in Cambustion Systems, Springer, Berlin, 1993.

16. Abdei-Gayed, R. C. Bradley, 0., and Lau, A K. C., Twen.ty-Secand Symposium {Intemational) on Combw­tion, The Cernbustion lnstitute, Pittsburgh, 1988, pp. 731-738.

17. !Y!iake-Lye, R C., and Hammer, J. A, Tweniy-Second Symposium (International) on Combwtion, The Com­bustion lnstitute, Pittsburgh, 1988, pp. 817-824.

18. Peters, N., private communication, 1991. 19. Favre, A, in Problems of Hydrodynamia IVUi Contin­

uurn Mecha~ Sedov 60th birthday, SIAM, Philadel­phia, 1969.

20. Joncs, W. P., and Whitelaw, J. H., Cambwt. F!ame 48:1-26 (1982).

21. Jones, W. P., in p,.,d.iction Methods far Turbulent F!DWf (W. KaUmann, Ed.), Hemisphere, 1980, pp. 380-421.

22. Sanders, J. P. H., and Lamers, A~ P, G. G .• Int. Cammun. Heat Mass Trans[. 19:851-858 (1992).

23. Williams, F. A, Cambustion Theory, Addison Westey, Redwood City, 1985, pp. 73-75.

24. Liew, S. K., Bray, K. N. C, and Moss, J. B., Combust. Flame 56: !99-213 (1984).

25. Bllger, R. W., in Turbulent Reacting Flows (P. A. Libby and F. A Williams, Eds.), Springer, Berlin, 1980, p. 73.

26. Wittmer, V., Ph.D. thesis, UniversWit Karlsruhe, Germany, 1980.

27. Gasman, A 0., and Pun, W. M., Imperia! College Mechanica! Engineers Report HTS (73/2, 1973.

28. Puri, I. K., Seshadri, K., Smooke, M. D., and Keyes, ,D. E., Combwt. Sci. Techno/. 56:1-22 (1987).

29. Smooke, !YL D., J. Camp. Phys. 48:72-105 (1982). 30. Rogg, B., Report CUED/A-THERlv10/TR39, Cam­

bridge, 1992. 31. Horch, K., Pb.D. thesis, Universität Karlsruhe, Ger­

many, 1978. 32. Eiclchoff, H., Lenze, B., and Leuckel, W., Twenlieth

Symposiw-n {lntemtlliona/) on Cambustian, The Corn· bustion lnstitute, Pittsburgh, 1984, pp. 311-318.

33. Lin, C K., Jeng, M. S., and Chao, Y. C., Exp. Flu.ids 14:353-365 (1993).

34. Pitts, W. M., Combust. Flame 76:197-212 (1989).

Recewed 17 November 1992; reuiud 8 lultil 1993

135

136

Chapter 8

Conclusions

Turbulent free jet diffusion flarnes have been investigated theoretically and nurnerically within the framework of k c turbulence models and the laminar flarnelet concept of turbulent cornbustion. The cornputational rnethods developed for the numerical sirnu­lation of the turbulent flarnes has been based on the 'TEACH'-like rnethod for elliptic (recirculating) [GOS73] and parabalie flows. Separate programs have been written for flames of an elliptic nature (lifted flarnes) and jets and stabie jet flarnes which are of a parabolic nature.

Several physical phenomena have been addressed. The turbulent transport of heat and mass, i.e. scalar transport, has been investigated

by further developing a non-equal scales model for scalar transport, basedon the Two­Scale Direct Interaction Approximation (TSDIA) of Yoshizawa [YOSSSJ. As a result, predictions of the turbulent Schmidt number and the mechanical to scalar time scale ratio showed a good qualitative agreement with data from experiments as well as results of second-order turbulence models.

The flamelet concept provides the detailed chemica! cornposition and local flame extinction properties. The concentration of NO, which is one of the unwanted chemica\ species in flarnes, was found to be strongly dependent on the strain rate. This casts doubt on methods for the prediction of NO that do not take strain rate variations into account, such as the 'Sirnple Chernical Reaction Mechanisrn' (SCRS). The sealing of NO concentrat ion with Reynolds number observed in experirnents can he explained with the flamelet rnethod if the strain rate associated with the small eddies is used to quantify flame-stretch.

Differential diffusion effects have been found to be important in laminar diluted hy­drogen flamelets where detailed transport properties are used. Temperatures predicted in the flamelets, which include detailed transport properties, are much higher in the fuel-lean regions then when all diffusivities are equal, such as in the SCRS. However, in turbulent diluted hydragen flames differential diffusion is not important when the Reynolds number is sufficiently high. Therefore the mean flame temperatures are pre­dicted too high when laminar flamelet data, including detailed transport, are used to simulate the fuel-lean regions of a turbulent flan1e. When in these regions the strain rate is sufficiently low, such that equilibrium chemistry prevails, then a method in which equal species diffusivities are assumed, is more appropriate.

The effect of intermittency, which is the phenomenon that a turbulent flow is al­ternately laminar and turbulent, has been found to be of minor importance for the

137

predictionf) of mean quantities such as temperature and species concentrations. The type of averaging (Favre or conventional) has only a smal! effect on the predicted

sca.lar variables. The effect of radiation was found to be important in hydracarbon (methane or nat­

ura! gas) flames and has been attributed to the preserree of radiating soot particles. A first model to account for soot-radiation within the flamelet concept has been devel­oped which, in the experiment considered, gives reasanabie correlation of measured and predicted temperatures.

Liftoff is a typical and intensive form of local flame extinction. For natura! gas flames this phenomenon has been successfully modelled with the flamelet concept. The strain rate, rather than the scalar dissipation rate, was found to be the most important stretch descrihing parameter. Further evidence for this condusion has been given by examination of the extinction behaviour of diluted laminar flamelets.

Recommendations for future work

In some flame experiments considered in the present study laminarizing effects caused by the increase of molecular viscosity with temperature appeared to be present. Since turbulence models presently in use are not able to account for these effects the predicted mixing rate in the near field of the flame is higher than in those experiments. This calls for research on turbulence models capable of capturing these effects. Besides the empiricallow Reynolds number corrections to C I' and C2 proposed by Jones and Launder [JON72], the RNG model [YAK92] might be useful in this regard since it incorporates low Reynolds number effects.

In predictions for flames the mixture fraction decays too slowly in the far field (gen­erally for x/D > 80). This leads to an overprediction of the temperature in this region and also calls for more efforts in the field of turbulence modeHing within the k- ê model.

Some problems in the non-equal scales model for scalar transport, such as a geometry dependenee of two coefficients and one coefficient being not dimensionless, have to be resolved. A salution has been proposed in terms of a multi-scale turbulence model.

The fiamelet model can be extended by allowing for transient effects, so as to imprave predictions of CO mass fractions. Furthermore, the phenomenon of differential diffusion in laminar flamelets and in turbulent flames deserves more attention. In partienlar investigations on ·the applicability of flamelets in turbulent flames, regarding differential diffusion effects, could be continued.

The effect of radiation in hydracarbon flames can only be modelled if the soot volume fraction is predicted. The model ihat is used in the present study is a first attempt within the flamelet concept and it could be improved by using the two-equation model proposed by Moss et al. [MOS88], for instance.

138

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148

Appendix A

Favre-averaging

Both the conventional averaged and Favre-averaged valnes are independent of time for times [t- ~T, t + ~T]. This gives, for instance,

and

Fairly obvious relations such as

=4>+1/J a- fj_ -if>=-4> ax fJx

can be derived easily from the definition of 7f;. Two important relations are:

if>l/J 1>-;j;+if>'l/J' + 4>"1/J".

The latter can be derived by writing </;1/J as p</>1/J/75 and using

pif>" 0

(A. I)

(A.2)

(A.3)

(AA)

(A.5)

Eq. (A.5) follows from pif> li'avre-average, i.e., J; = pif>lp.

+ pif>" = p;j; + pif>" and applying the definition of a

The remaing relations in Eq. (2.10) are now derived.

=</;- =1>-4> (A.6)

and because a Favre-average is time-independent this is

= -;p- J;. (A.7)

The time-average of the Favre-fluctuation </;" can also be expressed in density correlations in the following manner. Starting from Eq. (A.7) we have

prf; p

Now decornposing pif> into (p + p')( J; + </>") this leads to

149

(A.8)

(A.9)

which, with Eq. (A.7) leads to

<P" =

On the other hand, if p</J is decomposed into (p + p')(4i + </J') one finds

- 1 - -<P- =(fi<P + p'</J') =

p =- pif/

p p

(A.lO)

(A.ll)

The relation between Favre-averaged Reynolds-stresses and conventionally averaged Reynolds-stresses is a little more complicated. The Reynolds-stress pu:uj is equal to

With the notion that

Pu'u'. = • J

u;= û; +u?

where equation (A. 7) is used,the first term on the r.h.s. is rewritten as

P'u'u'- = ' J

= p'u"u" - p'u" u" 1 ) t J

(A.l2)

(A.13)

(A.l4)

The following expression for the Reynolds-stress on the !.h.s. of equation (A.l2) can also he derived

pu~uj p(u?- = pu"u'! + -pu'! u" t J t J (A.l5)

where use is made of p</J" 0. Substituting these expressions for the !.h.s. and first term on the r.h.s. of equation (A.l2) leads to

Pu 11u 11 + ~pu" u" p'u~'u"- p'u~' 1 J t J 1 J 1 P'u" u('+ -pu'u'­

J 1. • r

In this equation the first term is the Favre-averaged Reynolds-stress

Pu"u'! = p-u"u" t J : J.

If also the relation p'u:' -pu;' is used then the desired equation is obtained:

- -pu~' u'! ' J.

150

(A.l6)

(A.l7)

(A.18)

Appendix B

Numerical Methods

B.l Finite Volume Method

The Finite Volume Method (FVM), which is used by practicaHy all investigators working on numerical modeHing of turbulent flames 1

, is described extensively in the liter at ure [PAT80) and only a briefdescription is presented in this appendix.

The elliptic computer-code used in the present study is basedon the 'TEACH' code [GOS73), developed at Imperia! College in London. Versionsof this program are widely used by researchers on fluid dynamics [LIL82, CHE87, SPE87, FAI91, HWA93). The program has been rewritten and extended during the present investigation with fea­tures related to the incorporation of the jet, several turbulence models, scalar transport rnodels, the therrno-chemical part needed for the turbulent flame computations and var­ious in- and output utilities. The parabalie version differs from the elliptic one in the definition of the grid and in the thin shear flow approximation.

B.l.l General equations

A general form of the axisymmetric2 transport equations is

a. ) 1o( ) -z.;-(pur/J + -~ rpvr/J ux rur

(B.l)

where rP is the quantity transported, r is a generalized exchange coefficient and I) is a so-called souree term. The transport equations are integrated over the volume of the flow. The flow-domain is divided into 'patched' control volumes, the facesof which are orthogonal toeach other, see Fig. B.l. The boundaries of the control volumes coincide with the physical boundaries on the flow.

A balance equation for the convective-diffusive fluxes and souree term, after integra­tion of Eq. (B.l), is obtained as

(PeUerPe PwUwrPw) X r8r8c.p t (rnPnVnrPn r,p,vs!f>s) X 8x8c.p (B.2) orfJ orfJ orfJ or/J

(fe0

x le-fw 0x !w)xr8r8c.pt(rnfn0

r ln-rsf•ar !.)x8x8rp

+ll>p x r8x8r8c.p

1See for instanee the papers on numerical methods for flames in all proceedings of the Gombustion Symposia.

2The cartesian equations are obtaîned by substîtuting r l.

151

/ l...--L--"'--t-"----"--"

Control volume s

Syrrmetry axis

Figure B.l: Typical control volume (shaded) around gridpoint P with its surrounding gridpoints (E, W,N,S),

where liep is the azimuthal angle, the lower-case subscripts e, w, n, s denote evaluation at the control volume-faces, and rtixtirticp is the volurne tiV of the control volurne considered. This equation is discretized by evaluating the quantities on the control volume faces employing linear interpolation, assuming that the grid is equidistant or nearly equidistant

1 'I/Je 2( of'P +'I/JE) (B.3)

1 1/J,. = 2( 1/Jp + 1/JJV)

where 1jJ is p, u, v, r or rf; and the upper-case subscripts denote the gridpoints. The gradients in the diffusive terms are discretized by a second-order scheme:

81/J Ie= rPE - rPP ax lix. ärf; ar

1/Jw äx lixw ärf; I _ rPP 1/Js är ,- tir,

The discretized equation to be solved is

with Ap AErPE + Awr/Jw + ANrPN + Asr/Js + <l>ptiV

AE + Aw + AN + As.

(BA)

(B.5)

The coefficient Ak represent the influence of the value of rPk at the corresponding grid­point through convective and diffusive effects. The souree term <l>p in general is a

152

non-linear function of </> and is linearized3 as i.Pp Eq. (B.6) becomes

Su + Sp</>p. With this procedure

(Ap- Sp )</>P L Ano</>nb + Su. (B.6) nb

Here, the subscript nb denotes 'neighbours' which means E, W, N or S. In the present study the so-called 'hybrid-scheme' [PAT80] is used for the coefficients Anb· This is first order or second order accurate, depending on the local cell Reynolds ( or Peelet) number puóx/f:

1 1 AE = max(2 I Ce I, De) 2Ce

1 1 Aw = max(21 Cw I,Dw) + 2Cw (B.7)

1 1 AN max(21 Cn I,Dn)- 2Cn

1 1 As= max(

21 Cs j,D.) + 2C•

with Ap 'EAno nb

where the convection and diffusion coefficients are given by Ce = PeUerÓrÓ<p and De = (f e/ Óxe)róró<p and similar expressions for the other coefficients. When the local cell Reynolds number is larger than 2, diffusive fluxes are neglected. This procedure is known as 'upwind' differencing and it is used to stabilize the numerical algorithm. It is based on clear physical reasoning: when convection is very strong, the upstream cells will experience no diffusive influence from downstrearn cells. The resulting first order accuracy presents no problem when the grid spacing is smal! enough. Stabilization is also enhanced when all coefficients A are positive, and therefore the coefficient Sp should always be negative.

The equations of the type of Eq. (B.6) are solved by the Tri-Diagonai-Matrix­Aigorithrn (TDMA) [PRE92] which only involves the solution of a set of algebraic equa­tions by successive substitutions. To prevent divergence of the coupled set of equations, an under-relaxation metbod is employed.

Boundary conditions

The computational domain consists of 'live' gridpoints where the solution is cornputed. The boundary ('ghost') gridpoints are located just outside the physical domain such that the physical boundary is midway between the ghost gridpoints and the live gridpoints just inside the physical dornain. The ghost gridpoints are used to set boundary con­ditions. These can be 'Dirichlet' (fixed value) or 'Neurnann' (zero-gradient) boundary conditions. The former are implemented by setting the value at the boundary point to the boundary condition and applying a sufficiently strong convective flux. The latter are imposed by making the corresponding coefficient -A- zero, thereby breaking the conneetion between the live gridpoint and the boundary gridpoint.

Continuity equation and pressure

The density is either obtained frorn the fiamelet rnethod (if the mixture fraction and the strain rate are given) or from the perfect gas law if the SCRS is used. In the latter case the temper at ure is obtained frorn the mixture fraction.

3 Due to this linearization it is necessary to solve the set of equations in an iterative manner.

153

Since a 'pressure-equation' does not exist, and the continuity equation is an addi­tional equation which the computed veloeities must obey, the continuity equation may be used to obtain the pressure. In the present study the so-called 'SIMPLE' (Semi­lmplicit-Method for-Pressure-Linked-Equations) methad [PA TSO] is used to secure mass­conservation.

The veloeities u and v are decomposed into the computed veloeities (u* and v*) and velocity corrections ( du and dv). The same is clone for the pressure (p* and p'). Si nee the computed veloeities and pressure (u*, v• and p*) obey the discretized momenturn equations, equations relating du to p' and dv to p' can be found. Using the continuity equation, a pressure correction equation (for p') is obtained [PA TSO]. This p' (or conti­nuity) equation contains only convection effects and a souree term which represents an effective mass-deficit4 •

The order in which the equations in the 'SIMPLE' method are solved is:

L Compute u•, v• from the momenturn equations with a guessed pressure and density field.

2. Compute p' from the pressure-correction equation.

3. Update the pressure with p'.

4. Update the velocity with the velocity-corrections du and dv (obtained from u•, v• and p* equations).

5. Compute scalar variables such as k, t:, J, g, t:9 and {i.

6. Return to 1, until convergence is achieved.

B.1.2 Elliptic metbod

In the present study the elliptic method camprises the solution of all differential equa­tions, including pressure gradients and axial diffusion, on a rectangular grid (Fig. B.2). The grid is staggered for the axial and radial velocities, i.e., veloeities are computed in

Figure B.2: Reetangu/ar grid for !he elliptic mcthod. Only the gridlines for scalar vari­ables just inside and outside of the computational dom a in are .shown, including the ghost gridpoints.

4 This rnass-deficit îs 0 when the converged solution is obtained.

154

different control volumes than all other variables. This is because veloeities are needed to evaluate convective fluxes on the control volume-faces. Therefore the veloeities are computed on the control volume-faces of the scalar variables. The velocity gridpoints are located midway between the scalar gridpoints. The velocity control-volume faces are midway between the velocity gridpoints. Schematic representations of this staggered grid for the scalar variables and the axial and radial veloeities are given Figs. B.3 and B.4, respectively.

Figure B.3: Contr-ol volumes for- the axial velocity in the elliptic method.

Figure B.4: Control volume for the ra­dial velocity in the elliptic method. The ghost veloeities are located on the symme­try axis.

The grid is most dense near the nozzle (located bottom-left) and expands in axial and radial direction with an expansion factor smaller than 1.1. Boundary conditions at the nozzle are given by the 'initia!' profiles of velocity, turbulent energy and dissipation. The mixture fraction in the nozzle equals 1 and its fluctuation is 0. At the upper left si de ( entrance of coflowing air) the axial velocity equals the coflow velo city, the radial velocity is 0 and turbulence quantities k, e:, g and e:9 are 0. At the radial (free) boundary the axial velocity equals the coflow velocity; turbulence quantities are 0, and the radial velocity is computed from the continuity equation. The pressure correction is set to 0 at all boundaries. At the downstream boundary of the flow it is assumed that there is no recirculation and that convection outweighs diffusion: all coefficients Ae are 0, i.e., downstream gridpoints just outside the computational domain do not influence the upstream variables. It is noted that this boundary condition is equivalent to the para.bolized method. The coefficients A in the pressure correction equation are 0 at all boundaries.

The convergence criterion is that the sum of all control volume residues5 in the flow domain is less than about 10-3 .

the residue is defined in terms of Eq. (B.6).

155

B.1.3 Parabolized metbod

In the parabolized method axial ditfusion is neglected and pressure gradients in the momenturn equations are not considered in the present study. The neglect of axial ditfusion gives Ae = 0 and Aw PwUwrl5r8r.p. The grid is staggered for the radial velocity, while the axial velocity is computed in the same control volumes as the scalar quantities (Fig. B..5). In the parabolized method only two gridlines are considered during

Downstreem gridline

Scalarand

~~~•axral velocity control volume

Radial veloc1ty control volume

Figure B.5: Control volumes for scalar variables and veloeities in the parabolized method.

the computation: the current ( downstream) gridline and the upstream (left) gridline. A typical set of two gridlines including control volumes is shown in Fig. B.5. The grid expands in the radial direction in order to capture the expanding jet, which has been exaggerated in Fig. B.5 for clarity.

The expansion of the grid in axial direction is related to the spreading of the jet: 15x = stepwidth x y112 , where stepwidth is typically between 0.01 and 0.05. The radial expansion, gridspreadingratex/5x is so smal! that it does not affect the assumption of an orthogonal grid.

The convergence criterion is that computed variables at the current gridline do not change by more than a factor 10-6

, which usually amounts to about 3 iterations per gridline.

Boundary conditions at the nozzle and the coflow entrance are similar to those in the elliptic method while the boundary conditions at the free boundary are the same for all variables except for the radial velocity for which a no-gra.dient condition is applied.

B.1.4 Computing times

The elliptic method requires more than 10 to 20 times ( depending on the details of the computa.tion) more computing time as the parabolized method. Typical comput­ing times for the parabolized meibod on a 486 dx 33 MHz pc are: 5 rninutes for an isotherrnal jet and about 1 hour for a turbulent ftame computation with 10 flamelets.

156

The elliptic calculations have been performed on a Silicon Grapbics workstation, which is approximately 2 times faster than the pc.

B.2 Integration of the ,8-function

The numerical evaluation of the integral

(i> = l4>(J)P(J)df (B.8)

with r-1 (1 - nb-1 P(J) = !3( a, b) (B.9)

is considered. The beta-function f3(a, b) is defined as

f3(a, b) l r-1(1- nb-1df. (B.lO)

The coefficients a and b are determined from f and g by a "{ 1 and b ?(1 !) with 'Y = f ( 1 - !) / g 1. The function 4>(!) is supplied by a discrete set of points 4>(!;) with i = 1, N;. The most important condition to be fulfilled is normalization: J~ P(f)df = L For all variables 4>(!), varying sufficienty smooth with J, the precise form of P(f) (whether it is chosen to be a Gaussian or a beta-function) in Eq. (B.8) is unimportant [BILSO]. All variables of interest in the present study have this property. Therefore the condition of the normalization is far more important than the accuracy with which Eq. (B.S) is evaluated. To guarantee the normalization, the following metbod is used. The values of P(J;) are multiplied with the corresponding mixture fraction intervals 8 J; and subsequently are summed to yield

N;

s L:!t-1(1- Mb-18k (B.ll) i=l

This quantity now serves to normalize the numerically evaluated integral

1> ~ È4>(/ï)j;"-1(1 !i)b-18/ï 1=1

(B.12)

such that the normalization is satisfied. At the boundaries (i = 1 or i N;) the pdf P(f;) can become infinite when a < 1

orb< 1, respectively, in which case the integral is clone 'analytically'. When a< 1, the first term in the series of Eq. (B.ll) is approximated by writing the function (1 nb-1 in a 6-term Taylor series around f = 0 and integrating the resulting polynomial in f analytically. The same procedure is foliowed for an infinity at f 1 with a 6-term Taylor series for r-1 around f 1.

The numerical evaluation of the integral is performed in this way, except in the following situations. When the scalar varianee g is smaller than 10-7

, the beta-function is replaced by a delta fundion at f = 1 and (i> is obtained by linear interpolation between 4>(M and 4>(/ï+l)· When 1 < w-4 or 1 1 < w-4 the pdf is replaced with a delta function at f = 0 or f 1, respectively. Finally, when 7(1-7)/g < 1.0001, the scalar fluctuations are so large that the pdf is replaced with a delta function at f = 0 and a delta function at f = 1, their strengtbs given by a and b, respectively.

157

B.3 Newton iteration method

B.3.1 General description

The boundary value problem of the lamina.r f:lamelets described by Eqs ( 4.14,4.15,4.16 and 4.17) are solved with a computer program which is a precursor of the Cambridge code RUNJDL written by Rogg [ROG92]. lt was modilied in the group of professor Peters in Aachen, from whom it was obtained during this study. In the present study some additional effects such as thermal radlation and the correction velocity 1/" have been added. The numerical metbod is described in [SM082, SM083a, SM083b] and only a very brief description is given in this appendix.

The problem is essentially to find the salution of N + 3 non-linear differentlal equa­tions, with N denoting the number of chemica] species, of which only N 1 species are calculated. The (inert) nitrogen (N2 ) mass fraction equals 1 minus all other concen­trations. The other equations are the continuity, momentum, temperature and mixture fraction equations.

The stagnation point streamline is represented by M gridpoints ry[k], k = 1, ... , M and as an example the finite difference equation for the mass fraction Y; is

F[k] = Y;[k]- }~O[k] bt

2 ~] (B.l3)

where F[k] is the residual at gridpoint k, Y;0 is the mass fraction on the previous time step and bt is the time increase. Time derivatives are retained because time steps are used when the steady state salution cannot be obtained directly. From Eq. (B.13) it is seen that the diffusive terms are discretized second-order accurate. The convective terrns are discretized using the up wind scheme (first order accurate). This means that -</>[kj~ in Eq. (B.l3) is discretized as

_</>[kj dYi dTf {

-"(kjYi[kH]-Y;[k] '+' 'l[k+l]-lJ[k]

- 4> [kJ '--'f.+-'7F-:T"

if V< 0

if V> 0 (B.14)

The variables are ordered in a vector \IJ with M x (N + 3) elements, such that the first N +3 elernents correspond to the variables at the first gridpoint etcetera. The finite difference problern can then be written in terrns of this salution vector \IJ, the Jacobian J and the residue F a<>

(B.l5)

where J represents the change of the residual vector F due to a srnall perturbation in the salution vector \IJ, i.e.,

(B.16)

Here i and j denote the ith and jth element of the salution vector and residual vector, respectively. Since the finite difference equation only couples the salution in a gridpoint k to the gridpoints k + l and k 1, the Jacobian is aso-called block tridiagonal matrix.

1.58

This can be solved by a block-TDMA method which is equivalent to the scalar TDMA solution procedure [PRE92].

In order to stabilize the process, an under-relaxation method is employed by multi­plying the residual vector F in Eq. (B.l5) with a parameter À, which is smaller than 1:

(B.17)

Steady state cakulations are always started with À 1 and if the process does not converge À is halved until convergence is reached. Sirree most of the computing time is consumed by the (numerical) computation of the Jacobian J, the computational step of Eq. (B.l7) is repeated (without recomputing J) until the difference between pn+l and pn becomes too large. In that case a new Jacobian is cakulated and this process is repeated until convergence is achieved [SM082]. When a steady state solution cannot be obtained on a certain grid, time steps are required. After the time stepping procedure has successfully been completed the steady state calculation is started again.

The grid is adjusted dynamically in order to locate enough gridpoints near the flarne zone. Each time a steady state solution has been obtained, 'convergence' of the grid is checked and if necessary the grid is adjusted after which the steady state calculations are started again. This sequence is repeated until the location of the gridpoints satisfies the 'grid-tolerance' which is related to the nurnber of gridpoints in regions with large gradients and curvatures of the solution. The details are not discussed here.

B.3.2 Computational details

Computational details mainly concern the grid and solution procedure parameters.

The grid

A number of about 100 gridpoints suillees for an accurate calculation. These gridpoints are located between the left and right boundaries, i.e., in terms of the dimensionless coordinate ry, 17 E [T/L, T/R]· The numerical values for these parameters are about -2.5 (oxidizer side) and +1.2 (fuel side), respectively, but they may depend on the problem considered. When convergence cannot be achieved on a certain grid, it may be necessary to adjust these values. In order to prevent the location of too many gridpoints near these boundaries, minimum distances between the boundary gridpoint and the next gridpoint can be applied.

The final solution can be inspected to verify whether the grid is indeed dense in the appropriate regions. The numerical criterion whether the grid is 'converged', i.e., is dense enough in the appropriate regions, can he controlled by a number called the grid-tolerance.

The convergence criterion that is used to decide whether a steady state solution is converged is based on the maximumnorm of the residual vector F. It usually is taken to be w-s or 10-6

Initia! salution

It is very important to start a computation on a sufficiently good guess of the solution, otherwise convergence may not be achieved. This is especially important when the

159

boundary conditions orthestrain rate ae are modified. The changes in these parameters should be made in relatively small steps.

160

Appendix C

Two-Scale-Direct-Interaction­A pproximation

The Two-Scale-Direct-Interaction-Approximation (TSDIA) is an extension of the Direct­Interaction-Approximation (DIA) of Kraichnan [KRA59, KRA64] which is a method to evaluate two-time-two-point covariances and is valid in principle for homogeneous tur­bulence. The extended Two-Scale formalism developed by Yoshizawa [YOS83, YOS84b, YOS84a] takes account of the influence of the (inhomogeneities of the) mean field. In this extension a scale expansion separates the slow varlation of the mean field and the fast varlation of the fluctuating field.

A useful result is Eq. (2.50) for the scalar dissipation rate. In the following the derivation of this equation is presented following the method of the TSDIA.

The equation for the fluctuating mixture fraction, without spatial variations of D is

(C.1)

Now a scale separation with the scale parameter ó is introduced [NAY73], which leads to the decomposition of the instantaneous variabief into the slow (mean) part ](X, T) and the fluctuating part f'(x,X;t,T) [YOS82)

f(x,X;t,T)

u(x,X;t,T)

](X,T)+ f'(x,X;t,T)

u( X, T) + u'(x, X; t, T).

(C.2)

(C.3)

The number of independent variables is increased by the following definition of the derivatives [NAY73]

i)

a x i)

Equation (C.l) is rewritten up to order ó as

0 f' - 0 f' 0 I I 0 2 f' <lt + u1 -;;-- + -;;--( uJ) D~ = u ux1 ux1 ux1

8[ af' - af' I a1 a ( ~~~ '!') 2D a2f' l -aT Uj i)Xj - uj OXj - axj uj +ui + OXjOXj .

161

(CA)

(C.5)

(C.6)

In this equation the terms a~ i ujf' and 2D a:;K, can he neglected because they are & ( _!!!_]_'_ . much smaller than ax, ujf') and D &x,&xi, respectJvely.

Equation (C.6) will now he written in Fourier space with respect to the fast space coordinate x, while explicitly retaining the inhomogeneities in the mean field with respect to derivatives in the slow space coordinate X. To this end the fluctuating mixture fraction J'(x,X;t,T) is written as

J'(x, i; t, T) 1 J'( K, X; t, T) exp( -iK. x) (C.7)

where the wavenumber is denoted by K.. With this definition any derivative with respect to Xj is replaced by a multiplication with -iKj and Eq. (C.6) becomes

:/'(ft; t)- iK/flJf'(ft; t)-

iK3 i i ti(R p- ij)uj(p; t)J'(q; t) + DK2 J'(ié; t)

( Dj'(K.;t)

ti - DT uj(.ii:;t)aa!- LLti(.? J p q

(C.8)

where is the material derivative + and ti(.? - p- ij) is the Dirac delta­

function. It is possible to remove the convection effect by the large eddies iK/fid'(i?;t) from the equations by the transformation t) exp(iKjfijt)f(R.; t) [YOS84b]. The material derivatives then are defined as = exp( -iKjUjl) gT exp( iKjUjl) [YOS84b], which, however, does not make any difference for the final results. Therefore, in the following this convection effect is simply neglected while retaining the original material derivatives.

The fluctuating field J'(i?; t) is expanded around an isotropie basic field Jo( i?; t)

= J'( il; t) = (C.9)

n=O

and is inserted into Eq. (C.8). This leadstoorder 5° to (the r.h.s. vanishes)

:/~(il; t) + DK2 J~(il; t) - iKJ i i ti(il p- ij)uj,0 (p; t)f~(if; t) 0. ( C.IO)

To order ti this leads to

:/: ( il; t) + DK2 J; (R; t) i Kj i i ti(K.- p- ij)uj,0 (p; t)J: (ij; t)

r r <~ ~ ;;'\, l"(~ l aJ~(.?;t) _ '· <~·t) ay iKj }p},/' K p-q)uj,l(p;t JO q;t f)T U1,o K, f)XJ

if/(il p- ij)a~j uj,o(p; t)f~(if; t). (C.ll)

The equations for J~(K.; t) and f{ ( K.; t) can be written in closed form using the Greens' fundion Ge(K.; t, t') for Eq. (C.lO). This Greens' fundion is the solution of Eq. (C.lO) with a delta-fundion on the right-hand-side:

:tGe(il;t,t')+DK2Go(.?;t,t')- iiti(.? p-ij)uj,0Ge(K.;t,t') ti(t-t'). (C.l2)

162

The solution for J~(ii;t) then is

JH ii; t) i Kj!; i~ 6( ii p- ij) [,>0 dt1 Ge("ii; t, tr)uJ,1(p; h)f~(ij; t1)

lt dttGo(ïU,tt)uj,o(ii;tt)ààXJ lt dttGo("ii;t,tt)DD J~(ii;t1)--oo 1 -oo T

kfrti("K-fi ij)[,"" dttGe(ii;t,tt)a~/u1,o(fi;t!)f~(if;tt)). (C.l3)

Now an equation for g will be derived, basedon the expansion Eq. (C.9).

9 < !~!~ > +ti(< !~!~ > + < JU~ >) + 62(2 <JU~ > + < !~!~ > + < I~ I~ >) + order(63

)

where the brackets indicate an average over the waverrumher i!:

< !~!~ >= h < f~(ft;t)f~(-ii;t) > /6(0).

Now the scalar correlation function Qo (i; t, t') is defined as

Qo(ft; t, t') =< f~(ii; t)f~( -ii; t') > /8(0)

(C.l4)

(C.l5)

(C.l6)

where the brackets denote an ensemble average without averaging over K-. The delta function 6(0) appears because in fact the expression reads Qe("K;t,t')ti("ii + "ii') =< !~(ft; t)JMK,'; t') >. The basic field of order 6° experiences the least influence of the inho­mogeneities of the mean field while higher order terms are successively more sensitive to these inhomogeneities which are quantified by mean field gradients, see Eq.'s (C.lO) and (C.ll). For this reasou the basic field of order tJO is assumed to be isotropie [YOS84b]:

Qe(ii; t, t')

Ge(K,;t,t') Qe(K; t, t')

Ge(K; t, t'). (C.l7)

(C.l8)

A further simplification is obtained if the turbulence spectrum is in equilibrium and inertial range forms are applied [YOS84b]:

Qo(K;t,t') cre(K)exp(-w(K) I t t' I) Ge(K; t, t') = H(t- t')exp( -we(K)(t t'))

(C.l9)

(C.20)

with H(t- t') the Heavyside step function which is 0 for t < t' and 1 otherwise. The forms for w,we andere are [YOS84b]

w(K) ~ we(K) ~ EJ/3K213

cre(K) ~ Eg€-1/3,.-11/3.

(C.21)

(C.22)

The scalar varianee is now approximated by truncating the expansion (C.l4) at order 8, noticing that terms involving < u~f~ > and < uUö > are 0 [YOS83], and using Eq.'s (C.l3) and (C.l6):

(C.23)

163

Inserting the inertial range forms for Qo and Go this leads to an expression for the varianee g in terms of i 1, ê 9 and ê:

g

(C.24)

which gives exactly Eq. (2.38) in chapter 2

(C.25)

Demanding transferability, narnely i1 r-v g312ê 112ê;312, then gives the equation for E:9 ,

Eq. (2.50).

164

Appendix D

Transport Properties and Reaction Mechanism

D.l Diffusion veloeities

The approximation for the concentration ditfusion velocity reacis

D· ~ ---'-'V X xi ' (0.1)

in which Xi is the rnalar fraction of species i and D, is the mixture averaged ditfusion coefficient given in section 0.2

The approximation for the thermal ditfusion velocity reacis

- DT VT pY;T

with the thermal ditfusion coefficient nr (kg m-1 s-1 ) evaluated as:

(0.2)

(0.3)

in which M is the molar mass of the mixture M =

formula [KEE86],[CHA 70]. M,X, and RT is an empirica!

with

{

0.015 0.225

RT = 0.375 0.520 0.6

+(t 1)0.3 +(t 1.7)0.188 +(t- 2.5)0.097 +(t ·- 4.0)0.0264

t T

165

if tE [1; 1.7) if t E [1.7; 2.5] if t E [2.5; 4.0} if t E [4.0; 6.5] if t E [6.5; oo]

(0.4)

(0.5)

in which é( i) is the Lennard-Jones potential parameter f for species i in Kelvin. The values of this parameter for different species can be found in Table Bl of [BIR60] and [HIR54].

The thermal ditfusion coefficients already existed in the flamelet program, but the inclusion of the thermal ditfusion velocity was implemented in the present study.

D.2 Thermodynamic and transport properties

The mixture averaged thermodynamic and transport properties are given by the follow­ing equations. The mixture enthalpy is giveu by

N

h = .Z:.:h;Y; (D.6) i=l

The mixture specific heat is giveu by Eq. (1.11). The mixture heat couductivity is given by

1 ( N 1 ) À = 2 I: X; À; + r_N &

t:;::;;l t=l Ài

(D.7)

The mixture averaged diffusivity of species i is given by the Curtiss-Hirschfelder approx­imatiou

r_N c!.t_ j# v,.

D; 1- Y;

(D.8)

where Vji is the binary ditfusion coefficient of species j in species i or vice versa. The binary ditfusion coefficieuts are giveu by

4

(D.9)

The single component viscosity, couductivity and enthalpy are giveu by polyuomial expansious in the temperature:

4

In l'i = I: ak,i{In T)k-l (D.lO) k=l

4

In>..;= (D.ll)

(D.l2)

In the data file, accompanying the laminar flamelet programme, which contains the polynomial coefficients, the enthalpy coefficients cornprise 7 numbers. The seventh is the entropy, which is not needed in the calculations in this thesis. The specific heat for the individual speciescan be extracted frorn the enthalpy data using the calorie equatiou of state

(D.l3)

166

so dh;

= Cp,i(T) (D.14)

Here h? is the standard heat of formation; the negative of the heat released when the species i is formed from its elements. It is given at standard conditions of 1 atmosphere and 298!<.

Many of the coefficients mentioned in this appendix can be found in NASA reports [GOR71] or [KEE90], which partly use the JANAF tables [JAN71].

D.3 Reaction mechanism

The reaction mechanism for hydrogen oxygen is given in Table D.l, induding the NO production mechanism. The NO mechanism is not important with ~espect to the energy or temperature equation; the concentrations involved are very low. The third body coefficients used are given in Table D.4.

The chemical production rate of species i, m; in mass per unit volume per secoud is given by [WIL85]

(D.l5)

in which M. is the number of elementary reactions, 11r k is the stoichiometrie coefficient on the left hand side and <k is the coefficient on the' right-hand-side. Note that w; is the molar production rate from Eq. (4.16). Ru is the universa! gas constant with the value 8313.3J / (kmole K). The reaction rate kk of the k-th reaction is given by

(D.I6)

with Bk, ak and the activation energy Ek given in Table D.I. The activation energy is given in J /kmole. Sametimes the coefficients in these rate expressions are given in different units, for example kcal/mole.

The dirneusion of the frequency factor B follows from the dimensions of all other quantities appearing in Eq. (D.l5).

167

No. RP.:.rt.inn E a B

Hz 02 cha.in reactions 1 H + 02 ---> OH +0 2.000. 1014 0.00 70.300 2 OH+ 0 ---> H + 02 1.568. 1013 0.00 3.520 3 H2 + 0 OH +H 5.060 ·104 2.67 26.300 4 OH+ H ---> H2 + 0 2.222 ·104 2.67 18.290 5 H2 +OH ---+ JI20 + H 1.000. 108 1.60 13.080 6 H20 + H ---+ fh +OH 4.312. 108 1.60 76.460 7 OH+ OH -t Jl20 + 0 1.500. 109 1.14 0.420 8 }[20 + 0 --> OH+ OH 1.473. 1010 1.14 71.090

Jl 02 formation and consumption 9 Jl + Oz+ M' JI02 + M' 2.300. 101 ~ -0.80 0.000 10 JI02 + M' -t H +02 +M' 3.190. 1018 -0.80 195.390 11 H02 + H -t OH+ OH 1.500. 1014 0.00 4.200 12 H02 + H ---> }[2 + 02 2.500. 1013 0.00 2.900 13 H02 +OH !hO+ 02 6.000. 1013 0.00 0.000 14 Jl02 + }[ -> }[20 + 0 3.000. 1013 0.00 7.200

13 -15 H02+0 1.800 10 0.00 1.700 I Jl202 formation and consumption

16 HOz + H02 ..... H202 + 02 2.500. 1011 0.00 -5.200 17 OH+ OH +M' ..... fh02 + M' 3.250. 1022 -2.00 0.000 18 Hz02 + M' --> OH+ OH+ M' 1.692. 1024 -2.00 202.29 19 IhOz + H --+ H20 +OH 1.000. 1013 0.00 15.000

i 20 Ilz02 +OH --+ ll20 + H02 5.400. 1012 0.00 4.200 1 21 Il20 + Jl02 --+ HzOz +OH 1.802. 1013 0.00 134.750

Radkal recombination I 22 H + ll + M' --+ H2 + M' 1.800 · 10H' -1.00 0.000 i 23 OH+ Jl + M' --+ H20 + M' 2.200. 1022 -2.00 0.000 . 24 0 + 0 + M' ---+ 02 + M' 2.900. 1017 -1.00 0.000

NO production

125 Nz + 0 --+ NO+ N 1.900. 1014 0.00 319.300 . 26 NO+ N --+ N2 + 0 4.22047. 1013 0.00 4.253 i 27 NO+ H N +OH 1.300. 1014 0.00 205.850

128 N +OH -+ NO+ JI 4.80384. 1013 0.00 5.270

29 NO+O --+ N + Oz 2.400. 109 1.00 161.670 30 N + Oz --+ NO+O 1.13052. 1010 1.00 27.838 i

Tàble D.l: The hydragen reaction mechanism, including the NO schemc, used in the flamelet calculations. Most of the data are taken fmm Ref. {PET.93}.

168

No. ·~~~· "" E a B

•

Cl mechanism

co- co2 reactions

• 31 CO+ OH -+ co2 + H 4.400. 106 1.50 -3.100 .

1 32 COz + H -+ CO +OH 4.956. 108 1.50 89.760 C H consurnption

i 33 CH + Oz -+ HCO +0 3.000 ·1013 0.00 0.000! 34 CH + co2 -+ HCO +CO 3.400. 1012 0.00 2.900 i

HCO consumption 35 HCO +H -+ CO+ Hz 1 2.ooo . 1014 o.oo 0.000 36 HCO +OH -+ CO+ ll20 : 1.000. 1014 0.00 0.000 37 HCO + 02 -+ CO+ HOz • 3.000. 1012 0.00 0.000 38 HCO + M' -+ co + ll + liJ' ,7.100. 1014 0.00 70.300 39 CO+ ll + M' -+ HCO + M' 1.136 . 1015 0.00 9.970

C H 2 consumption 40 CHz + H Cll + Hz 8.400 ·109 1.50 1.400

Cll +Hz Cll2 + ll • 5.830 ·109 1.50 13.800 41 42 Cllz + 0 -+ CO+ ll +H 8.000. 1013 0.00 0.000 43 CH2 + Oz -+ CO+ OH +H 6.500 ·1012 0.00 6.300 44 Cllz + Oz -+ COz + H + ll 6.500. 1012 0.00 6.300

CllzO consumption 45 Cll20 + H -+ HCO + llz 2.500. 1013 0.00 16.700 I 46 Cll20 + 0 JICO +OH 3.500 ·1013 0.00 14.600 47 CHzO +OH HCO + llzO 3.000 ·1013 0.00 5.000. 48 CJ[zO + M' -+ JICO + H + M' 1.400. 1017 0.00 320.00

CH3 consumption 49 CH3 + H -+ CHz +Hz 1.800. 1014 0.00 63.000 50 Cllz +Hz -+ CH3+ H 3.680. 1013 0.00 44.300 51 CH3+ H CH4 3.050 ·1034 -6.5 32.010 52 CH3 + 0 CH20 + H 7.000. 1013 0.00 0.000 53 CH3 + CH3 -+ CzHB 5.730. 1051 -11.6 75.660 54 CH3 + Oz -+ CHzO +OH 3.400 ·lOll 0.00 37.400 55 CH4 + H -+ Cll3 +Hz 2.200 ·104 3.00 36.600 56 CH3 +Hz -+ CH4 + H 8.391. 102 3.00 34.560 57 CH4 + 0 -+ CH3 +OH ! 1.200. 107 2.10 31.900

Table D.2: The Cl mechanism ltsed in the fiamelet calculations. Most of the data are taken from Ref. [PET93}. The data of reactions 51 and 53 are different from the Lindemann farm in Ref. {PET93}, since pressure dependenee is not considered.

169

60 C2 H +llz 61 CzHz + H 62 CzH + Oz

63 HCCO + H 64 Cllz +CO 65 JICCO + 0

66 C2H2 + 0 67 C2H2 + 0 68 Czllz + Oll 69 Czll + llzO

70 Czll3 + ll 71 C2Jl3 + 02 72 Czll3

: 73 Czllz + ll

74 Czll4 + H 75 C2lh + ll2 76 Czll4 + 0

:n C2H4 +OH 78 Czll3 + H20 79 CzH4 + M'

80 Czlls + H 81 Cll3 + Cll3 82 CzHs + Oz 83 C2Ils 84 Czll4 + ll

85 CzH6 + ll 86 Czll6 + 0

l 87 Czll6 +OH

-t

-t

-t

-+

-t

..... -t

-t

-t

-t

-t

-t

-t

-t

-+

-t

-t

-+

-+

-t

HCCO + 0 i s:ooo. IQI3

JICCO consurnption CH2 +CO 13 000 · JO"' JICCO + H 2.361·1012

CO+ CO+ll 1.000. 1014

C2JI2 consumption CH2 +CO 14.100. 108

JICCO + H 4.300. 1014

C2ll +llzO . 1.000. 1013

Czllz +OH i 9.000. 1012

C2JI3 consumption C2ll2 + llz 3.000 ·101

"

Czflz + HOz 5.400. 1011

C2JI2 + H .5.720. 1030

Czll3 1.0.53. 1014

C2ll4 consumption

Czll3 + Hz 11.500 · 1014

CzH4 + ll 9.60.5. 1012

Cll3 +CO+ ll . 1.600 ·109

C2JI3 + HzO

1

3.000 ·1013

Czll4 + 0 ll 8.283 · 1012

Czll2 + ll2 + M' 2 . .500 · 1017

C2ll5 consumption C/l3 + CJI3 3.000. 10'" Czlls + 1I 3 . .547 ·1012

CzH4 + HOz 2.000. 1012

Czll4 + Il 1.600. 1041

Czfls 3.189 ·1013

CzHs consumption CzHs +Hz Czlls + 0/J Czfls + llzO

2 5.400 ·10 3.000 ·107

: 6.300 ·106

0.00 0 00 0.00

0.00 0.00 0.00

1.50 0.00 0.00 0.00

0.00 0.00

-5.10 0.00

···---···

0.00 0.00 1.20 0.00 0.00 0.00

0.00 0.00 0.00

-8.60 0.00

3.50 2.00 2.00

0.000 119 95 6.300.

0.000 -29.390

0.000

7.100 50.700 29.300

-15.980

0.000 0.000

188.73 3.39

42.700 32.640

3.100 12.600 65.200

319.800

0.000 49.680 20.900 .

218.110 !

12.610 I

21.800 21.400

1

.

2.700.

Table D.3: The C2 mechanism [PET93}. The data of reactions 72 and 83 are different from the Lindemann form in Ref. [PET.93} since pressure dependenee is not considered.

H20: 6.5 co 0.75 COz : 1.5

CIJ,I 6.5 02 0.4 Nz 0.4 ... . ...

Table D.4: Third body coefficients u.sed in the fiarnelet calculations

170

Appendix E

Definition of the Mixture Fraction

The definition of the mixture fraction, Eq. (4.20), is based onthefact that the fuel and oxidizer eventually will farm only C02 and H20. In this definition, which is valid if the fuel contains only C and H, the element mass fractions of C, H and 0 occur (Eq. (4.20)):

z 2ZcfliJc + ~ZH/1Hu + (Zo.o Zo)/Mo 2Zc,F/Mc + ~ZH,F/Mu -f-zo,o/Mo

(E.l)

Here Z; is the mass fraction of the element i and the indices F and 0 indicate that the mass fraction corresponds to the fuel stream or air stream respectively.

Firstly the mixture fraction is defined in terms of the conserved scalar, also called Shvab-Zel'dovich variabie [WIL85]

Ç = Yc$H"- ~ró2 s

where s is the stoichiometrie value in the reaction

CxHy +(x+ ~y)02-> xC02 + ~yH20 x1\1co 2 , y

1 kg CxHy + s kg 02-> M;;- kg C02 + 2

MF kg H20

with

s (2x + ~y)Mo.

MF ,

(E.2)

(E.3)

(E.4)

Now the rnass fraction of H2 and 0 2 need to be written in terrns of element mass fractions of H and 0. It can be shown that

if the following relations are used:

Yoz

171

1 -Zo s

(E.5)

(E.6)

(E.7)

(E.8)

Now Eq. ( 4.20) can be derived by using Eq. (E.2) and multiplying it with (2x + tY)/MF which gives the conserved scalar f,':

"' _ (2 1 ) Ze + Zu Zo '>- x+-y --2 MF Mo

(E.9)

With Z =Ze+ Zu (Eq. (4.2)) and

Ze (E.lO)

é,' becomes

(E.ll)

This conserved scalar must be normalized between 0 (air) and 1 (fuel) to give Eq. (4.20).

172

Summary

Turbulent isothermal jets and jet ditfusion flames, which are elementary flow con­figurations in most combustion applications, have been stuclied theoretically and com­putationally. The theoretica! aspects concerned the modeHing of turbulent transport of scalars, such as mass and heat, and the description of the turbulent flame with the larninar flamelet concept. In this concept the turbulent flame is an ensemble of laminar stretched one dirnensional flames which can be computed including detailed chemistry. This renders possible the prediction of the chemica! composition of a turbulent flame, which is useful in finding ways to diminish the exhaust of, for instance, NOx and CO in turbulent combustors. In addition the phenomenon of local fiame extinction can be predicted, which is essential to explain and model the lift-off behaviour of turbulent ditfusion fiames.

The turbulent transport of scalars was stuclied within the k- E class of turbulence models, where normally the scalar dissipation rate is assumed to be proportional to a velocity time scale and consequently the turbulent Schmidt number is a constant. This is not confirmed by experimental evidence. A scalar transport model has been developed in which a scalar dissipation rate equation, ba.<:~ed on the Two-Scale DIA theory, is solved. As a result, the predicted non-constant turbulent Schmidt number agrees qualitatively with experiments and the scalar varianee of the mixture fraction is predicted more accurately in an isothermal jet. The predicted scalar to velocity time scale ratio in a turbulent flame agrees reasonably with seeond-order model predictions in the literature.

The flamelet study has shown that the strain rate, rather than the generally used scalar dissipation ra te, quantifies the thermo-ehemieal non-equilibrium effects in a turbu­lent flame most aeeurately. This was eoncluded from both turbulent fiame computations for Harnes exhibiting lift-off, and from the queuehing behaviour of laminar counterflow Harnes ( Hamelets ). In particular, the strain rate of the small eddies, rather than the large eddy strain rate, was found to be most appropriate to model both the lift-off height in a natura! gas flame and the NO-sealing behaviour in a turbulent hydrogen flame.

173

Samenvatting

Turbulente isotherme vrije stralen en diffusievlammen, die de elementaire stromings­configuratie vormen voor de meeste verbrandings toepassingen, zijn theoretisch en nu­meriek onderzocht. De theoretische aspecten behandelen de modellering van het turbu­lente transport van scalaire grootheden, zoals massa en warmte, en de beschrijving van de turbulente vlam met de methode van laminaire 'fiamelets'. In het flamelet concept wordt de turbulente vlam voorgesteld als een set van laminaire één-dimensionale vlam­men. Deze laminaire vlammen kunnen, inclusief gedetailleerde chemie, worden berekend.

·Met deze data kan de lokale chemische samenstelling in een turbulente vlam numeriek volledig worden beschreven. Dit is nuttig bij pogingen om de uitstoot van bijvoorbeeld NO:c en CO uit turbulente verbrandingsprocessen te verminderen. Bovendien kan lokale vlam-uitdoving worden voorspeld. Dit is essentieel bij het verklaren en modelleren van het gedrag van turbulente diffusievlammen die dicht bij de brandstof uitstroom uitge­doofd zijn ('lift-off' vertonen).

Het turbulente transport van scalairen is onderzocht binnen het kader van het k - r;;

model, waarin normaal gesproken de scalaire dissipatie evenredig wordt verondersteld met een snelheids tijdschaal. Hierdoor is het turbulente Schmidt getal constant, wat in tegenspraak is met experimentele gegevens. Een model voor het transport van scalaire variabelen is ontwikkeld waarin een vergelijking voor de scalaire dissipatie, gebaseerd op de 'Two-Scale DIA' theorie, wordt opgelost. Het turbulente Schmidt getal, dat in dit geval berekend kan worden, is in goede qualitatieve overeenstemming met experi­menten. Bovendien wordt de scalaire variantie van de mengfractie in een isotherme straal nauwkeuriger berekend. De verhouding tussen de typische snelheids- en scalaire tijdschalen in een turbulente vlam, berekend met dit model, is in redelijke overeen­stemming met berekende resultaten uit de literatuur, gevonden met een tweede-orde turbulentie model.

De fiamelet-studie heeft aangetoond dat de 'strain-rate', in plaats van de algemeen aangenomen scalaire dissipatie, de thermo-chemische niet-evenwiehts verschijnselen in een turbulente vlam het beste quantificeert. Deze conclusie is gevolgd uit berekening­en aan turbulente vlammen met 'lift-off' en ook uit het uitdoof gedrag van laminaire tegenstroom vlammen (flamelets). In het bijzonder blijkt de strain-rate, geassocieerd met de kleinste werveltjes in plaats van met de grootste wervels, het gesebikst te zijn voor zowel de modellering van een turbulente aardgasvlam vlam met 'lift-off' als voor de modellering van het schalingsgedrag van NO in turbulente vlammen.

174

Acknowledgements

Bij het totstandkomen van het hier gepresenteerde werk hebben diverse mensen, in persoonlijk en/of inhoudelijk opzicht, een belangrijke rol gespeeld. Hiervoor wil ik in het bijzonder bedanken: Marjo, Antoon Larners, dr. Krishna Prasad, Arjen van Maaren, Wim van Helden, Hoaran Meng, leden van de groep WF en de afstudeerders Noud Smals, Sander Willems en Bart Schrnitz. De leden van de kerncornmissie, Antoon Lamers en dr. Krishna Prasad wil ik bedanken voor het proeflezen en hun suggesties.

175

7 juni 1965 1977-1982 1982-1984

September 1984-Februari 1990

Februari 1990-Januari 1994

Curriculum Vitae

Geboren te Roermond Havo, Rijksscholengemeenschap Roermond Atheneum B, Rijksscholengemeenschap Roermond Technische Universiteit Eindhoven Studie Technische Natuurkunde Afstudeerrichting Theoretische Natuurkunde Assistent in Opleiding aan de TU Eindhoven Faculteit Werktuigbouwkunde Vakgroep WOC.

176

Stellingen

behorende bij het proefschrift van

J .P .H. Sanders

Scalar Transport and Flamelet Modelling in Turbulent Jet Ditfusion Flames

l. Analytische theorieën voor turbulentie maken het vaak mogelijk het semi­empirische k - e model te reproduceren, als wel er zinvolle uitbreidingen aan toe te voegen. TSDIA theorie van Yoshizawa, hoofdstuk 2 van dit proefschrift.

2. Preferentiële diffusie vari waterstof kan resulteren in een vlamtemperatuur die honderden graden Kelvin hoger is dan de vlamtemperatuur berekend op basis van gelijke diffusiviteiten. In sterk verdunde, hoog turbulente, waterstofvlammen leidt dit in de brandstofarme gebieden van de vlam tot problemen bij de toepassing van het flamelet concept. Hoofdstuk 5 van dit proefschrift.

3. De scalaire dissipatie als maat voor 'flame-stretch' levert zowel bij ver­dunning van de brandstof als bij het voorspellen van 'lift-off' onrealistisch vlam-gedrag met betrekking tot uitdoof verschijnselen. Een betere maat voor 'flame-stretch' is de str&in rate. Hoofdstuk 7 van dit proefschrift.

4. De vorming van stikstofoxide (NO) is zeer sterk afhankelijk van de lokale temperatuur. Gezien de nauwkeurigheid waarmee in turbulente vlammen de temperatuur berekend en gemeten kan worden moet een quantitatief zeer goede overeenstemming van berekende en gemeten NO-concentraties in zo'n vlam op toeval berusten.

5. Bij het tonen van stukjes ongerepte wilde natuur op de televisie verdient het aanbeveling de geografische locatie daarvan niet kenbaar te maken.

6. De stupiditeit van extreem-rechts wordt nog eens extra benadrukt door de na aanslagen neergekalkte teksten die vaak blunders van taalfouten be­vatten. Aanslag op hOtelier in Zandvoort (N.H.) met achtergelaten tekst: 'auslander eraus', AD 29-9-1993.

7. Een belangrijke positieve eigenschap van de SGP, namelijk haar princi­pialiteit, heeft deze politieke partij verkwanseld door vrouwen bestuurlijke taken te ontzeggen maar tegelijkertijd aan te geven geen bezwaar te hebben tegen een vrouwelijke vorst.

8. Wanneer de makers van belastingwetgeving zouden beseffen, en accepteren, dat niet voor iedere uitzonderingssituatie een clausule moet worden opge­nomen, zou een veel hoger rendement van de belastingwetgeving mogen worden verwacht.

9. Gezien het feit dat Christenen behorend tot de Rooms-Katholieke kerk de absoluut grootste kerkgemeenschap ter wereld vormen, staat de vaak gebruikte naamgeving 'Christelijke ... ' voor protestantse instellingen in geen verhouding tot de werkelijkheid.

10. In de zich als 'landelijke' dagbladen presenterende kranten worden vaak termen als 'het centraal station' gebruikt voor het centraal station in bij­voorbeeld Amsterdam (Volkskrant), waarbij de plaatsaanduiding als in een regionale krant wordt weggelaten.

Deze kranten zouden een grotere nauwkeurigheid moeten betrachten met betrekking tot binnenlandse plaatsaanduidingen om recht te doen aan het etiket 'landelijk'.

11. De spreuk 'zachte heelmeesters maken stinkende wonden' is in Nederland zeker van toepassing op de aanpak van criminaliteit met lichamelijk letsel als gevolg. Deze aanpak staat in sterk contrast met de aanpak van vermogens­delicten.

12. De extra onderwijskorting op het AIO-salaris is een vorm van discriminatie aangezien een dergelijke korting niet geldt voor hoogleraren en universitaire docenten die hun vakkennis op peil willen brengen.

13. Wanneer de naam van een universitaire vakgroep niet in overeenstemming ia met het onderzoek dat er wordt verricht en bovendien een te algemeen of nietszeggend karakter heeft, duidt dit op een samenvoeging van werk­groepen die, op de keper beschouwd, niets met elkaar te maken hebben. Dit kan op zijn beurt duiden op een grootschalig inter-vakgroepsconftict bij de betrokken faculteit. Faculteit Werktuigbouwkunde, TUE.

14. De ongeloofwaardigheid van de overheid kan worden teruggebracht door een meer consistente wetgeving. Gedacht kan worden aan enerzijds het in overeenstemming brengen van de 'strengheid' van de belastingdienst en het gebrek daaraan bij de sociale diensten en anderzijds het minder snel wijzigen van eenmaal ingevoerde regelingen.

15. .Wanneer het versienummer van een commercieel computerpakket met meer dan één cijfer wordt aangeduid, zou het vervangen moeten worden door een beter pakket.