CFD-LES Modeling of Premixed Combustion Using a Thickened Flame Approach

Combustion and Flame 158 (2011) 1750–1767

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Combustion and Flame

journal homepage: www.elsevier .com/locate /combustflame

LES modeling of premixed combustion using a thickened flame approachcoupled with FGM tabulated chemistry

G. Kuenne a,b,⇑, A. Ketelheun a, J. Janicka a,b

a Institute of Energy and Power Plant Technology, Darmstadt University of Technology, Petersenstrasse 30, 64287 Darmstadt, Germanyb Center of Smart Interfaces, Darmstadt University of Technology, Petersenstrasse 32, 64287 Darmstadt, Germany

a r t i c l e i n f o

Article history:Received 23 September 2010Received in revised form 17 December 2010Accepted 8 January 2011Available online 7 February 2011

Keywords:Turbulent premixed combustionTabulated chemistryThickened flameLarge eddy simulationSwirl burner

0010-2180/$ - see front matter � 2011 The Combustdoi:10.1016/j.combustflame.2011.01.005

⇑ Corresponding author at: Institute of Energy aDarmstadt University of Technology, PetersenstraGermany. Fax: +49 6151 16 6555.

E-mail address: [email protected] (G. K

a b s t r a c t

Flamelet Generated Manifolds (FGM) tabulated chemistry is used in combination with a thickened flameapproach to perform Large Eddy Simulation (LES) of premixed combustion. Two-dimensional manifoldsare used to describe the chemistry by the mixture fraction and progress variable. Simulations of one-dimensional flames have been used to verify the coupling of the tabulated chemistry and the LES solverwhere important features like the grid dependence of flame propagation are carefully addressed. Finally,the method is applied to the turbulent flame of a premixed swirl burner including the complex geometryof the swirl nozzle. Results of the velocity, species and temperature are compared with experimentaldata. Thereby different efficiency functions are used to show the sensitivity related to this model param-eter. Some aspects regarding dynamic thickening, numerical accuracy and computational efficiency arealso addressed.

� 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction chemistry for LES) model by Fiorina et al. [5] to allow a proper

Lean premixed combustion is of increasing importance in manyindustrial applications (such as land based gas turbines [1],aero-engines [2] and automotive engines [3]) regarding the pollu-tant formation due to the lower peak temperature when comparedto non-premixed systems. The occurrence of unsteady phenomenalike flashback and combustion instabilities makes these flamesdifficult to control and hence it is desirable to predict theirbehavior using LES. Since a flammable mixture exists before com-bustion occurs the flame speed is a decisive parameter and needsto be reproduced by the simulation. This is difficult because thereaction can not be resolved on typical computational grids andmodels are required to ensure the correct flame propagation. Inaddition, the model needs to account for the interaction ofunresolved vortices with the flame. Therefore the flame speedenters the model as an a priory known parameter, such as in theG-equation or Flame Surface Density concept, or results on itsnatural way from the integration of the chemical source term.Regarding the latter strategy the use of a filtered chemical look-up table has been proposed very recently by Vreman et al. [4]which has been refined in the F-TACLES (filtered tabulated

ion Institute. Published by Elsevier

nd Power Plant Technology,sse 30, 64287 Darmstadt,

uenne).

description of the filtered flame structure and propagation. Thisapproach seems very attractive but not much experience has beengained with it yet. Another well established strategy to resolve theflame on LES meshes is to artificially thicken the flame (ArtificiallyThickened Flame (ATF)-model).

The ATF-model is very attractive since its theoretical derivationcontains no restricting assumptions regarding the flame topology.Due to its universal validity it has been applied to premixed flames[6], lifted flames [7] as well as ignition sequences [8] where finiterate chemistry effects are important. In order to minimize themodeling effort introduced by the modified flame turbulence inter-action the thickening factor is limited to resolve the length scalesof major species only. Therewith the complexity of the chemicalscheme is in general limited to strongly reduced mechanisms(1–3 steps). Of course a small number of reactions is desirablefor computational efficiency, but important flame characteristicsare difficult to reproduce. As mentioned by Selle et al. [6] andSchmitt et al. [9] especially at higher equivalence ratios errorsoccur regarding the flame speed and burnt gas temperature.Boudier et al. [10] adjusted the preexponential constant to obtainthe correct flame propagation but the burnt gas temperatureremained to high in rich regions.

In the present work the ATF concept is combined with a tabula-tion strategy to include detailed chemistry effects into the LES.Regarding premixed flames two similar tabulation approachesdeveloped simultaneously called FPI (flame prolongation ofintrinsic low-dimensional manifolds, ILDM [11]) [12,13] and FGM

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G. Kuenne et al. / Combustion and Flame 158 (2011) 1750–1767 1751

(flamelet generated manifolds) [14,15] proved to be well suited toaccurately describe the flame structure and propagation. It will bedemonstrated that important flame characteristics can be repro-duced over the whole range of equivalence ratios by transportingtwo controlling variables for mixing (mixture fraction) and reac-tion progress (CO2 mass fraction) combined with the FGM table.Compared to reduced mechanisms this method is thought to be apromising alternative when using artificial thickening. Of coursemore a priori knowledge about the reaction is required when usingtabulated chemistry. For example, if heat losses are important thatneeds to be accounted for by an additional table dimension while itnaturally arises in the Arrhenius law of a reduced mechanism. Inaddition, the freely propagating flame obtained from the one-dimensional detailed chemistry solution is only an approximationfor the stretched and curved flame in a turbulent flow field. But thisflamelet assumption is not very restrictive as investigated by Poin-sot et al. [16] by means of DNS where the domain of flamelet mod-eling has been found to be much larger than expected fromclassical combustion diagrams. Further arguments related to themass burning rate of a stretched and curved flame in combinationwith the Lewis number are given in Section 2.1.

The outline of this paper is as follows. In Section 2 the chemistryreduction using FGM and the LES solver will be introduced. Thecoupling of these will be investigated and the quantitative errorbehavior of flame propagation on coarse grids will be explainedby means of theoretical derivations and 1-D numerical simula-tions. In Section 3 the ATF concept will be summarized and the im-pact of dynamic thickening based on a flame sensor on the flameturbulence interaction is illustrated. Some interesting findingsregarding the numerical treatment of the thickening procedureare also discussed. Finally in Section 4 the model is applied to amethane-air swirl burner at atmospheric pressure where the flameis stabilized by a central recirculation zone which is common prac-tice in gas turbine combustors. To allow for a fully developed tur-bulent flow the upstream geometry including the bluff-body swirlnozzle of full complexity was included in the computationaldomain.

2. Coupling of tabulated chemistry with the LES solver

2.1. FGM tabulation

Like many other reduction strategies the technique of flameletgenerated manifolds aims at describing detailed chemistry by onlya couple of controlling variables. This section is restricted to themanifold construction of a methane-air flame used in this work.A more fundamental description of the theory and its verificationcan be found in [14,15]. For the table generation, first a one-dimen-sional freely propagating premixed flame at constant equivalence

Fig. 1. Evaluation of different progress variables (see Eqs. (1) and (2)) using the results frGridpoints). Left: Source term thickness. Right: Corresponding error (Eq. (3)) when nodensity q, filled symbols: error in temperature T.

ratio is simulated using the detailed chemistry one-dimensionalflame code Chem1D [17] with the GRI 3.0 reaction scheme [18].This computation is repeated for different equivalence ratios tospan a two-dimensional manifold which can be parameterized bythe mixture faction and a reactive scalar as progress variable.

Within this work the non-normalized mass fraction of carbondioxide is used as progress variable even though it is not strictlymonotonic in rich regions (/ > 1.2). In the literature different pro-gress variables (Ypv) built by a linear combination of several(weighted) species mass fractions have been suggested to meetthe requirement of being monotonic for all equivalence ratios.Two of them being often used are

Ypv;1 ¼ YCO2 þ YCO½19;13� ð1Þ

Ypv;2 ¼YCO2

WCO2

þ YH2O

WH2Oþ YH2

WH2

½20;4;21� ð2Þ

where W denotes the molar mass of the corresponding species.Unfortunately these progress variables have a much thinner reac-tion zone compared to YCO2 alone. This is illustrated on the leftof Fig. 1 where the thickness of the source term (dð _xpvÞ) of the dif-ferent progress variables defined by its full width at half maximumis shown for various equivalence ratios. In the region around stoi-chiometry the source term of YCO2 can be resolved using only halfof the resolution compared to Eqs. (1) and (2) which is highlydesirable. Hence, we decided to neglect the decomposition ofCO2 into CO at high temperature levels by using only the mono-tonic part of the CO2 mass fraction to allow for a unique relationto the state variables extracted from the table. The resulting errorregarding the prediction of density and temperature defined to bethe maximum difference between the value at the turning point ofthe CO2 mass fraction (index CO2,tp which is the last monotonicentry) and the correct value obtained in the flamelet behind theturning point

�ðqÞ ¼max qCO2 ;tp

� qflamelet

� �qCO2 ;tp

ð3Þ

is shown on the right of Fig. 1. As one can see Ypv,1 and Ypv,2 are themore accurate progress variables to describe the complete flamestructure for all equivalence ratios while—as expected—deviationsfrom the correct value in the rich region can be observed if YCO2 isused (the error in flame propagation speed is also below 10%). Butthis error is small compared to the additional modeling uncertaintyintroduced by increasing the amount of artificial thickening (seeSection 3) to resolve the flame.

The resulting manifold is shown in Fig. 2 where the chemicalsource term of carbon dioxide is shown as a function of the twocontrolling variables (equivalence ratio has been used instead of

om 1-D simulations of a methane-air flame at T = 300 K, p = 101,325 Pa (GRI 3.0, 300n-monotonic parts of the progress variable are neglected. Open symbols: error in

Equi

vale

nce

ratio

0

0.5

1

1.5

2

CO2 mass fraction0

0.050.1C

O2

sour

ce te

rm

0

50

100

Temperature22001900160013001000700400

Fig. 2. Manifold of a methane-air flame defined by two controlling variablesshowing the source term of carbon dioxide (kg/m3 s) colored with temperature (K).

Fig. 3. Flame structure represented by density (q), chemical source term ( _xCO2 ) andCO2 mass fraction (YCO2 ) obtained on four different grids with FASTEST incomparison with the detailed chemistry solution (Chem1D).

1752 G. Kuenne et al. / Combustion and Flame 158 (2011) 1750–1767

mixture fraction in the picture) which are transported by the LESsolver. Outside the flammability limits the extrapolation techniquegiven by Ketelheun et al. [22] has been applied to obtain a com-plete manifold for all chemical states. The resulting table consistsof 901 � 1011 entries for mixture fraction and progress variableequidistantly distributed to allow a fast, non-searching tableaccess.

Throughout this work the Lewis number is assumed to be unityalthough especially at higher equivalence ratios there are non-neg-ligible differences in the laminar flame speed when compared tocomplex transport [23]. Besides being a common assumption[20] which simplifies the thermo chemical description by mixturefraction and progress variable there is good reason to do so regard-ing the flame turbulence interaction. As mentioned by van Oijenet al. [15] without the Lewis number being unity variations inthe element mass fractions and the enthalpy due to preferentialdiffusion occur. As a result, the mass burning rate decreases furtherthan can be described by a single progress variable when the flameis stretched by turbulence. Regarding this issue de Swart et al. [24]have shown that the effective Lewis number of a methane-air mix-ture is close to one due to partly canceling preferential diffusion ef-fects. Using this assumption the effect of stretch and curvature onthe mass burning rate has been investigated for the corrugatedflamelet regime [25] and the thin reaction zone regime [26] andthe resulting error has been found to be within 5% when using asingle progress variable.

2.2. The LES solver

The three-dimensional finite volume code FASTEST uses block-structured, hexahedral, boundary fitted grids to represent complexgeometries. Regarding the velocity, spatial discretization is basedon multi-dimensional Taylor series expansion [27] to ensure sec-ond order accuracy on arbitrary grids. To assure boundedness ofscalar quantities the TVD-limiter suggested by Zhou et al. [28] isused where the value on the cell face is obtained by its downwind(index D) and upwind (index U) nodes by2

Uface ¼ UU þ12

BðrÞðUU �UUUÞ ð4Þ

with the limiter function

1 The memory requirements for the eight variables (source term, viscosity anddensity plus five post-processing variables like temperature) stored in the table are6MB for each CPU.

2 for simplicity written for an equidistant grid here.

BðrÞ ¼r 3rþ1ð Þ

rþ1ð Þ2: r > 0

0 : r 6 0

(r ¼ UD �UU

UU �UUU: ð5Þ

An explicit Runge–Kutta scheme is used for the time advancementof momentum and species mass fractions where the temperaturedependent transport coefficients are taken from the chemistrytable. The code solves the incompressible, variable density,Navier–Stokes equations where an equation for the pressure correc-tion is solved within each Runge–Kutta stage to satisfy continuity.The solver is based on an ILU matrix decomposition and uses thestrongly implicit procedure [29] to take advantage of the block-structure. Sub grid fluxes of momentum are accounted for by theeddy viscosity approach proposed by Smagorinsky [30] where themodel coefficient is obtained by the dynamic procedure of Germanoet al. [31] with a modification by Lilly [32]. A gradient approach hasbeen chosen for the sub-grid flux of scalar quantities with aturbulent Schmidt number of 0.7.3

2.3. Investigation of the coupling

For a quantitative assessment of the coupling, one-dimensionalflames have been computed with the LES solver. The aim is to ob-tain the necessary conditions regarding the grid spacing which al-low a proper representation of the flame structure and propagationspeed. Therefore several simulations over a range of equivalenceratios have been carried out using successive coarsening startingfrom a very fine grid. Figure 3 shows the grid dependence of theflame structure for a equivalence ratio of / = 0.8 where the

3 Except in the reaction layer where the unresolved scalar flux is treated by thembustion model.
co

Fig. 4. Left: Grid dependence of the flame speed slD. Different unburnt gas velocities illustrate the numerical uncertainty if resolution requirements are not considered. Right:Grid dependent deviation from the correct flame speed normalized using the unburnt gas velocity.

Fig. 5. Four time steps of chemical source term and CO2 mass fraction to illustrate the stagnation of the flame when using insufficient resolution.

Fig. 6. Flame position and propagating speed over time showing the processstarting on the unstable part and finally stagnation on the stable part of the sourceterm (gray line: unburnt gas velocity).


Chem1D solution has been added as a reference. Here the density(q), CO2 source term ( _xCO2 ) and CO2 mass fraction (YCO2 , the trans-ported control variable) through the flame front are shown for fourdifferent grid sizes. One can see that above Dx � 0.17 mm strongdeviations occur and especially the source term is poorly predicted.Since the propagation speed of the flame behaves as sl /

R1�1 _xdx

errors are expected which can be seen on the left in Fig. 4 wherethe flame speed is shown as a function of the grid spacing. Theflame speed slD introduced here will be used in the remainder ofthis work to distinguish between the effective flame propagationobtained on the numerical grid and the correct laminar flamespeed sl. It was obtained by three different methods leading to con-sistent results. First, from the fulfillment of continuity

slD ¼ub � uuquqb� 1

ð6Þ

where ub and uu denote the burnt and unburnt gas velocity respec-tively. Second, from spatial propagation

slD ¼ uu �xf t2ð Þ � xf t1ð Þ

t2 � t1ð7Þ

where xf denotes the flame position determined by YCO2 ¼ 0:5YeqCO2

attime t1 and t2, respectively. Third, from the scalar transport equa-tion which yields in the flame reference frame

sl ¼1

quYCO2 ;b

Z 1

�1_xCO2 dx: ð8Þ

To ensure that the results are independent of boundary conditionsdifferent inlet velocities—below as well as above the correct flamespeed of 24.8 cm/s—were used. As expected, above Dx � 0.17 mmthe flame speed is neither independent of the grid size nor of the in-let velocity. It is interesting to note that the flame speed converges

towards the unburnt gas velocity for very coarse grids. This isemphasized on the right of Fig. 4 where the grid dependent devia-tion from the correct flame speed has been normalized using theunburnt gas velocity. The reason for this behavior is the increasingerror of the midpoint rule used to evaluate the source term in thecontext of the finite volume method which is

sl ¼1

quYCO2 ;b

Z 1

�1_xCO2 dx � slD ¼

1quYCO2 ;b

Xi

_xCO2 ;iDx;i ð9Þ

where the index i runs over all control volumes. For largeDx� 0.17 mm mainly one grid point at the flame front (index ff)contributes to the sum leading to

0 1 2 3t (ms)

20

22

24

26

28

30

32

0.0 0.2 0.4 0.6xf (mm)

slsl

(cm

/s)

Fig. 7. Oscillating behavior of the flame speed during a 1-D simulation with uu > sl as a function of flame position xf (left) and time (right). Blue: computed flame speed slD (Eq.(9)) used in the transport equation; Red: approximation of slD using a sine function (see Eq. (13)). For visualization purpose a coarse grid producing a large error has beenchoosen. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


slD �1

quYCO2 ;b_xCO2 ;ff Dx;ff : ð10Þ

Using this equation the behavior of unphysical flame stagnation willbe explained in the following. The 1-D simulation to supplementthe theory uses a typical LES grid size of Dx = 0.63 mm. The unburntgas velocity is uu = 50 cm/s (above the correct flame speed of24.8 cm/s). The initial field to start the simulation is the correctflame structure obtained on a sufficiently fine grid. This field wasthen interpolated onto the coarse grid. The behavior of the first timesteps now depends on the location of the contributing node toEq. (10). In this example the flame is positioned such that the nodewith the major contribution is on the ascending part of the chemicalsource term (Fig. 5, t = 0 ms, the area of the bars represents the sumgiven by Eq. (9)). Since the distinction of ascending and descendingpart is important in this illustration they are colored blue and red,respectively, in Fig. 5. The corresponding flame position xf and flamespeed sl given by Eq. (9) are plotted in Fig. 6 over the simulationtime. Since the flame speed in the initial time step is below the un-burnt gas velocity the flame starts to move backward which furtherdecreases the flame speed according to Eq. (10) in the next timesteps (see Fig. 6 and t = 0.02 ms in Fig. 5). Therefore, the ascendingpart of the source term is unstable since it reduces the flame speedif the flame is already below the unburnt gas velocity and vice versa.At t = 0.1 ms the flame has moved further backwards such that themajor contribution of the source term is on the descending part onegrid point to the right. The descending part of the source term be-haves inversely to the ascending part and therefore the flame stag-nates for t ?1 as can be seen in Fig. 6. Since the maximum value of_xCO2 through the flame front given by the chemistry table will not

be exceeded in the simulation this derivation only holds for suffi-ciently coarse grids. Or, in other words, for every unburnt gas veloc-ity there exists a minimum grid size Dx,min to stagnate the flameaccording to

(a)

Fig. 8. (a) Local error as defined in Fig. 7 as a function of the ratio of gridspacing to the cerror following from the deviation in absolute flame speed as a function of the unburnt

slD ¼ uu ¼1

quYCO2 ;bmaxð _xCO2 ÞDx;min ð11Þ

as observed on the right in Fig. 4.However, the above derivation just makes clear how the flame

propagation is dominated by numerical uncertainties if the originalflame thickness is not resolved. Of course we need to stay below acertain grid size but instead of judging from visual impression inFig. 4 we want to give a more precise definition of the acceptableerror. Due to the shape of the source term the evaluation ofEq. (9) will oscillate when the flame passes a grid point wherethe amplitude is only a function of the grid spacing. To illustratethat, Fig. 7 shows the transient evolution of the flame speedobtained during a 1-D simulation. On the left the definition ofthe local error �D has been added to be the largest deviation fromthe correct flame speed sl. Using this definition several simulationsover a large range of equivalence ratios, grid spacings and unburntgas velocities have been carried out to obtain a universal relation ofthis error to the grid spacing which is given in Fig. 8a. Here the rel-ative error �D/sl is a unique function of the grid spacing related tocharacteristic flame length scales. Hence if we want the local error�D to be below 10% of the correct flame speed, the grid size shouldnot exceed

Dx;max � 0:7 � dð _xCO2 Þ � 0:3 � dðYCO2 Þ ð12Þ

where dðYCO2 Þ is the flame thickness obtained by the gradient of theCO2 mass fraction and the thickness of the chemical source termdð _xCO2 Þ is represented by its full width at half maximum. To assessif the definition of the local error is a sufficient measure for theresolution required we furthermore need an estimate how it affectsthe absolute flame speed saD which describes the movement of theflame relative to the combustor geometry. Therefor some additionalinformation has been added to Fig. 7. The simulation shown hereuses a unburnt gas velocity twice the correct flame speed and a grid

(b)

haracteristic flame length scales dðYCO2 Þ (dashed) and dð _xCO2 Þ (solid line). (b) Globalgas velocity.


spacing of Dx,max (Eq. (12)) < Dx < Dx,min(Eq. (11)) and hence theflame gets pushed back. While the oscillation is relatively symmet-ric with respect to the flame position xf it gets distorted in time. Thishappens because the larger contributions reduce the difference be-tween the flame speed slD and the unburnt gas velocity and willtherefore contribute a larger portion to the absolute flame speedwith respect to time (this is decisive since the flame position fol-lows from the time integration of saD). Obviously the frequencyand therewith the error in absolute flame speed (saD � sa)—whichwe call the global error—is a function of the unburnt gas velocitywhich now needs to be quantified. Besides the 1-D simulationswe approximated the flame movement to follow

@xf

@t¼ saD ¼ uu � slD � uu � ½sl þ oþ Asinð2pfxf Þ� ð13Þ

to derive an analytic relation between the local and the global error(The offset o and amplitude A are defined in Fig. 7, note that the sinefunction has been defined with respect to xf so the frequency f is noadditional unknown). As can be seen in Fig. 7 the approximation ofslD does fit very well to the real error behavior. The analytic solution(taken from a standard handbook of mathematics, e.g. [33]) of Eq.(13) is plotted together with simulation results in Fig. 8b. Here,two regions can be identified. As outlined above for uu � sl < �Dthe flame will not move and hence the global error increases line-arly. For higher unburnt gas velocities the flame will start to moveand the error decreases due to partly cancelling effects of the oscil-lating integration error. With increasing uu the error becomes moresymmetric in time and hence the global error further decreases un-til a constant value caused by the offset is reached. Hence, the max-imum global error will not exceed the local error and therefore Eq.(12) provides an appropriate criterion for the necessary resolution.The corresponding maximum grid size is plotted in Fig. 9 for theone-dimensional and three-dimensional case—where the correctflame propagation in a diagonal direction through the cell needsalso to be ensured—as a function of the equivalence ratio. If this grid

Fig. 9. Maximum grid size according to Eq. (12) to ensure the correct flamepropagation normal (1-D) and diagonal (3-D) through a cell as a function ofequivalence ratio (/).

Fig. 10. Flame speed as a function of equivalence ratio (/) obtained with the LESsolver FASTEST using two controlling variables in comparison with the detailedchemistry solution (Dx 6 Dx,max).

size is not exceeded the Chem1D solution can be reproduced veryaccurately by the LES solver using transport equations for the con-trolling variables. This has been verified for different equivalenceratios as shown in Fig. 10. Here the flame speed obtained byChem1D and the LES solver are compared and a very good agree-ment can be observed.

3. Combustion modeling

As shown in the previous section important properties of theflame can only be captured by the LES solver below a certain gridsize (Dx � 0.1 mm), which is far below typical grid spacings usedin simulations of complex geometries (Dx � 1 mm). If these prop-erties are important as in premixed combustion where the flameposition is determined by its propagation speed, additional model-ing is required. In this work, the ATF-model is used in combinationwith the FGM approach to ensure an accurate coupling on arbitrarymeshes. This approach is illustrated in the following. In addition,some numerical aspects of the model will be discussed.

3.1. The ATF approach

The principle of the ATF-model is based on a coordinate trans-formation that is applied to the scalar transport Eq. (14) to thickenthe flame front and therewith make it resolvable on coarse grids.

@qYk

@tþ @

@xðquiYkÞ ¼

@

@xqDk

@Yk

@x

� �þ _xk ð14Þ

This idea has been initiated by Butler and O’Rourke [34] andO’Rourke and Bracco [35] where Eq. (14) is expressed in terms ofniðxiÞ ¼ F � xi and sðtÞ ¼ F � t finally leading to

@qYk

@s þ@

@niðquiYkÞ ¼

@

@niqFDk

@Yk

@ni

� �þ

_xk

F ð15Þ

which is a pure mathematical operation without any assumptions.If one just solves Eq. (15) with respect to t and x instead of s andn, Yk gets thickened by F compared to the solution of Eq. (14).Accordingly it is possible to resolve the flame and hence obtainthe correct propagation speed on a grid of Dx if the thickening factorsatisfies F P Dx=Dx;max. To verify this, Fig. 11 represents the griddependency as introduced on the left in Fig. 4 for a flame thickenedby the factor five where the flame speed slD is now grid independentup to a grid spacing five times coarser compared to the unthickenedflame. The drawback of this modeling approach is the modifiedinteraction of the flame front with the turbulent flow field. It hasbeen shown by Poinsot et al. [16] and Meneveau and Poinsot [36]by means of DNS that the efficiency of a vortex to wrinkle the flamedepends on the ratio of its radius r and the flame thickness d whichis obviously decreased by F . Therefore an efficiency function E hasbeen derived [37–39] to increase the flame speed and compensate

Fig. 11. Grid dependence of the flame speed slD using a thickening factor of five.Filled symbols: constant thickening using F ¼ 5, open symbols: dynamic thicken-ing using Fmax ¼ 5.


for the lost flame surface caused by the thickening. Since the flamespeed and thickness follow

sl /ffiffiffiffiffiffiffiffiD _xp

; d /ffiffiffiffiffiffiffiffiffiffiD= _x

qð16Þ

the diffusion as well as the source term are multiplied by E to keepthe thickness constant resulting in the final transport equation

@qYk

@tþ @

@xiðquiYkÞ ¼

@

@xiqFEDk

@Yk

@xi

� �þ EF

_xk ð17Þ

Fig. 12. Comparison of a dynamic and constant thickened flame using Fmax ¼ 5 and F ¼is shown using the right y-axis. Solutions have been obtained using a very fine grid (lin

Fig. 13. Flame thickness obtained by the CO2 mass fraction (YCO2 ), temperature (T), dendynamic (points) thickening. The thickness of the chemical source term ( _xCO2 ) is repres

which propagates a flame of thickness d1l ¼ Fd0

l with the turbulentflame speed sT ¼ Esl. In this work the efficiency functions derived byColin et al. [38] as well as Charlette et al. [39] are used. Both formu-lations utilize DNS of flame vortex interaction to characterize theunresolved flame surface and have been implemented in the finitevolume code of Section 2.2. They are summarized in the following.

3.1.1. The original formulation by Colin et alColin et al. [38] defined the efficiency function to be the ratio

between the wrinkling factor (i.e. the flame surface divided by its

5 respectively. An unmodified flame has been added. The flame sensor X (gray line)es) and a resolution according to Dx ¼ Fmax � Dx;max (points).

sity (q), and viscosity (l) for different thickening factors using constant (lines) andented by its full width at half maximum.


projection in the propagation direction) of the original flame ofthickness d0

l and the thickened flame of thickness d1l (Eq. (18))

which is a natural consideration to account for the lost flame sur-face. This wrinkling factor is based on the ratios of the test filterDe � 10Dx and the flame thickness, and the velocity fluctuationu0De

acting on the test filter De and the laminar flame speed s0l .

E ¼N d0

l

� �Nðd1

l Þ¼

1þ aCDe

d0l

;u0De

s0l

!u0De

s0l

1þ aCDe

d1l

;u0De

s0l

!u0De

s0l

ð18Þ

with CDe

d0;1l

;u0De

s0l

!¼ 0:75 exp �1:2=

u0De

s0l

� �0:3" #

De

d0;1l

!2=3

ð19Þ

The velocity fluctuation u0Deis based on a similarity assumption

leading to u0De¼ 2D3

x jr � ðr2~uÞj where the Laplace operator is eval-uated on a 4Dx stencil because the amplitude of its discrete Fouriertransform represents a good filter for the relevant length scales offlame vortex interaction. The parameter a can be estimated withthe turbulent Reynolds number according to Eq. (20) where b is amodel constant in the order of unity.

a ¼ b2 lnð2Þ

3cmsðRe1=2t � 1Þ

; cms ¼ 0:28; Ret ¼u0lt

mð20Þ

0 0.01 0.02 0.03 0.04

0

0.005

0.01

0.015

0.02 400350300250200150100500-50

F=2 (Constant)

0 0.01 0.02 0.03 0.040

0.005

0.01

0.015

0.02 400350300250200150100500-50

F=5 (Constant)

0 0.01 0.02 0.03 0.04

0

0.005

0.01

0.015

0.02 400350300250200150100500-50

F=5 (Dynamic)

Fig. 14. Three snapshots of two-dimensional flame vortex interaction to illustratethe reduced flame turbulence modification by the dynamic thickening. Contour ofthe vorticity (1/s) to visualize the vortex and isolines of chemical source term tomark the reaction zone. The snapshots correspond to t = 0.04 s in Fig. 15.

3.1.2. The power-law formulation by Charlette et alThe power law flame wrinkling model by Charlette et al. [39]

describes the unresolved flame surface (i.e. the efficiency function)in terms of an inner and outer cutoff scale which represent therange of length scales that require modeling regarding the flameturbulence interaction.

ED

d0l

;u0Ds0

l

;ReD

!¼ 1þmin

D

d0l

;Cu0Ds0

l

" # !c

ð21Þ

Following the relationship between strain-rate and the energy spec-trum the function C has been obtained by numerical integrationand the fit

CfitD

d0l

;u0Ds0

l

;ReD

!¼ ððf�a

u þ f�aD Þ

�1=aÞ�b þ f�bRe

h i�1=bð22Þ

is provided for practical use with:

fu ¼ 427Ck

110

� �1=2 18Ck

55

� �u0Ds0

l

� �2

ð23Þ

fD ¼27Ckp4=3

110D

d0l

!4=3

� 1

0@

1A

24

351=2

ð24Þ

fRe ¼9

55exp �3

2Ckp4=3Re�1

D

� � 1=2

Re1=2D ð25Þ

a ¼ 0:6þ 0:2 exp �0:1ðu0D=s0l Þ

� �� 0:2 exp �0:01ðD=d0

l Þ� �

; b ¼ 1:4ð26Þ

Herein Ck = 1.5 is the Kolmogorov constant, D ¼ Fd0l is the filter size

and ReD ¼ 4ðD=d0l Þðu0D=s0

l Þ the corresponding sub-grid turbulentReynolds number. The velocity fluctuation u0D is obtained assuggested by Colin et al. [38] and the exponent is set to c = 0.5according to the non-dynamic formulation of the model.

3.2. Dynamic thickening

To ensure that pure mixing is accurately predicted the thicken-ing should be limited to the flame front. Since the burner presented

in Section 4 includes the mixing of pure air with the premixedreactants as well as with the burnt gas the use of a dynamic thick-ening introduced by Legier et al. [7] is necessary. The idea of a spa-tially varying thickening factor has already been introduced byButler and O’Rourke [34] and it is demonstrated in [35] that thescalar transport equations remain invariant under the transforma-tion to ensure the correct flame propagation. The procedure is touse a more general definition of the new coordinate n(x) normalto the flame according to

nðxÞ ¼Z x

Fðx0Þdx0 ð27Þ

where the local thickening factor follows to be @n@x ¼ FðxÞ. If Eq. (27)

is unique the spatial derivatives can be exchanged using

@

@x¼ @

@n@n@x¼ FðxÞ @

@n: ð28Þ

Further assuming that the spatial variation of species is dominatingin the flame normal direction the time derivative can be expressedin terms of the absolute flame speed via

@

@t¼ @xF

@t|{z}uu�sl

@

@x: ð29Þ

Inserting Eqs. (28) and (29) into Eq. (14) yields

Fig. 16. Increase of turbulent flame speed sT by the efficiency function based onconstant and dynamic thickening.

Fig. 15. Reduced total reaction rate x (lines) and available flame surface k (symbols) during the flame vortex interaction shown in Fig. 14.

4 Since not all parameters of the simulations done by Colin et al. [38] are known thesults are not identical.


@qYk

@n@xF

@tþ @

@nðquYkÞ ¼

@

@nqFðxÞDk

@Yk

@n

� �þ

_xk

FðxÞ ð30Þ

which compared to the original equation

@qYk

@x@xF

@tþ @

@xðquYkÞ ¼

@

@xqDk

@Yk

@x

� �þ _xk ð31Þ

predicts the same absolute flame speed @xF@t in the—flame resolving—

coordinate n(x). No physics have been changed so far. The step ofartificial thickening is now to solve Eq. (30) with respect to x insteadof n which elongates the coordinate n onto the x-grid. Thereby theflame gets thickened and, if F varies spatially, the flame structuregets distorted as will be illustrated below. To dynamically deter-mine the thickening factor we use the formulation for the flamesensor X suggested by Durand and Polifke [40] to detect the flameand apply the thickening according to:

X ¼ 16½cð1� cÞ�2; c ¼ YCO2=Yeq:CO2; F ¼ 1þ ðFmax � 1ÞX: ð32Þ

This local thickening ensures the correct flame propagation upto a grid size similar to using Fmax everywhere (also shown inFig. 11) but since the thickening varies through the flame frontthe flame structure is not identical. This is illustrated in Fig. 12where the flame structures of a dynamically thickened flame usingFmax ¼ 5 and a constantly thickened flame with F ¼ 5 are com-pared. The preheating and oxidation zone of the dynamically thick-ened flame are thinner while the sensor is almost one in thereaction zone ensuring a correct flame propagation according toEq. (9). Two resolutions are shown in Fig. 12. To exclude numericaluncertainties a very fine mesh (resolving a flame with F ¼ 1) hasbeen used to assess the impact onto the flame structure whenEq. (32) is used in Eq. (17). The second resolution satisfiesDx ¼ Fmax � Dx;max which is the more relevant case for practicaluse. Using a constant thickening both resolutions produce thesame result because all gradients are smooth compared tothe mesh. Differences occur when the dynamic thickening is

applied. While the source term is well resolved (as with constantthickening) the preheating zone remains very sharp and getsslightly flattened because of the resolution limit. The impact ofthe dynamic thickening procedure on the flame turbulence interac-tion is twofold.

First, since the overall flame is thinner, the flame becomes moresensitive to vortices again. An unthickened flame has been addedto Fig. 12 to illustrate that especially the change of density andviscosity—which play a major role regarding the mechanism ofdissipating a vortex—is not as modified as in a flame withglobal thickening since it mostly occurs in the preheating zone.Figure 13 shows the flame thickness represented by different vari-ables. In the upper two graphs the gradient of the variables hasbeen used to obtain the thickness dF

gradient of the respective layer be-tween the burnt and unburnt state. The two graphs below showthe corresponding thickness represented by the distance between15% and 85% (dF

15%�85%) of variation through the flame front. Thesegraphs have been added since the smooth s-shape of the variablesthrough the flame front is partly distorted by the dynamic thicken-ing procedure and hence the gradient may not be the optimalchoice to characterize the flame thickness. In addition, the evalua-tion of the gradient on the coarse mesh introduces an additionalerror. The threshold of 15–85% has been chosen since it leads toa similar thickness as the gradient for the constantly thickenedflame. As expected the constantly thickened flame behaves linearlywith the thickening factor. This only holds for the dynamicallythickened flame when the thickness dF

15%�85% is used. This demon-strates that it is more appropriate in this case. Like in Fig. 12, twogrids have been considered to show its impact. Comparing the tworesolutions, the thickness is increased on the coarse mesh which isconsistent with the visual impression from Fig. 12. Nevertheless,the dynamically thickened flame is still much thinner than the con-stantly thickened flame. To illustrate the impact of this behavior,two dimensional simulations of flame vortex interaction have beencarried out. Following previous works [38] the stream functiongiven by Poinsot et al. [16] has been used to initialize the velocityfield. The comparison of three flames interacting with a vortex ofsame strength is given in Figs. 14 and 15. Figure 14 shows snap-shots taken at the same instant of time where the above mentionedinfluence of dynamic thickening can be observed. Regarding theconstant thickening the results are similar4 to the simulations doneby Colin et al. [38] to derive the efficiency function: The thickerflame gets less distorted by the vortex. If the same thickening factoris used with a dynamic thickening (last snapshot in Fig. 14) anincreased interaction of the flame with the vortex can be observedwhich is quantified in Fig. 15. Here the total reaction rate x (inte-grated over the computational domain x ¼

R_xdA) together with

re

Fig. 19. Substep strategy for the scalar transport applied within a 3-stage Runge–Kutta scheme.

(a) (b)

Fig. 17. Illustration of spatial interpolation: (a) Influence of the thickening procedure on the limiter function B(r), centered scheme has been added as the optimum, (b) CO2

mass fraction through the flame front at the computational nodes and the corresponding faces (F ¼ 5).

Fig. 18. Left: Ratio of diffusion and convection number for the burnt/unburnt (red/blue) state with/without thickening (full/dashed). The molecular diffusion coefficient anddensity have been taken from the burnt and unburnt values of a methane-air flame (/ = 0.8). Right: Corresponding theoretical speedup and number of substeps appliedwithin a 3-stage Runge–Kutta scheme for the ATF simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version ofthis article.)


the available flame surface k (obtained from the iso-surfaceYCO2 ¼ 0:5Yeq

CO2) normalized with the values of a freely propagating

flame are plotted. As expected by visual impression from Fig. 14,the dynamically thickened flame with Fmax ¼ 5 behaves like anintermediately thickened flame between the simulations using aconstant thickening with F ¼ 2 and F ¼ 5. This observation is inagreement with the study done very recently by Auzillon et al.[41] where the effect of thickening on the flame structure anddynamics was investigated in the context of the F-TACLES modelof Fiorina et al. [5]. They showed, that by filtering the flame by agaussian function the thermal thickness of the flame is much lessincreased compared to the thickness of the chemical source termwhich yields significant improvements regarding the prediction offlame turbulence interaction. As shown by Colin et al. [38] thedifference between different thickening factors in Fig. 15 is less pro-nounced for higher ratios of vortex velocity to the laminar flamespeed.

Second, together with the thickening factor, both efficiencyfunctions (Eqs. (18) and (21)) tend towards unity outside the flamewhich is a necessary condition since otherwise Eq. (17) would stillbe modified in regions of pure mixing. Hence, for a given velocityfluctuation u0De

, the efficiency function is not constant throughthe flame front5 which influences its impact on the propagationspeed. To quantify this effect one-dimensional simulations havebeen carried out where the velocity fluctuation u0De

has been set toa constant value (of course u0De

would be zero in a 1-D flame ifu0De¼ 2D3

x jr � ðr2~uÞj is used) yielding a given efficiency Emax ¼EðX ¼ 1Þ. Figure 16 shows the results of these simulationsobtained with different values for Emaxðu0De

Þ. As one can see thedynamically thickened flame is slower than the flame with constant

5 The preheating zone gets slightly compressed by this.

thickening which uses E ¼ Emax everywhere. The results only slightlydepend on the thickening factor or the formulation for the efficiencyfunction used, so only one graph is shown here.

The second of the effects mentioned above can be interpreted asa reduction of the efficiency function to account for the reducedmodification of flame turbulence interaction. For this work, it is as-sumed that the efficiency function arising from the dynamic proce-dure yields appropriate values of turbulent flame propagationcorresponding to an effective flame thickness of the dynamicallythickened flame. Further knowledge about the impact of thedynamic thickening procedure onto the efficiency function isdesirable but necessitates DNS studies of flame turbulence interac-tion which is beyond the scope of this work.

3.3. Numerical aspects of the thickening procedure

During this work several influences of the thickening procedureon the stability and accuracy of the scalar transport have been

Fig. 20. Geometry of the Tecflam bluff-body swirl nozzle.

Coflow (pure air)

Inlet (methane-air)

300mm

717m

m 117m

m

Measurement Planes

Monitoringpoint

Fig. 21. Dimensions of the computational domain with a cut-out of the ellipticallysmoothed mesh. For visualization purposes, the mesh is coarser than the one usedin the simulations.

6 Only for very high velocities or if the speed of sound is restrictive for theumerical scheme the Courant number may dominate again.


observed. A positive feature of the model is the reduction ofgradients. Compared to the mesh the original flame is like a sharpinterface and hence gets flattened by numerical inaccuracies as onecan see in Fig. 3. To keep the solution stable the limiter function(Eq. (4)) reduces the order of spatial accuracy which increasesnumerical diffusion to avoid unphysical oscillations. The introduc-tion of an explicit thickening reduces this numerical diffusion andtherewith uncertainties influencing the flame propagation. InFig. 17a the limiter function is shown with and without thickening

to illustrate the improved order of accuracy through the flamefront. Even on this relatively fine grid of Dx = 0.3 mm the deviationfrom a pure centered scheme is much smaller for the thickenedflame. In Fig. 17b the CO2 mass fraction of the thickened flame isshown together with the values given by Eq. (4) on the control vol-ume faces indicating almost 2nd order interpolation.

Besides this expected behavior of spatial interpolation, instabil-ities regarding the time integration occurred after the thickeningprocedure was applied. Following classical stability analysis (e.g.[42]) basically two parameters—the diffusion number D and theCourant number C—arise, drawing restrictions for the explicit timeintegration.

D ¼ DDt

D2x

; C ¼ uDt

Dxð33Þ

The exact value of these numbers determining the largest possibletime step Dt depends on the stability function of the correspondingscheme as well as on the influence of non-linearities (e.g.D(U)).However, in this section we just want to point out how the ATF-model may alter the scalar transport equation to be diffusiondominated and how we dealt with it to keep the method computa-tionally efficient. Without the ATF-model, the flow is in generaldominated by convection and hence the Courant number is decisivefor the time step size. But the diffusion coefficient D is increased bythe thickening factor and efficiency function (Eq.(17)). Assumingtypical values of F � 10 and E � 3 this leads to DATF ¼ EFD � 30Dindicating that the diffusion number may exceed the Courant num-ber as is illustrated on the left of Fig.18. Here the ratio of the diffu-sion number and Courant number (Eq.(33)) in the burnt andunburnt state of a methane-air mixture (/ = 0.8) are shown withand without application of the ATF-model. This plot has been ob-tained assuming a velocity field yielding a fluctuation ofu0De¼ 2 m=s at a reference grid spacing of 0.5 mm. The velocity to

obtain the Courant number has been set to 5 m/s. Hence the situa-tion corresponds to the application of different grids for the sameflow configuration. A different velocity field would shift the graphsbut the behavior would remain the same.6 Below Dx � 0.1 mm theflame is fully resolved and the function follows / 1/Dx while withincreasing grid size the flame needs to be thickened (F / Dx) andhence the ratio remains constant. Further coarsening of the grid in-creases the efficiency function (through the increase of the thicken-ing factor as well as the velocity fluctuation u0De ) which convergestowards E / F 2=3 / D2=3

x (when using Eq.(18)). This causes rising

n

0 2 4 6 8

0 1 2 3 4 5 6r/R (-)

1mm

0 2 4 6 8

10mm

0 2 4 6 8

20mm

0 2 4 6 8

30mm

0 2 4 6 8

60mm

0 2 4 6 8

90mm

0 2 4 6 8

umean (m/s)

120mm

0 1 2 3

0 1 2 3 4 5 6r/R (-)

1

2

3

1

2

3

1

2

3

1

2

3

1

2

1

2urms (m/s)

0 2 4 6 8

0 1 2 3 4 5 6r/R (-)

0 2 4 6

0 2 4 6 0

2

4

6

1 2 3 4

1 2 3 4 0

2

4wmean (m/s)

0

1

2

3

0 1 2 3 4 5 6r/R (-)

1

2

3

1

2

3

1

2

1

2

1

2

1

2wrms (m/s)

Fig. 22. Isothermal flow. Mean and fluctuating part of the axial (u) and azimuthal (w) velocity obtained by the LES (lines) in comparison with experimental data (dots).R = 15 mm (radius of the buff-body).


ratios of the diffusion and convection number with increasing gridsize which strongly deviates from the behavior without the ATF-model where diffusion becomes less important. This influence ofthe combustion model on the scalar transport equation results instability restrictions on the maximal possible time step size, makingthe simulation computational too expensive. Increasing the numberof stages of well established Runge–Kutta schemes does not seem tobe promising since the computational costs rise while the stabilityregion is only slightly extended (e.g.[43,44]). Since only the trans-port equation for the scalars is modified, the momentum equationalone would still be stable and diverges only as a result of the incor-rect prediction of density and viscosity due to unstable scalar trans-port when the time step is not sufficiently decreased. Therefore,multiple sub steps for the scalar transport equation assuming a con-stant velocity in the convective term are used within a time step (inthis work within every Runge–Kutta stage) as illustrated in Fig. 19.This procedure is very efficient since only for the correction of thevelocity a system needs to be solved to satisfy continuity. Sincethe solver takes most of the computation time it is desirable to keepthe time step as large as the momentum equation allows, whichwas possible with the substep strategy. The resulting speedupdefined by the ratio of computing time needed with and withoutsubsteps to simulate the same pyhsical time is shown on the rightof Fig. 18. The graph has been obtained by extrapolating the speed-up gained for the swirl burner of Section 4 using six substeps with agrid size of Dx = 0.5 mm in the reaction zone whereby the comput-

ing time for the individual code sections has been measured. Hence,outside this region its a theoretical speedup that has not been ver-ified. Several one and two-dimensional test cases of reacting andnon-reacting flows showed that the method is at least as accurateas without substeps while allowing larger time steps when diffu-sion becomes important.

4. Application to a premixed swirl burner

4.1. Configuration and numerical setup

The configuration investigated is the Tecflam swirl burner of theInstitute of Energy and Power Plant Technology as shown inFig. 20a. Measurements of the velocity were performed bySchneider et al. [45] using Laser Doppler Anemometry with mag-nesium oxide as seeding material. Raman/Rayleigh scattering hasbeen used by Gregor et al. [46] to obtain the temperature and mainspecies distribution. The instantaneous data of this study also ver-ified the flamelet-like behavior of the flame structure. As illus-trated in Fig. 20a, the air enters the configuration from thebottom where methane is injected using a perforated ring line.Then the methane-air mixture (temperature = 300 K, equivalenceratio = 0.83) is deflected by 90� to enter the radial and tangentialchannels (see Fig. 20b) where the swirl is generated. Hereafterthe fuel moves upward again, passes the annulus around the bluff

Fig. 23. Illustration of instantaneous flame turbulence interaction. (a) Isosurface of the chemical source term and a slice extracted at z = 0 mm showing the temperature field(K). (b) Slice extracted at x = 15 mm above the bluff body showing the axial velocity (m/s) and isolines of temperature (increasing to the centerline: T = 350, 1560 and 1840 K)marking the preheating and the reaction zone.

02468

0 1 2 3 4 5 6

r/R (-)

1mm

02468

10mm

02468

20mm

02468

30mm

02468

60mm

02468

90mm

02468

umean (m/s)

120mm

0 1 2 3

0 1 2 3 4 5 6

r/R (-)

1

2

3

1

2

3

1

2

3

1

2

3

1

2

1

2urms (m/s)

0 1 2 3

0 1 2 3 4 5 6

r/R (-)

-1 0 1 2 3 0 1 2 3 4 0 1 2 3 4

0 1 2 3

0

1

2

0

1

2vmean (m/s)

0

1

2

0 1 2 3 4 5 6

r/R (-)

1

2

3

1

2

3

1

2

3

1

2

3

1

2

1

2vrms (m/s)

Fig. 24. Reacting flow. Mean and fluctuating part of the axial (u) and radial (v) velocity obtained using the efficiency function of Charlette et al. (solid lines) and Colin et al.(dashed lines) in comparison with experimental data (dots). R = 15 mm (radius of the buff-body).


body and enters the unconfined section. Here the Reynolds numberbased on the bulk velocity (5 m/s) and the hydraulic diameter

(Dh = 30 mm) is Re = 10,000. The swirl number which representsthe ratio of azimuthal and axial momentum at the nozzle exit is:

0 2 4 6 8

0 1 2 3 4 5 6

r/R (-)

1mm

0 2 4 6

10mm

0 2 4 6

20mm

0

2

4

630mm

1 2 3 4

60mm

1 2 3 4

90mm

0

2

4wmean (m/s)

120mm

0

1

2

3

0 1 2 3 4 5 6

r/R (-)

1

2

3

1

2

3

1

2

1

2

1

2

1

2wrms (m/s)

0.000.010.020.030.040.050.06

0 1 2 3 4 5 6

r/R (-)

10mm

0.000.010.020.030.040.050.06

20mm

0.000.010.020.030.040.050.06

30mm

0.000.010.020.030.040.050.06

zmean (-)

60mm

0.000

0.005

0.010

0.015

0.020

0 1 2 3 4 5 6

r/R (-)

0.000

0.005

0.010

0.015

0.020

0.000

0.005

0.010

0.015

0.020

0.000

0.005

0.010

0.015

0.020zrms (-)

Fig. 25. Reacting flow. Mean and fluctuating part of the azimuthal velocity (w) and mixture fraction (z) obtained using the efficiency function of Charlette et al. (solid lines)and Colin et al. (dashed lines) in comparison with experimental data (dots). R = 15 mm (radius of the buff-body).


S ¼R Ra

Riðuwþ u0w0Þr2dr

DhR Ra

Riðu2 þ u02Þrdr

� 0:7 ð34Þ

As depicted in Fig. 21 the nozzle is placed concentrically inside a co-flow of pure air issuing with 0.5 m/s. The dimensions and measure-ment planes (velocity components have been measured in planesranging from 1 mm up to 120 mm above the swirler exit and spe-cies data is available at 10 mm up to 60 mm) are also given inFig. 21.

The flame, which stabilizes by the recirculation of hot gasesabove the bluff body, has a thermal power of 30kW and coversthe range of 1 < Ka < 4 and 4 < Da < 20 which leads to the regimediagram classification of a thickened wrinkled flame.

The block-structured grid contains 3.2 million control volumesand has been elliptically smoothed to obtain a better orthogonality.The grid has been refined towards the near nozzle region whereasit gets coarser with increasing distance to spare cells. Since theflame is very compact it was possible to maintain almost cubiccells with an edge length of 0.5 mm covering the reaction zone.Hence a spatial constant thickening factor of Fmax ¼ 5 was usedin combination with the dynamic thickening outlined in Section3.2. A time step size of Dt = 18ls has been chosen leading to a Cou-rant number of C3D ¼ 0:77 (temporal average of spatial maximum).

7 The three-dimensional complement to C in Eq. (33).

The diffusion number, increased by the thickening procedure, wasabove D3D ¼ 3 and therewith exceeding the stability limit of the em-ployed Runge–Kutta scheme which is D3D < 0:63. Therefore, accord-ing to Section 3.3, six substeps have been applied within every timestep of the momentum equation to keep the simulation stable. Itshould be noted that the use of dynamic thickening restricts thedestabilizing effect of the ATF-model to a smaller spatial region(i.e. the flame front) and due to the dependency of the diffusionnumber on the grid size temporal variation will occur on non-uniform grids. Nevertheless, despite this stabilizing effect thesimulation became unstable without using the substep strategy orunaffordable small time steps. Previous studies of this burner[47–49] showed that the flow field of the isothermal case can be pre-dicted sufficiently accurate with the RANS approach while largedeviations from experimental data occurred in the reacting caseeven with tuning of model parameters. Especially the flame turbu-lence interaction eludes a time averaged description and requires ahigher spatial and temporal resolution. For this reason LES is a prom-ising tool to improve the prediction capabilities of CFD in such react-ing flows.

4.2. Results

Since this work focuses on combustion, only a short summary ofthe isothermal case will be given in order to show that thenon-reacting flow field can be well predicted by the LES and the


numerical setup used. A more extensive study regarding theisothermal case of this configuration using RANS and LES can befound in [48]. The main part of this section will be devoted toassess the capability of the combustion model in predicting thistype of reactive flow. Therefore all three velocity components aswell as species mass fractions and temperature will be comparedwith experimental data.

4.2.1. Isothermal caseIn Fig. 22 the simulation results of axial and azimuthal veloci-

ties are compared with experimental data. As is typical for thistype of swirl flow, the velocity field starts to expand right afterthe nozzle exit, which produces a positive pressure gradient inthe axial and radial direction, high enough to form a central recir-culation zone as intended for flame stabilization. In the first mea-surement plane (1 mm) directly at the nozzle exit the mean andfluctuating part of the velocity are well predicted. This indicatesthat the inclusion of upstream geometry is sufficient to allow theturbulent structures to form without artificial forcing. Regardingthe time averaged quantities excellent agreement can be observedin all planes further downstream. The fluctuating part is also wellpredicted even though it is slightly underestimated in most of themeasurement planes. In general, this is expected since only the re-solved part of the fluctuations is shown but since the amount ofsub-grid fluctuations is unknown it would be too speculative to de-vote the deviations to it. Overall the spreading of the turbulent

Fig. 26. Reacting flow. Mean and fluctuating part of the CO2 mass fraction (YCO2 ) and temColin et al. (dashed lines) in comparison with experimental data (dots). R = 15 mm (rad

swirling flow and the size and intensity of the recirculation zoneare accurately predicted which is a necessary requirement to as-sess the combustion model in the reacting case.

4.2.2. Reacting caseTo give an illustration of the flame stabilization and instanta-

neous burning behavior two snapshots are shown in Fig. 23. Asone can see from the iso-surface of the chemical source term inFig. 23a the flame has stabilized above the bluff body where themethane-air mixture issuing from the annulus is consumed. A slicecolored with the temperature has been added to illustrate theburnt and unburnt state. As expected in a turbulent flow fieldthe iso-surface is strongly distorted by vortices. Even though theflame turbulence interaction acting on the smaller scales is sup-pressed by the thickening procedure the increased flame surfacedue to wrinkling is significant. Figure 23b shows a slice extractednormal to the axial direction (i.e. perpendicular to Fig. 23a) to illus-trate how the flame is embedded in the turbulent flow field. Goingfrom higher radii to the centerline the contour of the axial velocityreflects the almost laminar coflow of 0.5m/s, the turbulent flowissuing from the annulus around the bluff body and the centralrecirculation zone. The isolines of temperature added in Fig. 23bto mark the preheating and reaction zone show that the flamehas stabilized in the inner shear layer where—in a time averagedsense—the turbulent flame speed matches the velocity componentnormal to the flame front. As one can see the reaction zone is

perature (T) obtained using the efficiency function of Charlette et al. (solid lines) andius of the buff-body).


almost of constant shape while the thickness of the preheatingzone varies strongly. This should be viewed more as a qualitativeillustration rather than a quantitative assessment because someof the deviations are devoted to three-dimensional effects, judgingfrom Fig. 23a, a non-negligible amount.

In Figs. 24–27 the simulation results are compared with exper-imental data. Since the efficiency function is the main modelparameter, different formulations as proposed by Charlette et al.[39] and Colin et al. [38] have been used in these simulations toinvestigate its impact on the results. The model parameter a inEq. (20) has been set to a constant value of 0.08 based on measuredturbulent scales. In general, the velocity and scalar fields are wellpredicted by both simulations. Compared to the isothermal caseall velocity components are increased by the thermal expansionwhich is reproduced by the simulation very well. Except for theoverestimated radial velocity, only minor deviations from the mea-surements exist which cannot be clearly attributed to the combus-tion or turbulence model.

As in the experiment the mixture fraction (Fig. 25 right) dropsin the radial direction caused by the transition from the swirledannular methane-air jet into co-flowing air. The high standarddeviation—which is almost identical in the simulation and experi-ment—indicates that the mixing layer is dominated by turbulentdiffusion. Since the mixture fraction is a non-reacting scalar theslight deviation from the measurements in radial direction can

0.00

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5 6

r/R (-)

10mm

0.00

0.01

0.02

0.03

0.04

0.05

20mm

0.00

0.01

0.02

0.03

0.04

0.05

30mm

0.00

0.01

0.02

0.03

0.04

0.05

YCH4,mean (-)

60mm

0.00

0.01

0.02

0 1 2 3 4 5 6

r/R (-)

0.00

0.01

0.02

0.00

0.01

0.02

0.00

0.01

0.02

YCH4,rms (-)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Fig. 27. Reacting flow. Mean and fluctuating part of the CH4 and O2 mass fraction (YCH4 anet al. (dashed lines) in comparison with experimental data (dots). R = 15 mm (radius of

only be caused by the overestimated radial velocity. Spreadingand broadening of the mixing layer in downstream direction arewell predicted. It should be noted that most of the fuel consump-tion occurs at constant equivalence ratio and only the outer edgeof the flame burns under stratified conditions by the dilution withthe co-flowing air.

On the left of Fig. 26 the CO2 mass fraction is compared with themeasurements. It is of major interest since it is the transported var-iable to describe the reaction where the combustion model entersexplicitly. Since the flame is very sharp the distribution is almostcompletely caused by the flame turbulence interaction resultingin a wide turbulent flame brush. Indeed a bimodal distributionwith only very few intermediate states has been reported in [46].Except that a small radial offset exists and the gradient at higheraxial positions is slightly overpredicted the flame brush was wellcaptured in the simulations. As expected from the thickening pro-cedure the standard deviation is too low at 10 mm because moreintermediate states exist. The deviation from the experiments de-creases at the higher axial positions since the ratio of the flamebrush thickness to the flame thickness increases. At higher radialpositions a second maximum of the CO2 mass fraction exists, com-ing along with a second peak in the standard deviation. This phe-nomenon caused by the recirculation of hot gases over the outerswirler plate has only been partly captured by the simulation.However, the large scatter of the experiments indicates that it is

.00

.05

.10

.15

.20

.25

0 1 2 3 4 5 6

r/R (-)

.00

.05

.10

.15

.20

.25

.00

.05

.10

.15

.20

.25

.00

.05

.10

.15

.20

.25

YO2,mean (-)

0.00

0.02

0.04

0.06

0.08

0 1 2 3 4 5 6

r/R (-)

0.00

0.02

0.04

0.06

0.08

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0.08

0.00

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YO2,rms (-)

d YO2 ) obtained using the efficiency function of Charlette et al. (solid lines) and Colinthe buff-body).

Fig. 28. Frequency spectrum taken at x = 1 mm, r = 20 mm.


a rare event with large time scales introducing statistical uncer-tainties (i.e. the samples available from the measurements as wellas the simulation may be insufficient to quantify it correctly).

On the right of Fig. 26 the temperature is compared with exper-imental data. As outlined in Section 2.1 the temperature is only apost-processing step and indicates whether the connection to thetwo controlling variables can describe the physical situation. Ingeneral, the agreement with the measurements is comparable tothe CO2 mass fraction, but differences exist regarding the burntgas temperature. The main reason for this deviation is thought tobe the water-cooled bluff body with a wall temperature of 353 K.The heat loss is obvious in the measurements at 10 mm wherethe temperature drops to 1730 K on the centerline above the bluffbody. As expected, this cannot be captured by the simulation sincethe zero-gradient boundary condition for the controlling variablescorresponds to an adiabatic wall. This could be improved by addingenthalpy as a controlling parameter into the table and solving thecorresponding transport equation (e.g. [15,19]). As reported in [46]the temperature drop can be observed at the higher axial position,too due to the long residence time of the fluid in the recirculationzone.

To further assess the capability of the chemistry reduction topredict the species distribution the mass fractions of CH4 and O2

representing the reactants of the system are compared with themeasurements in Fig. 27. The mean profile of the methane massfraction reflects its consumption by the chemical reaction on theleft and the mixing with the co-flowing air on the right comingalong with a double peak in the standard deviation. Therebythe fluctuation induced by the flame turbulence interactionsignificantly exceeds that of the pure mixing situation. At the high-er axial positions both the CH4 and O2 mass fraction exceed themeasurements indicating insufficient fuel consumption which isconsistent with the CO2 mass fraction being too low in this region.

Comparing the efficiency functions, a clear trend regarding allflow quantities can be observed when going over from the formu-lation given by Colin et al. [38] to that of Charlette et al. [39](marked with arrows in Figs. 24–27). As mentioned in [39] Eq.(21) results in higher values than Eq. (18) when the model constantb in Eq. (20) is not adjusted according to the turbulent Reynoldsnumber (which is not possible since the local turbulent Reynoldsnumber is unknown in LES). In agreement with this, the meanflame position is shifted to a more forward position (e.g. position30 mm in Fig. 26 and CH4 mass fraction at 60 mm in Fig. 27). Asa result the axial and azimuthal velocity at 10–30 mm increasedue to the thermal expansion through the flame front. In addition,the radial velocity is slightly increased since the flow is directedfurther outwards by the forwarded flame position. In general, thedifferences between the results obtained with the two formula-

tions are small and hence the model uncertainty will not influencethe results crucially.

Finally, measurements of the temporal autocovariance will beused to further assess the simulations. These measurements havebeen done by Schneider et al. [45] to quantify the precessing vortexcore (PVC) which rotates above the bluff body. Under isothermalconditions, this coherent structure contributes a significant partto the fluctuations while it vanishes in the reacting case. Hence itis interesting if the simulations are able to capture this transientphenomenon and its suppression by viscous forces. As in theexperiment, the axial velocity has been monitored at the swirlerexit (the position is given in Fig. 21) to construct the spectra basedon the autocovariance Ruu = hu(t)u(t + s)i (h � i denotes time averag-ing, s is the temporal shift, the velocity has been monitored for 3 scorresponding to 114 rotations of the PVC). Results are given inFig. 28 where an excellent agreement of the simulations and theexperiment can be observed. The magnitude of energy associatedwith the corresponding frequency is well predicted. Regardingthe PVC, a distinct peak at 38 Hz, almost identical in the simulationand experiment, can be observed in the isothermal case. Eventhough the monitoring point is positioned in the cold flow up-stream of the flame front, this peak does not exist in the reactingcase since the large coherent structure is not able to form underthis condition.

5. Conclusion

The well established thickened flame approach has been cou-pled with two-dimensional tabulated chemistry to include detailedchemistry effects into the simulation. The correct couplingwith the LES solver has been verified and a quantification of theresulting error when neglecting resolution requirements revealedunacceptable numerical inaccuracies. A separated treatment oftime integration for scalars and velocity improved the computa-tional efficiency to obtain results in an acceptable amount of time.Findings regarding the dynamic flame thickening have been givenbut its impact on the efficiency function cannot be quantified with-out three-dimensional DNS. The application of the model to aswirled turbulent premixed flame showed good agreement withexperimental data.

Acknowledgments

Computations have been performed on the National Supercom-puter of the ’Leibniz-Rechenzentrum der Bayerischen Akademieder Wissenschaften’ (HLRB-II Project ID pr47ve) and on theHessian High Performance Computer (HHLR) in Darmstadt. We


gratefully acknowledge financial support by the German ResearchCouncil (DFG) and the German Science Council (WR) through theCollaborative Research Center SFB 568 and the Cluster of Excel-lence initiative Center of Smart Interfaces.

References

[1] F. Biagioli, Combust. Theor. Model. 10 (2006) 389–412.[2] W. Lazik, T. Doerr, S. Bake, R.v.d. Bank, L. Rackwitz, in: ASME Turbo Expo. Conf.

Proc., 2008, pp. 797–807.[3] F. Zhao, M.C. Lai, D.L. Harrington, Prog. Energy Combust. Sci. 25 (1999) 437–

562.[4] A.W. Vreman, J.A. van Oijen, L.P.H. de Goey, R.J.M. Bastiaans, Flow Turbul.

Combust. 82 (2009) 511–535.[5] B. Fiorina, R. Vicquelin, P. Auzillon, N. Darabiha, O. Gicquel, D. Veynante,

Combust. Flame 157 (2010) 465–475.[6] L. Selle, G. Lartigue, T. Poinsot, R. Koch, K.U. Schildmacher, W. Krebs, B. Prade, P.

Kaufmann, D. Veynante, Combust. Flame 137 (2004) 489–505.[7] J.P. Legier, T. Poinsot, D. Veynante, in: Proceedings of the Summer Program

2000, Center for Turbulence Research, 2000, pp. 157–168.[8] M. Boileau, G. Staffelbach, B. Cuenot, T. Poinsot, C. Brat, Combust. Flame 154

(2008) 2–22.[9] P. Schmitt, T. Poinsot, B. Schuermans, K.P. Geigle, J. Fluid Mech. 570 (2007) 17–

46.[10] G. Boudier, L. Gicquel, T. Poinsot, Combust. Flame 155 (2008) 196–214.[11] U. Maas, S. Pope, Combust. Flame 88 (1992) 239–264.[12] O. Gicquel, N. Darabiha, D. Thvenin, Proc. Combust. Inst. 28 (2000) 1901–1908.[13] B. Fiorina, O. Gicquel, L. Vervisch, S. Carpentier, N. Darabiha, Combust. Flame

140 (2005) 147–160.[14] J.A. van Oijen, L.P.H. de Goey, Combust. Sci. Technol. 161 (2000) 113–137.[15] J.A. van Oijen, F.A. Lammers, L.P.H. de Goey, Combust. Flame 127 (2001) 2124–

2134.[16] T. Poinsot, D. Veynante, S. Candel, J. Fluid Mech. 228 (1991) 561–606.[17] Chem1D, A one-dimensional laminar flame code, developed at Eindhoven

University of Technology, <www.combustion.tue.nl/chem1d>.[18] G.P. Smith, D.M. Golden, M. Frenklach, N.W. Moriarty, B. Eiteneer, M.

Goldenberg, C.T. Bowman, R.K. Hanson, S. Song, WCG Jr., V.V. Lissianski, Z.Qin, GRI-Mech 3.0, <www.me.berkeley.edu/gri_mech/>.

[19] B. Fiorina, R. Baron, O. Gicquel, D. Thevenin, S. Carpentier, N. Darabiha,Combust. Theor. Model. 7 (2003) 449.

[20] A.W. Vreman, B. Albrecht, J.A. van Oijen, L.P.H. de Goey, R.J.M. Bastiaans,Combust. Flame 153 (2008) 394–416.

[21] W.J.S. Ramaekers, J.A. Oijen, L.P.H. Goey, Flow Turbul. Combust. 84 (2009)439–458.

[22] A. Ketelheun, C. Olbricht, F. Hahn, J. Janicka, in: ASME Turbo Expo Conf. Proc.,2009, pp. 695–705.

[23] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, second ed.,R.T. Edwards, Inc., Philadelphia, USA, 2005.

[24] J. de Swart, G. Groot, J.A. van Oijen, J.H.M. ten Thije Boonkkamp, L.P.H. de Goey,Combust. Flame 145 (2006) 245–258.

[25] J.A. van Oijen, G. Groot, R.J.M. Bastiaans, L.P.H. de Goey, Proc. Combust. Inst. 30(2005) 657–664.

[26] J.A. van Oijen, R.J.M. Bastiaans, L.P.H. de Goey, Proc. Combust. Inst. 31 (2007)1377–1384.

[27] T. Lehnhaeuser, M. Schaefer, Int. J. Numer. Methods Fluids 38 (2002) 625–645.[28] G. Zhou, L. Davidson, E. Olsson, in: Fourteenth International Conference on

Numerical Methods in Fluid Dynamics, 1995, pp. 372–378.[29] H.L. Stone, SIAM J. Numer. Anal. 5 (1968) 530–558.[30] J. Smagorinsky, Monthly Weather Rev. 91 (1963) 99–164.[31] M. Germano, U. Piomelli, P. Moin, W.H. Cabot, Phys. Fluids A 3 (1991) 1760–

1765.[32] D.K. Lilly, Phys. Fluids A 4 (1992) 633–635.[33] I.N. Bronstein, K.A. Semendjajew, Handbook of Mathematics, third ed.,

Springer-Verlag GmbH, Heidelberg Germany, 1997.[34] T. Butler, P. O’Rourke, in: International Symposium on Combustion, vol. 16,

1977, pp. 1503–1515.[35] P.J. O’Rourke, F.V. Bracco, J. Comput. Phys. 33 (1979) 185–203.[36] C. Meneveau, T. Poinsot, Combust. Flame 86 (1991) 311–332.[37] C. Angelberger, D. Veynante, F. Egolfopoulos, T. Poinsot, in: Proceedings of

the Summer Program 1998, Center for Turbulence Research, 1998, pp.61–82.

[38] O. Colin, F. Ducros, D. Veynante, T. Poinsot, Phys. Fluids 12 (2000) 1843–1863.

[39] F. Charlette, C. Meneveau, D. Veynante, Combust. Flame 131 (2002) 159–180.

[40] L. Durand, W. Polifke, in: ASME Turbo Expo Conf. Proc., 2007, pp. 869–878.[41] P. Auzillon, B. Fiorina, R. Vicquelin, N. Darabiha, O. Gicquel, D. Veynante, Proc.

Combust. Inst. 33 (2010).[42] C. Hirsch, Numerical computation of internal and external flows,

Fundamentals of Numerical Discretization, vol. 1, John Wiley & Sons, Inc,Hoboken, USA, 2001.

[43] J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The InitialValue Problem, John Wiley & Sons, Inc, Hoboken, USA, 1991.

[44] J.C. Butcher, Numerical Methods for Ordinary Differential Equations, seconded., John Wiley & Sons, Inc, Hoboken, USA, 2008.

[45] C. Schneider, A. Dreizler, J. Janicka, Flow Turbul. Combust. 74 (2005) 103–127.[46] M. Gregor, F. Seffrin, F. Fuest, D. Geyer, A. Dreizler, Proc. Combust. Inst. 32

(2009) 1739–1746.[47] E. Schneider, A. Maltsev, A. Sadiki, J. Janicka, Combust. Flame 152 (2008) 548–

572.[48] F. Hahn, C. Olbricht, C. Klewer, G. Kuenne, R. Ohnutek, J. Janicka, in: Proc. of the

ISTP19, 2008.[49] G. Kuenne, C. Klewer, J. Janicka, in: ASME Turbo Expo Conf. Proc., 2009, pp.

369–381.

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CFD-LES Modeling of Premixed Combustion Using a Thickened Flame Approach

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