Study of a Filtered Flamelet Formulation for Large Eddy ...

Study of a Filtered Flamelet Formulation for Large EddySimulation of Premixed Turbulent Flames

Christophe Duwig

Received: 16 March 2007 /Accepted: 13 August 2007 /Published online: 28 September 2007# Springer Science + Business Media B.V. 2007

Abstract Despite significant advances in the understanding and modelling of turbulentcombustion, no general model has been proposed for simulating flames in industrialcombustion devices. Recently, the increase in computational possibilities has raised thehope of directly solving the large turbulent scales using large eddy simulation (LES) andcapturing the important time-dependant phenomena. However, the chemical reactionsinvolved in combustion occur at very small scales and the modelling of turbulentcombustion processes is still required within the LES framework. In the present paper, arecently presented model for the LES of turbulent premixed flames is presented, analysedand discussed. The flamelet hypothesis is used to derive a filtered source term for thefiltered progress variable equation. The model ensures proper flame propagation. The effectof subgrid scale (SGS) turbulence on the flame is modelled through the flame-wrinklingfactor. The present modelling of the source term is successfully tested against filtered directnumerical simulation (DNS) data of a V-shape flame. Further, a premixed turbulent flame,stabilised behind an expansion, is simulated. The predictions agree well with the availableexperimental data, showing the capabilities of the model for performing accuratesimulations of unsteady premixed flames.

Keywords Turbulent premixed combustion . Large eddy simulation .

Flamelet model . Filtered flame

1 Introduction

Today, the major challenge of combustion devices is to meet environmental requirementswhile fulfilling growing energy needs. Thermal power technologies are widely used forelectricity production but the enforcement of new laws on harmful emissions will require

Flow Turbulence Combust (2007) 79:433–454DOI 10.1007/s10494-007-9107-1

C. Duwig (*)Division of Fluid Mechanics, Department of Energy Sciences, Lund University,Box 118, Lund 22100, Swedene-mail: [email protected]

new technologies to make thermal power plants cleaner and more efficient. For example,the use of lean premixed burners in gas turbines allows the temperature in the device to bereduced and nitric oxide emissions (NOx) to be lowered. However, while operating in leanpremixed modes, new stability problems appear (e.g. flashback, thermo-acoustic insta-bilities). Thus, important issues are control of the combustion process and the reduction ofpollutant production.

Significant advances in the understanding and modelling of turbulent combustion havebeen obtained over the past 30 years [1, 2] but no general model has been presented forpredicting inherently unsteady turbulent combustion. For a long time, time-accuratesimulations were not realistic and predictions were restricted to the Reynolds-averagedNavier-Stokes (RANS) framework. Recently, the increasing power of computing resourcesbrought forward the possibility to accurately resolve the time properties of the flow usinglarge eddy simulation (LES).

The philosophy of LES is to resolve the large scales of flow (including most of theturbulent kinetic energy) and to model only the smallest scales. The modelling activity haslead to different subgrid models for turbulence [2, 3]. However, in turbulent combustion,oxidation reactions occur usually at scales much smaller than those resolved in LES.Therefore, an extension of LES to premixed combustion requires subgrid modelling of thecombustion process. Early attempts to adapt RANS combustion models to the LESframework were presented (see [1, 2]) but new closures have also been proposed andpreliminary results are encouraging. At this point, it is worth mentioning that Boger et al.[4] derived their algebraic relation similarly to the Bray-Moss-Libby (BML) model (see [1,2] for a description of the original model within the RANS framework) to LES.

One may deal with the closure problem following two avenues, namely, resolving theflame front or capturing the flame front dynamics without resolving it. In the first category,Colin et al. [5] used the thickened flame method and artificially increased the flamethickness in order to be able to resolve the front directly on the LES grid. An efficiencyfactor (often modelled as a subgrid wrinkling factor) is introduced to correct the flame/turbulence interaction. So far, this approach has been used together with simple chemistrymodels to predict combustion instabilities [2, 5]. Another possibility has been suggested byKerstein [6] and has been extended for combustion application by Chakravarthy and Menon[7]. The so-called linear eddy model (LEM) locally reduces the flame front to a 1D problemoriented in the direction of the temperature gradient. The flame front is resolved along thenormal to the front and turbulence effects are included using stochastic re-arrangementevents. Recently, Sankaran and Menon [8] used the LEM together with complex chemistryand accurately predicted temperature fields in a gas turbine combustion chamber. Themodel enables the simulation of any type of flame, including extinction/re-ignition, butsuffers from very high computational costs.

The second method consists of capturing the front displacements without resolving it. Itoften is a direct extension of the flamelet assumption that has been successfully applied inthe RANS framework [1, 2]. The flamelet assumption relies on the fact that the reactionoccurs in layers thinner than the Kolmogorov length scale. For premixed combustion, itmeans that the flame front is wrinkled but that eddies do not enter the flame region nor alterthe flame structure significantly. The validity of the flamelet assumption has been studiedand discussed before [1, 2] and this assumption is today a successful tool for buildingcombustion models.

Within the flamelet regime, two major classes of closures have been suggested. Firstly, alevel-set approach that uses a scalar G iso-level for tracking the flame front [1, 9]. The levelzero of the scalar G denotes the flame front and propagates at the turbulent flame speed.

434 Flow Turbulence Combust (2007) 79:433–454

The subgrid flame speed is not known and requires modelling. In addition, coupling of theG-equation with the Navier-Stokes equations requires an additional model for the chemical/heat source terms as a function of G. The last step is not straightforward but is expected tohave some influence on the flame dynamics. Despite these limitations, this technique hasbeen widely used for the LES of combustion chambers, e.g. [9].

A second class of flamelet models are based on the filtered progress variable c (denotingthe normalised temperature or species mass fractions). As will be shown in detail later,unclosed terms (reaction and transport) need to be modelled. The filtered c-equation sourceterm is closed using the filtered flame surface density Σ (i.e. the flame surface per unit offilter box). Σ is computed from a transport equation [10, 11] or from an algebraicexpression [4, 10, 12, 13]. The non-resolved transport is modelled by two terms: thegradient and the counter-gradient transport terms [4, 10, 12]. Modelling separately thetransports and reactions raises additional constraints for ensuring proper propagation. It can,for example, be done using the Kolmogorov-Petrovski-Piskunov analysis [2, 4]. Anotherdrawback of these approaches is that the flame front is assumed to be infinitely thincompared to the filter size. It limits considerably the applicability of the model. However,these models have been shown to be suitable for simulations of industrial devices, such asgas turbine combustion chambers [13].

Recently, Duwig and Fuchs [14] suggested a formulation accounting for finite flamethickness. They performed LES of turbulent flames using this new flamelet formulationbased on the filtered c-equation [14, 15]. In addition, Fureby [16] compared this model toother techniques and highlighted the potential of the approach. The first results are veryencouraging but some theoretical aspects need to be tested and discussed. For instance, themodelling of the production term π would benefit from comparisons with direct numericalsimulation (DNS) data. In addition, a comparison with available combustion models wouldhighlight similarities and advantages of the present approach.

In this paper, the flamelet formulation used in [14, 15] is presented and discussed indetail. The formulation provides a closure that simulates the flame propagationconsistently with the filtered flame structure. After presenting the closure problem, theformulation is described. The effects of subgrid scale (SGS) turbulence are included usinga subgrid wrinkling factor. It accounts for finite flame thickness effects by using asuitable diffusion term. The analysis of the diffusion term leads to the definition of a non-dimensional number a describing the filtered flame structure. The differences between thepresent model and existing ones are also discussed. The validation is performed in twosteps. Firstly, model predictions are tested versus DNS data using explicit filtering.Finally, the capabilities of the formulation of simulating turbulent flames are shown bypresenting computations of a premixed unsteady turbulent flame stabilised behind anexpansion.

2 Governing Equations

The basic equations describing the motion of a fluid are the conservation equations ofmomentum, mass and energy [2]. The system of equations has to be closed with anequation of state. If the characteristic speed of the fluid is considerably smaller than thespeed of sound, the acoustical effects are decoupled from the flow, simplifying the systemof equations (low Mach number assumption). Under this assumption, the density is afunction of temperature and composition only. In addition, turbulent flows are inherentlyunsteady and there is no physical significance to a single instantaneous flow-field.

Flow Turbulence Combust (2007) 79:433–454 435

Therefore, one seeks to describe the statistical properties of the dependent variables andapply a low-pass filter to the governing equations, written as:

@ ρ@t

þr � ρeuð Þ ¼ 0 ð1Þ

@ ρeu@t

þr � ρeueuð Þ ¼ �rP þr � ρeueu� ρuuþ μreuð Þ ð2Þ

@ρeY i

@tþr � ρeueY i

� �¼ r � ρeueY i � ρuYi þ ρDireY i

� �þ w i ð3Þ

@ρ eT@t

þr � ρeu eTÞ ¼ r � ρeu eT � ρuT þ ρDTreT� �þ wT

ð4Þ

rT ¼ P

R

� �ð5Þ

where u is the velocity vector, T the temperature, P the pressure, ρ the density, μ theviscosity, R the specific ideal gas constant, Di the diffusivity of species i, Yi the massfraction of species i and ωi the reaction term of the species i. The overbar denotes the spatialfiltering operation and the tilde the density-weighted filtering:

eQ ¼ ρQρ

ð6Þ

In order to close the above system of partial differential equations, an SGS model isneeded. The SGS terms include the effect of the unresolved scales on the resolved ones. Inaddition to the modelling of SGS turbulence (i.e. turbulent transport), an expression for thefiltered chemical reaction term is required. Since the LES filter size is, ideally, on the orderof the Taylor micro-scale, chemical reactions often occur at much smaller scales than thefilter size. Consequently, combustion requires SGS models. The problem of combustionchemistry is that a comprehensive mechanism of oxidation involves hundreds of species. Itis unrealistic to resolve hundreds of additional species transport equations. Thus, instead ofconsidering the individual species, a simplification is necessary. The combustion process issummarised by a single scalar, a progress variable which may be related to the fuel massfraction, temperature or density field. Equations 3 or 4 can be normalised and re-written interms of a variable c satisfying c=0 for the fresh gases and c=1 for the burnt gases. Thegoverning equation for c is:

@ ρec@t

þr � ρecec� � ¼ SGSc þr � ρDTrc� �þ wc ð7Þ

SGSc ¼ r � eρeuec� ρucð Þ ð8Þ


Equation 7 describes the dynamics of the flame, not resolving the flame front itself, butthrough a filtered flame coordinate (the filtered progress variable). Although the flame frontis much thinner than the LES filter (or mesh spacing when implicit filtering is used), theflame position/displacement is obtained via the filtered progress variable field.

Simplifying the combustion process to a single variable is equivalent to summarising thecombustion process as a global reaction. The approach is common in numerical simulationsof reacting flows as it reduces the computational burden to a single extra equation [2, 4,10–19]. However, the right hand side (RHS) in Eq. 7 requires closure, which is presented inthe next section.

3 Flamelet Closure Problem

The classic strategy is to discuss separately the unclosed terms in Eq. 7 and propose aseparate modelling for each of them. However, the final objective is to provide a globalclosure to simulate flames. A convenient way of closing Eq. 7 is to consider a function πthat satisfies the following:

@ ρec@t

þr � ρeuec� � ¼ ρ0D0r2ecþ π ec; D0ð Þ ð9Þ

Equation 9 defines the function π. The closure problem focusses on obtaining a suitablemodel for the function π together with a constant effective diffusion ρ0D0 (to be determinedlater) in order to ensure correct flame propagation. From Eqs. 7 and 9, we see that thefunction π models both subgrid transport and subgrid reaction. As π is defined to be a sumof filtered terms, one expects it to be distributed at least over a filter length, i.e. it cannot besteeper than any filtered quantity in Eq. 9. In addition, π acts as a source term and enforcesthat the progress variable ranges from 0 (fresh state) to 1 in the burnt gases. Consequently,the integration of π across the filtered flame front should be related to the integral burningspeed, i.e. it is positive. However, the local value of π is not necessarily positive. Anothercondition is that the balance π diffusion should lead to a correct filtered flame propagationspeed.

3.1 Modelled flame

For closing Eq. 9, a simple distribution for modelling π is first chosen (following Duwig etal. [14, 15]), namely, a Gaussian function in space so that p ¼ a � exp � 6�X 2

Δ2

� �; where X is

the distance normal to the flame front, α is a coefficient and Δ is the filter length. π will bedistributed over a filter length and will satisfy the integral condition. We note that, althoughthe term π is not necessarily positive, the closure model enforces π≥0 and this assumptionwill be verified in Sect. 5.

In order to study the filtered flame propagation, the filtered flame displacement speed SΔ(defined with respect to the fresh gases) and the fresh gas density ρu are introduced.

We have by definition:

ρuS$ ¼@ ρec@t þr � ρ euec� �

recj j ¼ ρ0D0r2ecþ π X ; D0ð Þrecj j ð10Þ


Considering a planar flame, Eq. 10 is written in the coordinate X normal to the flamefront:

ρuS$decdX

¼ ρ0D0d2ecdX 2

þ p X ; D0ð Þ ð11Þ

In Eq. 11, SΔ expresses that subgrid turbulence accelerates the flame by increasing itssurface. The increase of flame surface is traditionally modelled using the wrinkling factorΞ, so S$ ¼ ΞSL; where SL denotes the planar laminar flame speed. The introduction of thewrinkling factor appears in many combustion models (e.g. [2, 4, 5, 9, 10, 12–16]). MakingEq. 11 non-dimensional by introducing a distance normalised by the filter length Δ,x ¼ X=$; gives:

decdx

¼ρ0D0

$2d2ecdx2 þ p X ; D0ð Þ

ρuSLΞ$

¼ ρ0D0

ρuSLΞ$

d2ecdx2

þ α � $ρuSLΞ

exp �6x2� � ð12Þ

Possible boundary conditions are (only two of them are necessary):

ecx!�1 ! 0;ecx!þ1 ! 1;decdxx!�1

! 0;d2ecdx2 x!�1

! 0 ð13Þ

The integral condition leads to:

α ¼ ρuSLΞ

ΔRþ1

�1exp �6x2ð Þdx

¼ ρuSLΞ$

ffiffiffiffi6

π

rð14Þ

Finally, introducing the non-dimensional number a, Eq. 12 gives:

decdx

¼ ρ0D0

ρuSLΞ$

d2ecdx2

þffiffiffiffi6

p

rexp �6x2

� � ¼ 1

a

d2ecdx2

þffiffiffiffi6

p

rexp �6x2

� � ð15Þ

Equation 15 describes the structure of a modelled filtered flame. The modelled filteredflames’ structure depends only on the parameter a. In the next section, the structure of themodelled filtered flames is studied and important properties of the model are presented.

3.2 Modelled filtered flame structure and important properties

Before solving Eq. 15, the meaning of the parameter a is discussed. The parameter acompares the propagation/integral reaction rate and the effective diffusion. Low values of aimply that the burning rate is low compared to diffusion fluxes. Consequently, the filteredflame should have a “thick” preheating zone (non-reactive layer). On the other hand, highvalues of a give a strong reaction rate compared to the diffusive flux and the flame will bevery “thin” (the preheating zone tending to zero thickness).

In the limit of a laminar flame, the expression of a can be simplified and we havea~Δ= Dth=SLð Þ~Δ=δl; where Dth denotes the thermal diffusivity and δl is the flamecharacteristic length. We note that a>>1 corresponds to the infinitely thin flame assumption(c=0 or 1) considered by Boger et al. [4], while a<<1 corresponds to a flame much thickerthan the filter size. We also note that, even if the flame is infinitely thin (a>>1), the filteredflame front thickness will never be below the filter length.


Solving Eq. 15 leads to a unique solution for a given a, namely:

ec � xð Þ ¼ a �ffiffiffiffi6

π

r�Zþ1

�1exp azð Þ

Zþ1

�1exp �6s2 � as

� �ds

0@

1Adz ð16Þ

The influence of the parameter a on the filtered flame front thickness is shown inFig. 1a. For a>>1, the flame thickness tends to the filter size. As plotted in Fig. 1b, thefiltered c-level corresponding to the reaction layer moves from 1 to 0.5 as a increases. Forlow values of a, the diffusion scale is much larger than the reaction scale. Consequently, thereaction occurs in a small layer compared to the diffusion zone, the layer getting closer tothe level c=1 as the diffusion increases. When a increases, the diffusion decreases until itreaches zero (corresponding to the case of an infinitely thin flame front) and the filteredc(x=0) iso-level tends toward 0.5. Figure 2 shows the representation of the reaction rate inthe c-space for different values of a. The reaction zone centre (corresponding to themaximum reaction rate) is moved towards the burned gases while a decreases. In the casea>>1, π recovers the parabola-like shape used by Boger et al. [4]. It is worth noting that,while the flame position (iso-level x=0) moves in the filtered c-space when a varies, it isfixed in the geometrical space (i.e. the coordinate x).

The present analysis introduces the parameter a for describing the subgrid flamethickness accounting for a finite laminar flame thickness. It is an improvement compared tothe models quoted previously [4, 10, 12], which correspond to the limiting case a→∞.

3.3 Effect of curvature on a laminar flame—determination of the parameter a

One considers the limiting case of a filtered curved laminar flame propagating at a speed Sd,defined with respect to the unburnt mixture. Projecting the c-equation on the normal to theflame front (defined positive if directed towards the unburnt mixture) gives:

ρuSd ¼@ρec@t þr � ρeuec� �

recj j ¼ ρ0D0d2ecdX 2 � ρ0D0κ dec

dX þ p X ; D0ð ÞdecdX

¼ ρuSL 1� κ$a

� �ð17Þ

Fig. 1 a Filtered progress variable (c) versus distance from the flame front normalised by the filter size (x)for different values of the parameter a together with the Π term. b ec�level corresponding to the maximum ofΠ (i.e. ec (x=0)) versus the parameter a


where κ is the filtered flame front curvature. It can be inferred from Eq. 17 that thethickening of the flame front by filtering affects the propagation of the filtered flame andthe speed becomes proportional to the curvature of the flame front divided by a. Acommonly used correlation relating the displacement speed with the planar laminar flamespeed uses the Markstein length L; Sd=SL(1−κ·L) [2]. From Eq. 17, the followingexpression a ¼ $=L is obtained and is fully consistent with the analysis presented in theprevious section. We also note that the flame displacement speed of the filtered flamerecovers the laminar flame speed for planar flames. As expected, the filtered flame frontpropagates at the same speed as the corresponding laminar flame.

In the flamelet regime, the flame structure is not affected by turbulence and laminarquantities (such as Markstein length) are constant. Consequently, the analysis of thecurvature term enables to determine the value of a (and, consequently, the effectivediffusion ρ0D0) in the flamelet regime to be a ¼ Δ=L: It is an improvement compared to[14], where a was estimated without justification.

3.4 Closure for 3D flames

The previous sub-sections presented a model for the function π(X, D0). However, thevariable X is not known and it is preferable to express the closure as a function of thefiltered progress variable. Figure 2 shows the shape of the function π mapped as a functionof a and of the filtered progress variable ~~c: From Fig. 2, a library for storing π/α=Π ( ec; að Þis constructed. Knowing the local value of the progress variable value and the number a, theec�equation is closed. The evaluation of a is done in the limit of a curved laminar flamewith a ¼ $=L and is consistent with the flamelet hypothesis assuming that turbulence does notmodify the flame structure, i.e. a is constant in the flamelet regime and does not depend on theturbulence level. This assumption limits the range of applicability of the present model.

The closed ec�equation reads as follows:

@ρec@t

þr � ρ euec� �¼ ρuSLΞ$

ar2ecþ ρuSLΞ

$Π ec; að Þ ð18Þ

Fig. 2 Π term plotted in the filtered progress variable space for several values of a; 2D (right) and 3D (left)representations


4 Comparisons with Other Combustion Models

4.1 Analogy of momentum and mass/heat transport

From Eq. 18, we note that the closed c-equation right-hand side consists of two terms. Thefirst term resembles an effective diffusion, which is the most popular approach formodelling SGS scalar transport using an analogy with the SGS transport of momentum.The subgrid scale diffusivity D$ð Þ is assumed to be proportional to the subgrid scalekinematic viscosity v$ð Þ; introducing the turbulent Schmidt number ScT. In the presentcase, the effective diffusivity is modelled using the wrinkling factor. The subgrid scalewrinkling factor is expressed as a function of the subgrid scale fluctuations u$; oftenmodelled from the subgrid scale viscosity [2].

Comparing the subgrid scale diffusivity with the effective diffusivity in Eq. 19 gives:

ρD$~ρuSL Ξ � 1ð Þ$

a¼ ρuu$$

a~ρuu$a

ð19Þ

where Damköhler’s expression [2] for the subgrid wrinkling factor has been used.The subgrid scale diffusivity obtained from Eq. 19 recovers the analogy scalar/

momentum transport with ScT∼a. The present approach and analysis allow refining the ideaof analogy between heat/mass and momentum transport in premixed flames.

In addition, for high turbulence levels, the expression proposed by Damköhler is knownto overestimate the wrinkling. More sophisticated correlations have been proposed tocompute the subgrid scale wrinkling. In this case, the turbulent Schmidt number would be afunction of a, u$; SL and the filter size Δ. For example, Colin et al. [5] proposed tomultiply u$ by an efficiency function Γ. This function is computed from DNS data tocorrect the influence of small turbulent eddies. Using the present flamelet approach givesScT∼a/Γ(SL). Through this relation, the chemistry affects the turbulent Schmidt number viathe laminar flame speed. Unsurprisingly, an increase of SL (i.e. a decrease of Γ) leads to anincrease of the Schmidt number. In other words, the efficiency function Γ traduces that theflame is less sensitive to the smallest eddies and, therefore, less sensitive to turbulence.

4.2 Relation to the thickened flame technique

In the present approach, the filter acts as a thickening procedure and allows representing theflame front on the computational grid. The diffusion term increases with the filter size whilethe reaction rate decreases. The filter size plays the same role as the thickening factor F in thethickened flame model” (TFM) proposed by Colin et al. [5]. However, in LES context, thefiltering operation is also applied to the velocity fields, unlike the thickening operation.Unlike Arrhenius expressions, the π term is not stiff in physical space but is distributed overone filter length. We use Δ∼2h, where h is the cell size, while the TFM spreads the flametypically over 5–6h. On the same grid, the filtered flamelet technique allows a betterrepresentation or less thickening of the filtered flame front.

4.3 Differences with the coherent flame technique

The present approach is based on the c-equation closed by an algebraic expression. Its formis similar to the coherent flame (CF) technique. Considering the limit a→∞, Duwig [17]showed that the present technique predicts the parabola-shaped flame surface density curve


predicted by Boger et al.’s model [4] and was supported by their experimental data. Inaddition, they ensured the unicity of the solution by setting the diffusion coefficientaccording to the Kolmogorov-Petrovski-Piskunov analysis. In the present formulation, thediffusion coefficient is derived from physical arguments and yields a unique solution for allvalues of a.

Another advantage of the present formulation comes with accounting for finite flamefront thickness. Most of the other c-equation closures [4, 12] assume that the flamethickness is infinitely thin compared to the filter size, limiting the applicability of thesemodels for simulating real flames. The present approach extends the CF technique,introducing the non-dimensional number a.

5 Comparison with Direct Numerical Simulation

A straightforward validation of the technique consists of extracting the Π term from a DNSdata set and to compare it to the model. We consider the DNS data obtained by Vervisch etal. [18] on a premixed V-flame. The flame is stabilised behind a hot wire and is constantlydisturbed by turbulence. Tabulated complex chemistry is used to ensure a goodrepresentation of the flame structure [18, 19]. The computations were performed in 2D.Turbulence is known to be an intrinsic 3D phenomenon and 2D simulations do not captureseveral features of the turbulence (e.g. vortex tubes). However, Vervisch et al. [18] pointedout that their 2D results correctly represent the interaction of premixed flame sheet withvorticity. These data were used successfully for model analysis [18, 19]. In the presentpaper, the effect of filtering on the flame structure is studied. The V-flame data consisted ofa well resolved wrinkled realistic flame and is suitable for testing the model.

Table 1 summarises the two flames used in this section. Figure 3 shows an instantaneousdensity field for flame 1. The flame is anchored at the hot wire and develops behind it.Vortices wrinkle the flame, creating cusps in the flame front. The DNS data (progressvariable, density and velocities fields) were filtered explicitly using a Gaussian kernel.Figure 3 also shows the density-weighted filtered progress variable for a filter size of fourtimes the laminar flame thickness (Δ/δ=4). The filtering operation smooths out the progressvariable gradients, distributing the flame front and removing details of the flame topology,especially in the high-curvature regions.

The normalised Π term was extracted from the filtered DNS data using Eqs. 9 and 18 sothat:

π ¼ ρuSLI$

9 ec; að Þ ¼ @ρec@t

þr � ρ euec� �� ρuSLI$

ar2ec ð20Þ

Table 1 Flame properties from the direct numerical simulation (DNS) data

Case Turbulent velocityfluctuation/laminar flamespeed

Turbulent length scale/laminar flame thickness

TurbulentReynoldsnumber

Laminar flamethickness/meshspacing

Flame 1 2.5 4.6 37 ∼15Flame 2 3.75 4.6 55 ∼15


with the filtered quantities being obtained by explicit filtering using Eq. 6 and the localsubgrid wrinkling factor was computed as:

I � rcj jrcj j ð21Þ

It is worth noting that Eq. 21 does not yield singularities (cusps or region with relativelyhigh curvature) when the denominator is zero. Therefore, Eq. 21 only provides an estimateof the wrinkling factor. The parameter a was evaluated as a=Δ/L, with L being estimated tobe approximately half of the laminar flame thermal thickness δ. Table 2 summarises thedifferent cases with varying filter sizes.

Figures 4 and 5 show the Π function extracted from the DNS data compared to themodel prediction. For relatively low filter length (or a<6), the scatter points are mainlyconcentrated around a curve. For a=2, the model over-estimates Π at low values of ec andunder-estimates Π otherwise. For instance, the DNS data scatter indicate that Π is negativefor ec < 0:4; which is not possible to capture with the present model. For a=4, the modelcaptures well the shape of the Π curve. In particular, the location of the maximum atec0:75� 0:8 is well predicted. However, one notices that most of the DNS data points areabove the model prediction line. For larger filter sizes (a>6), the scatter plots are morespread out. The global shape of the curve, in terms of magnitude and location of themaximum, is recovered by the model. However for a=16, the DNS data points hardlyexhibit any clear trend that could be used to model the Π term. As pointed out above, theestimation of the subgrid wrinkling factor might fail locally and one may seek to representΞ·Π instead. Figure 6a shows Ξ·Π in the ~~c space for a=8. The data set is similar to the onepresented in Fig. 5b but the scatter points are less spread out, displaying a relatively clear

Fig. 3a, b Flame 1. a Instantaneous progress variable field. b Density-weighted filtered progress variablefield with Δ/δ=4

Flame Filter size/laminar flame thickness: Δ/δ a

1 1 21 2 41 4 81 8 162 2 42 4 8

Table 2 Specifications of thefiltering tests performedusing DNS data


curve. The point distribution fits between the two lines predicted using the model, assumingΞ=1 and Ξ=3. Actually, the Ξ distribution for this particular case (not presented here)indicates that most of the Ξ values are between 1 and 2. It corresponds to the distribution ofthe scatter plots shown in Fig. 6a. Finally, Fig. 6b shows the mean Π curve (i.e. averagedover the different points for a given ecÞ obtained from DNS data together with the model

Fig. 4a–d Π function vs. density-weighted filtered progress variable for different filter sizes applied onFlame 1. a Δ/δ=1. b Δ/δ=2. c Δ/δ=4. d Δ/δ=8. The filtered DNS data and the model predictions are shownas dots and lines, respectively

Fig. 5a, b Π function vs. density-weighted filtered progress variable for different filter sizes applied onFlame 2. a Δ/δ=2. b Δ/δ=4. The filtered DNS data and the model predictions are shown as dots and lines,respectively


prediction for a=8. The model under-estimates the Π term for ec < 0:4 and for ec > 0:7 andpredicts a maximum at ec ~ 0:65 compared to ec ~ 0:75 for the curve extracted from DNSdata. However, the model is shown to be accurate for a between 4 and 8 and is suitable forclosing the ec�equation:

6 Large Eddy Simulation of a Turbulent Premixed Flame

6.1 Presentation of the LES tool

The proposed flamelet formulation was implemented into a Cartesian finite difference LEScode solving a low Mach number formulation of the Navier-Stokes equations [20]. TheCartesian finite differences techniques are fast, easy to code and accurate. These advantagesmake them suitable for the testing of LES of reacting flows. Spatial derivatives arediscretised using a fourth-order central scheme, except for the convective terms, which arediscretised using a fifth-order weighted essentially non-oscillatory scheme (WENO) [21].The WENO scheme ensures high-order accuracy in the smooth regions and enables tocapture accurately steep gradients in the flame region. Since the present technique enablescapturing the filtered flame front on 2–3 grid cells, the last point is of particular importancefor ensuring stability of the numerical technique. A second-order accurate upwind schemeis used for computing the time derivatives. The code is fully implicit and uses multigridacceleration, as well as local refinements [20].

For closing the Navier-Stokes equations, the filtered structure function model (FSF) [22]has been used.

Beside the Navier-Stokes and continuity equations, Eq. 18 is solved for capturing theflame front. The physical quantities, such as laminar flame speed and unburnt/burnt density,are taken from a 1D laminar detailed computation. The open-source code CANTERA [23]is used together with the San Diego 05 chemical oxidation mechanism [24].

The Π term is stored in a library (e.g. in Fig. 2) and read when needed. The filtereddensity was computed from the progress variable using the ideal gas law and assuming thatthe pressure does not change significantly across the flame (low Mach number assumption).The subgrid scale flame wrinkling is computed following Fureby [12] usingI ¼ u$=SLð Þ2�Dx; where Dx is the fractal dimension. uΔ is computed by taking the curlof the high-pass-filtered velocity [5] in order to take into account thermal expansion and

Fig. 6a, b Filtered Flame 2 with Δ/δ=4. a Ξ·Π function vs. density-weighted filtered progress variable. bMean DNS data together with the model predictions


large-scale fluctuations. The fractal dimension Dx is computed from a correlation followingFureby [12].

The vortex core is visualised using criteria based on the second largest eigenvalue of thevelocity derivatives tensor proposed by Jeong and Hussain [25] (the so-called λ2 technique).The vortex core is approximated by the region where the eigenvalue λ2 is negative.

6.2 Presentation of the flame

A premixed flame at low Mach number in a dump combustor is studied. The geometrycorresponds to an experimental test rig located at the Ecole Nationale Supérieure deMécanique et d’Aéronautique in Poitiers, France [26, 27]. Laser Doppler anemometry(LDA) measurements were performed within the European project MOLECULES [27]. Themeasurements were conducted in the inlet channels and in the combustors for both reactingand non-reacting cases.

The combustor consists of two 3-m long channels discharging into a 2-m longcombustor. The channel height is 30.4×10−3 m. The two channels are separated by a10×10−3-m thick splitter plate that ends 70.4×10−3 m upstream from the dump plane. Thedump plane consists of two symmetric backward-facing steps and the step height ish=29.9×10−3 m. The depth of the geometry is 150.5×10−3 m. In the present paper, we focuson an atmospheric of perfectly premixed propane/air flame, referred as C1 [27]. For thiscase, the fuel/air equivalence ratio was set to 0.75, the fresh gas temperature was 273 K, thebulk velocity was U0=11.5 m/s, the average Reynolds number was Re=25,000 and theKarlowitz number was estimated to be below 1.5, indicating that the choice of flamelet-based combustion models is valid [1, 2]. This flame has been used previously [12, 19]for testing combustion models and is relatively well documented, making it a suitable testcase.

The uncertainty regarding the velocity field measurement in the flame region wasestimated to be between 6% and 8% [26, 27]. In addition, one should estimate theuncertainty resulting from the time averaging procedure. For flow-varying density, asystematic error is introduced as simulations typically consider Favre averages, whereasLDA data are expected to be close to Reynolds averages. Denoting the time averagingoperation by ⟨ ⟩, a possible estimate of the systematic error (SE) on a variable u isSE ¼ ρu

.ρ

D E� ρuh i

.ρD E

: For the present simulation, the SE for the streamwisevelocity was found to be below 5% at distances lower than 7h from the expansion plane.Further downstream, it increases up to about 15% at a distance of 10h.

Modelling of the incoming turbulence is a difficult but important task for LEScomputations. Steady inflow conditions ensure a correct incoming mass flow but do notmimic many features of turbulence. However, in some cases, strong shear appears in theflow and generates some turbulence. In addition, the flame C1 is characterised by a stronginstability, resulting in large periodic variations. Using a triple decomposition technique ofthe LDA turbulent data, Bruel et al. [26, 27] pointed out that the streamwise velocityfluctuations consist of a low-frequency sinusoidal part, together with a seemingly randomhigh-frequency part. The inflow velocity profile may be written in a general form asfollows:

eu x; tð Þ ¼ U xð Þ þ A xð Þ � UFLUCT x; tð Þ þ U0B sin 2p � f � tð Þ ð22Þ

where U denotes the time-averaged inflow velocity, UFLUCT generates a seeminglyturbulent fluctuation, A and B are amplitudes and f is the pulsation frequency. The averaged


inflow velocity was taken from LDA measurements [26, 27]. The amplitude A was set sothat the fluctuation amplitude matched the seemingly random part of the fluctuation asobtained from the LDA measurements. The excitation frequency was set to f=50 Hz,corresponding to St=f *h/U0∼0.13 and the amplitude B was set to 0.27, corresponding to theresults of the triple decomposition. The seemingly random fluctuation (UFLUCT) wasgenerated using a digital-filter-based technique [28].

A mass conservative zero-gradient outflow boundary was used. No-slip adiabatic wallswere enforced.

The computational grid contained ∼1,300,000 points, with an averaged mesh spacing of∼1.6×10−3 m. A grid sensitivity analysis [29] showed that this resolution was suitable forperforming LES. It is worth stressing that the computational geometry covered only half ofthe real geometry in the spanwise direction. The major motivation was to reduce the CPUcost and is supported by previous work [12, 19, 29]. With the present resolution, a is closeto 6, with the filter size Δ computed as Δ∼2*V0.333, where V is the volume of thecomputational cell. We note that a∼6 was tested in Sect. 5, leading to good agreementbetween the present model and the filtered DNS data. The model is then used within itsrange of validity.

The time step was computed using a CFL number of ∼0.2, chosen to preserve timeaccuracy. The time-averaged quantities were computed over ∼16,000 time steps(corresponding to ∼16 flow-through times).

Fig. 7a, b Time-averaged flame in the ORACLES rig. a 2D cut of the time-averaged temperature field [K]predicted by LES. b Direct light visualisation [26, 27], obtained with an exposure time of 1/50 s


6.3 Results and discussion

Figure 7 shows the time-averaged temperature field obtained by LES and a direct lightvisualisation of the flame [26]. The flame stabilises behind the steps and is symmetric. Itconsists of a V-shape flame brush pointing towards the exit of the combustor. The flamebrush lies in the shear layer between the central jet and the wakes of the steps. The flamebrush thickness covers almost all of the combustor at x/h∼7 downstream of the expansion.Both the experimental visualisation and the mean temperature field show the samequalitative flame pattern.

Figure 8 shows time-averaged quantities plotted along crosswise lines. The progressvariable field shows relatively steep gradients close to the expansion but smooths outfurther downstream. The recirculation zones behind the steps contain burnt gases andstabilise the flame. For x/h<7, the progress variable at the centreline is zero and the flamebrush divides into an upper and a lower part. Further downstream, the two parts merge andthe flame brush covers almost the whole cross-section of the combustor. Figure 8 alsoshows averaged velocity components obtained by LDA [26, 27] and LES. The agreementbetween the two data sets is on the order of the experimental uncertainty, indicating that thepresent LES tool captures the flame accurately. Both results show that the jet discharges

Fig. 8 Time-averaged quantities: LES vs. LDA [26, 27]. Top: progress variable field. Centre: streamwisevelocity component. Bottom: crosswise velocity component


into the combustor, creating two recirculation zones behind the steps. The recirculationzones are relatively short (∼3h) compared to the flame length. The streamwise velocityprofiles are relatively flat between x/h=4 and x/h=7. At x/h∼7–10, a clear minimum isobserved at the centreline. It results from the thermal expansion that accelerates burnt gasesin the upper and lower parts of the combustor. It is worth noting that the density ratio isclose to 8 for that particular case. Consequently, an accurate prediction of the flame brushlocation and thickness is needed to obtain good velocity component prediction. Figure 8indicates that the predicted velocity components agree well with the LDA measurements,suggesting that the flame brush is also well predicted, although there are no experimentalmeasurements for direct comparison.

Figure 9 shows profiles of the root-mean-squared (RMS) of the velocity fluctuations aswell as the Reynolds stress components Ruv. The overall agreement between the LES andLDA [26, 27] data is reasonable. Close to the expansion plane, i.e. x/h<6, the agreement isbelow the experimental uncertainty. Further downstream, the agreement is less good and theLES over-estimates the fluctuation levels. In order to understand the reason for the

Fig. 9 Second-order velocity moments: LES vs. LDA [26, 27]. Top: stream/cross-wise Reynolds stresscomponent. Centre: RMS of the crosswise velocity fluctuation. Bottom: RMS of the streamwise velocityfluctuation


discrepancies, a snapshot of the flame front is presented in Fig. 10. The flame is attached tothe corner of the steps. The flame surface extends downstream and is strongly wrinkled.Figure 10 also shows the vortex cores captured by the λ2 technique [25]. Large-scale vortextubes are visible in the channels resulting from vortex shedding in the wake of the splitterand from the periodic pulsation. These tubes are convected into the combustion chamberand, eventually, encounter the flame about 7h downstream of the expansion. At thislocation, the large tubes break down into smaller tubes, resulting in large flame wrinkling.This point explains the increase of velocity fluctuation intensity at x/h=7–10. Figure 11shows the instantaneous progress variable field at three instants during an acoustic cycle.The pictures show a strongly pulsating flame. A large pocket of fresh gas is injectedperiodically and travels downstream. The flame undergoes a large flapping motion, asreported by Nguyen and Bruel [26]. Further downstream, the fresh gas pockets break downand these gases are burnt. For a given point located downstream of x/h∼7, this mechanismimplies large fluctuations with hot (i.e. high streamwise velocity) and cold (i.e. relativelylow streamwise velocity) fluid parcels passing alternatively. Recalling that the density ratioacross the flame front is ∼8, it shows that the time-averaging technique might significantly

Fig. 10a, b Snapshot of the 3D flame. a Flame surface (iso-level c=0.5). b Vortex cores (iso-surface of λ2[25]) and flame trace (lines c=0.5)


affect the results. In other words, the LES results are Favre-averaged data, while the LDAare not. The resulting difference is expected to be relatively high in the case of strongintermittency and large density changes, explaining why LES predictions are almost twotimes higher than the LDA measurements.

Regarding the average streamwise velocity, SE defined previously provides an estimateof the discrepancies due to averaging. Recalling that SE was found to be below 5% for x/h<7 with 6% measurement uncertainty (MU), the total error would be ~11%, which islarger than the discrepancies between the LES and LDA data sets. Further downstream,these discrepancies increase but SE increases also with a peak up to about 15% at x/h=10.The differences between the LES and LDA data sets stay, therefore, below the sum of theSE and MU.

Fig. 11a–c 2D cut of the instantaneous progress variable field at three phase angles � within an acousticcycle. a �=0. b �=2π/3. c �=4π/3


Finally, the contribution of the unresolved scales to the present simulation is estimated.Figure 12 shows the local instantaneous subgrid wrinkling factor distribution. Thedistribution shows that the subgrid wrinkling is most likely below 3, with a plateaubetween 1 and 2 and an exponential decrease above 2. The tail of the distribution extendsup to 5. One may compare this value to the turbulent flame wrinkling, which is estimated tobe between 10 and 40 (computed from the time-averaged velocity and progress variablefields). These results indicate that most of the flame wrinkling is resolved during thecomputation and only a small fraction is modelled. Consequently, the subgrid wrinklingmodel is not expected to have a significant influence upon the result as reported by Fureby[12].

7 Summary

The present paper is devoted to the study of a flamelet model for the simulation ofpremixed turbulent flames using large eddy simulation (LES). A non-dimensional number ais introduced that compares the filter length and the flame characteristic thickness. Thisnumber enables to account for finite flame thickness, extending the validity range oftraditional flamelet models. Further, the model was compared to other available combustionmodels and the benefits of the present formulation were highlighted.

The model closure predictions were also compared to filtered direct numericalsimulation (DNS) data fields, showing that the model closure is able to predict accuratelythe production term Π for a between 4 and 8. This makes the model a suitable base fordeveloping new models for the LES of complex industrial flames (e.g. in gas turbinecombustors), which would typically lie within this range (i.e. the LES filter length being∼2–6 times the flame thickness).

Finally, the model was used to simulate a premixed turbulent flame. The LES grid wasfine enough to ensure that a∼6, making the present closure accurate. The predictions werewell in line with the experimental data, with discrepancies often below the experimentaluncertainty. It is concluded that the present formulation accurately simulated unsteadypremixed turbulent combustion within the flamelet regime using a single scalar forcapturing the flame front.

Fig. 12 Distribution of the localinstantaneous subgrid scalewrinkling Ξ in the combustor atthe three instants shown inFig. 11


Acknowledgements The author thanks Prof. Luc Vervisch and Dr. Pascale Domingo (CORIA, Rouen,France) for providing the V-flame DNS database. The author also thanks Dr. Pascal Bruel (University Pau/CNRS, France) for making available the LDA data.

This work was financially supported by the Swedish Energy Agency STEM within the program“Termiska processer för el-produktion.” The computations were run on HPC2N and LUNARC facilitieswithin the allocation program SNAC.

The routine used to generate the time-dependent inflow boundary condition was kindly provided byDr. Markus Klein (TU Darmstadt, Germany).

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http://www.cantera.org

http://maeweb.ucsd.edu/~combustion/cermech/olderversions/sandiego20050615/

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