DNS simulations of lean premixed flame kernels with flamelet generated manifolds: An overview

P. Wesseling, E. O˜nate and J. P´eriaux (Eds) c

° TU Delft, The Netherlands, 2006

DNS SIMULATIONS OF LEAN PREMIXED FLAME

KERNELS WITH FLAMELET GENERATED MANIFOLDS;

AN OVERVIEW

Rob J.M. Bastiaans, Jeroen A. van Oijen and L. Philip H. de Goey

Eindhoven University of Technology, Department of Mechanical Engineering P.O.Box 513, 5600 MB Eindhoven, The Netherlands

e-mail: r.j.m.bastiaans@tue.nl web page: http://www.combustion.tue.nl

Key words: Turbulent combustion, DNS, Premixed, Flamelet generated manifolds Abstract. In this paper recent developments in the application of the method of flamelet generated manifolds (FGM) to simulations of turbulent premixed combustion are consid-ered. Initially the method was developed for kinetic reduction for simulations of laminar combustion using stationary solvers. Recently the method was extended to direct numer-ical simulations of turbulent combustion. A lean premixed flame kernel was taken for a number of studies, investigating the application of the FGM technique. To that end the laminar mass burning rate was assessed in the context of the influence of stretch. Also a theoretical stretch model was involved in this test. It was found that the theoretical stretch was reproduced even up to the thin reaction zones regime. Validation also includes backsubstitution of stretched flamelets from the DNS into a detailed chemistry code. In two-dimensional cases a direct validation is performed with detailed chemistry for both unity Lewis numbers and physical Lewis numbers. Finally a flamelet theory of stretch effects in subgrid scales is evaluated from the DNS.

1 INTRODUCTION

1.1 Background

Turbulent premixed combustion of gaseous fuels is one of the most important energy conversion processes today, but its physics are not fully understood. On the one hand a detailed knowledge is required to understand the behavior of the conversion process and the efficiency and formation of pollutants. On the other hand this insight is needed to obtain the parameterizations that are essential in developing accurate models for large scale simulations1,2_{. One of the most important processes to understand is the physics of}

The flame surface area is very important in simulations with time or volume averaging, which is the goal in simulations of large practical systems. Both of these properties will be given attention to in the present overview.

A major part of the current activities in turbulent combustion is based on the laminar flamelet concept. Using this idea, it is assumed that the chemical time scales in the flame are generally much smaller than the time scales in the turbulent flow. This concept leads to the introduction of a flame front as a thin layer propagating through the surrounding flow field, while the turbulent eddies are larger than the flame thickness. A classification of the different available models for premixed turbulent combustion has been presented recently by Peters3_{. Early contributions are due to Damk¨ohler}4 _{and Spalding}5_{. A}

well-known model has been developed over the years by Bray, Moss and Libby6 _{and is based}

on the assumption that the flame front is infinitely thin. Other interesting and promising models are the coherent flame model7,8 _{and the G-equation model}3_.

One of the most important ingredients of all flamelet models is the description of the influence of flame stretch on the behavior of the flame. On the one hand, it is well known that flame stretch has an important influence on the local mass burning rate of the thin instantaneous flame front. Flame stretch effects together with curvature effects are, on the other hand, responsible for the creation and destruction of flame surface area7 _{and flame}

wrinkling3_{. There are several ways to take stretch effects on local mass burning rates into}

account. This can be done e.g. by using models derived from direct numerical simulation (DNS) data, measurement data or data arising from asymptotic theories9,10_{. In case of}

the G-equation, the weak-stretch limiting behavior found in asymptotic methods is used3_.

It is doubtful whether the weak-stretch theories can be used for the highly stretched turbulent flames in the corrugated flamelet and thin reaction zones combustion regimes. In this framework, De Goey and Ten Thije Boonkkamp11 _{introduced a new flame stretch}

theory based on a stretch field defined by a flame volume, rather then a flame surface. The theory gives an expression for the mass burning rate as function of stretch, based on the new definition. The result should be valid for combustion in which the flamelet assumption holds. This means that the deviations from locally one-dimensional behaviour are very small. This can be tested with Direct Numerical Simulations (DNS). In this context, by analyzing experimental data, De Goey et al.12 _{found that the theory is valid even in the}

thin reaction zones regime. Also the experimental study of Shepherd et al.13 _indicates

that even when the smallest turbulent scales are smaller then the flame thickness, there is no significant flame broadening.

For applications in turbulent cases, De Goey14_{, extended the theory to the influence}

of strong stretch to the turbulent mass burning rate in a volume averaged sense, i.e. a contribution to subgrid modelling in Large Eddy Simulations (LES).

1.2 DNS with of flame kernels with FGM

The disadvantage of using three-dimensional (3D) DNS is that the application of detailed chemical kinetic mechanisms is still limited, due to the very high computational costs that are involved. This is why the number of full 3D DNS studies with detailed chemistry is quite limited in literature and most of them involve combustion with a simple kinetic scheme. For this reason, we want to use a chemical reduction technique which is still able to describe the relevant physics we want to study. For this purpose we use the recently developed chemical reduction technique, referred to as the Flamelet Generated Manifold (FGM) method, to model the chemistry in the flame, Van Oijen et al.15,16_{. The research}

on the present subject started when we connected FGM to a fully compressible DNS code, Bastiaans et al.17_{, for a suitable problem, defined by expanding premixed spherical flame}

kernels, Groot18_.

The advantage of the problem definition of a spherical flame kernel is that the spherical periodicity facilitates high resolution statistical sampling. Additionally, at the same time, all boundaries are outflow boundaries, which is an advantage for the numerical imple-mentation. Here a disadvantage is the mean background curvature, which varies in time, starting from a high value and decreasing. Studying flat turbulent flames (like e.g. Bell et al.19_{, in which it was found that flame wrinkling is the dominant factor for increasing}

the turbulent flame speed) would be a good alternative because it does not suffer from background curvature and it does not require interpolation for statistical sampling. Also the flame does not reach the computational boundaries easily and long time integrations can be performed. Very recently we started work on this setup De Swart et al.20_{, but here}

we will only discuss spherical flames.

By using FGM the chemical kinetics is solved in advance and parametrized in a table. The state of the reactions, including diffusion, are assumed to be directly linked to a small set of progress variables. The conservation equations for the progress variables are solved in the DNS, with the unclosed terms being retrieved from the table. This allows the use of detailed chemical kinetics without having to solve the individual species conservation equations. FGM can be considered to be combination of the flamelet approach and the intrinsic low-dimensional manifold (ILDM) method of Maas and Pope21 _{and is similar to}

the Flame Prolongation of ILDM (FPI) introduced by Gicquel et al.22_{. FGM is applied}

similarly to ILDM. However, the thermo-chemical data-base is not generated by applying the usual steady-state relations, but by solving a set of 1D convection-diffusion-reaction equations describing the internal flamelet structure. The main advantage of FGM is that diffusion processes, which are important near the interface between the preheat zone and the reaction layer, are taken into account. This leads to a very accurate method for (partially) premixed flames that uses fewer controlling variables than ILDM.

1.3 Study of turbulent premixed flame kernels

com-bustion in an Otto engine is a good example of this kind of structure in practice and this could on itself already be a very good reason for studying turbulent flame kernels. The initial turbulence is decaying, as is the case in many practical combustion applications (discontinuous-flux devices, but not in continuous-flux applications like in jet engines).

Here a lean premixed turbulent expanding flame kernel is studied. Lean premixed combustion is becoming the method of choice for ground based gas turbine combustors due to several advantages. The high percentage of air results in complete combustion, reducing emissions of hydrocarbons and carbon monoxide. The excess air also results in lower combustion temperatures and as a consequence low emissions of nitrogen oxides. Therefore, significant research on turbulent premixed combustion has been performed under lean conditions, e.g. some of the studies already mentioned13,19_.

Many researchers have used flame kernels for the study of premixed turbulent com-bustion. Studies of particular interest, besides those based on DNS-FGM that will be mentioned later on in this overview, are the researches in Jenkins and Cant23_{, Gashi}

et al.24_{, Th´evenin}25 _{and Jenkins et al.}26_{. The study of Th´evenin is associated with rich}

flames, which are not addressed here. In Jenkins and Cant23_{the evolution of shape}

param-eters were studied in terms of flame normals and curvatures by means of DNS combined with single step chemistry. One of their conclusions is that at low turbulence intensities there is a tendency to favor spherical over cylindrical curvature. In Gashi et al.24_{, the}

numerical simulations of Jenkins and Cant23 _{were extended and supplemented by}

exper-imental PLIF observations for both methane and hydrogen combustion at stoichiometric and lean conditions. The result of the study is a qualitatively good agreement between the simulations and experiments. In a very recent case of Jenkins et al.26_{a spherical flame}

kernel in the thin-reaction zones regime was assessed by means of DNS and simple chem-istry. They analyzed correlations of curvature, tangential strain rates and displacement speeds and it was found that the existence of preferential curvature has a significant effect on the displacement speed.

Another study, which also gives an overview of many related papers, is given in Lipat-nikov and Chomiak27_{. In this study k − ² simulations have been performed and compared}

to measurements in terms of flame speed. Here the turbulent length scale is taken as an important parameter and relatively long time scales are considered, starting at a time where the present simulations have already finished.

1.4 Organization of the overview

since the parameter space is quite large and it is not possible to perform a large set of 3D detailed chemistry calculations for all those cases. Important questions that have to be answered are: what is the performance of the FGM method under stretch and curvature and partial premixing, does it represent the acoustics with sufficient accuracy, up to what Karloviz numbers can the method be applied, how many controlling variables are needed and can it account for preferential diffusion effects?

To circumvent the need for a very large computing power we started with a number of tests in lower physical dimensions, compared to 3D time dependent cases. First, 2D laminar and steady cases given by a Bunsen type of flame and a partially premixed case15,16_{. In steady cases the fields can be compared directly with each other. Secondly,}

a 3D unsteady laminar expanding flame kernel was tested with respect to the burning velocity as a function of flame radius, basically a 1D problem18_{. Third, it was assessed}

whether the acoustics18 _{and the burning velocity}28 _{was the same with FGM compared to}

detailed chemistry in a 1D case. To go over to real turbulent cases, which means three-dimensional and time dependent cases, we then involved flamelet theory11 _{to assess the}

mass burning rate as a function of stretch and curvature for which a consistent picture is still missing today (see e.g. Echekki and Chen29_{). Here a flamelet tracking approach was}

conducted in which at different instants in time the entire flame kernel was sampled by means of calculating tracking paths30 _{normal to the flame surface.}

Using the theory11 _{as a reference, a number of studies were conducted for unity Lewis}

Numbers. To be able to focus on the direct effects of stretch and curvature and to avoid that the results are obscured by related effects due to preferential diffusion, we restricted the analysis to unity Lewis numbers. For this special case, the local mass burning rate is only changed due to flame stretch and not due to the combined effect of flame stretch and preferential diffusion. Although the flame stretch theory11 _{includes the}

latter effect, the unity Lewis number assumption makes it easier to apply the flame stretch theory and to understand the results. However, we realize that the coupling of stretch and curvature with non-unity Lewis number transport processes is an important issue in premixed turbulent flames (see e.g. de Swart et al.31 _{and Lipatnikov and Chomiak}32_).

Therefore, the work presented can be considered as a first step to study the role of flame stretch in turbulent flames. The next step is to include preferential diffusion effects.

It was the aim of these studies to evaluate whether the theory and simplified versions of it, including the asymptotic versions, are able to describe the influence of flame stretch and curvature on the local burning velocity in a highly turbulent flame. In a first few papers, van Oijen et al.33 _{and Bastiaans et al.}28,34_{, we studied spherically expanding flames in the}

corrugated flamelet regime and the lower part of the thin reaction zones regime of the premixed combustion diagram. For these flames the theory described the effect of flame stretch and curvature on the local burning velocity very well. A next study30 _{was carried}

stretch theory is tested severely due to the very high local stretch rates. In this study the DNS results were also evaluated by 1D flamelet calculations with detailed chemistry and flame stretch and surface area profiles taken from the DNS. In a later study35 _the

thin reaction zones regime was explored in more detail and the evolution of the flame surface was taken into account. Also studies36_{were performed to investigate the required}

dimension of the manifold. It seemed that for the case studied, 1D manifolds result in small but significant errors which can be diminished up to an order of magnitude.

At the moment progress is made by doing 2D simulations with detailed chemistry at different locations in the thin reaction zones regime. The results of these simulations can be compared directly to results obtained with FGM. Here higher dimensional implementa-tions of FGM will be included as well. The detailed chemistry calculaimplementa-tions are performed for Lewis numbers equal to unity, but also for physical, but constant, Lewis numbers. Also, with respect to LES computations, the 3D DNS-FGM results are analyzed37_{. As}

explained in the first section here also a theory is involved, describing effects of strong stretch for the turbulent propagation of the flame front. For this study good results are already obtained.

The paper is organized as follows. First the governing equations, as well as the numer-ical method that is used in all the studies is given. Here also the chemistry and initial conditions will be discussed. Then the results of the mentioned studies will be discussed and the paper ends with some conclusions on the work that is performed so far.

2 NUMERICAL SETUP

2.1 Governing equations

Freely expanding flames are modelled in a turbulent flow field using DNS. More detailed information about the DNS program can be found in Bastiaans et al.17 _{and Groot}18_{. The}

governing equations are,

with α indexing the species involved and σij = µ Ã ∂ui ∂xj + ∂uj ∂xi − 2 3δij ∂uk ∂xk ! ; p = ρRT. (2)

The viscosity of the mixture, µ is computed with Sutherland’s law and the ratio of the thermal conductivity and the heat capacity at constant pressure (λ/cp) is assumed to be

a function of temperature only, Smooke and Giovangigli38_{. The species heat capacities}

are tabulated in polynomial form. Unity Lewis numbers are assumed for all species in order to prevent differential diffusion effects from obscuring the direct effects of stretch and curvature on the mass burning rate. For DNS-FGM the summation term in the temperature equation comes entirely from the manifold as well as the source term in the species equation. In that case the latter equation is used as a progress variable, which defines the primary entry in the FGM look-up table.

2.2 Numerical method

The equations are in fully compressible form and are solved in a three-dimensional cubic computational domain with a typical length of 12 mm and 254 grid points uniformly distributed in each direction. This gives a mesh size of approximately 0.0472 mm in each direction. The equations are discretized with a finite difference method on a collocated grid. For the spatial discretization of second derivatives, the sixth order accurate compact finite difference method of Lele39 _{is used. Also first derivates in diffusion terms (for}

non-constant coefficients) are calculated with a sixth order accurate compact scheme39_{. First}

order derivatives connected to advection are treated by a compact upwind fifth order finite difference method, developed by de Lange40_{. Here an asymmetry parameter of}

ra= 5/9 is used. The upwind scheme provides some damping but it has better dispersive

characteristics, which results in an accurate and stable method. For comparison the spectral characteristics of both the compact sixth order central scheme and the compact fifth order upwind scheme are displayed in figure 1.

The time integration is performed explicitly with a compact storage third-order Runge-Kutta method. A time step of order 10−8 _{is used to satisfy the stability criteria, in FGM}

limited by the acoustics. When applying detailed chemistry the accompanying stiffness results in a time step that should be one order of magnitude lower28_.

The boundary conditions are modelled with the Navier-Stokes Characteristic Boundary Conditions (NSCBC) of Poinsot and Lele41_{. The initial flame kernels are expanding}

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 z/π Re(z ′ )/ π Ref U5 C6 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 z/π Im(z ′ )/ π U5 C6

Figure 1: Resolution characteristics of spatial discretization schemes, modified wavenumber as function of the input wavenumber. Left: Real part showing dispersion, Right: Imaginary part, showing the damping of the scheme. Compact sixth order central scheme for a first derivative, C6, showing zero damping, compact fifth order upwind scheme with asymmetry ra = 5/9, U5, showing damping but

higher resolution characteristics.

2.3 Chemistry

The manifold used in the present researches is based on the GRI3.0 kinetic mechanism with 53 species and 325 reversible reactions42_{. The scaled mass fraction of carbondioxide,}

which is monotonically increasing, is used as the single controlling variable (progress variable) for most cases. Since pressure, enthalpy and element mass fractions are constant in these flames, they are not needed as additional controlling variables. A large portion of the terms and parameters in the governing equations, equation (1), are given by the manifold. These items include the source term for the progress variable(s), the source term for the temperature equation, which is given by all the terms in the summation over species (the large term in square brackets), the viscosity, conductivity, specific gas constant, the heat capacity at constant volume and the ratio of heat capacities. The full 1D equations for detailed chemistry are solved with CHEM1D43,44 _{to construct the}

manifold.

The chemistry is chosen due to the large interest in the power industry in lean premixed combustion engines and there is detailed knowledge of its chemical kinetics. Therefore premixed combustion of a methane/air mixture is used, with an equivalence ratio of φ = 0.7. An additional advantage for lean methane chemistry is that the flame speed and the flame thickness are equal for the full GRI3.0 kinetics, and for the case in which these kinetics are used with a Lei = 1 approximation (as can be observed from simulations

using CHEM1D43,44_{, displayed in figure 2). We will use the Le}

i = 1 approximation in

0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 25 30 35 40 GRI 3.0 Burning velocity, s l , [m/s] Equivalence ratio, φ [−] 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10 −3 GRI 3.0 Gradient thickness, δf , [m] Equivalence ratio, φ [−]

Figure 2: Burning velocity and flame gradient thickness from simulation of laminar methane air com-bustion as function of equivalence ratio based on GRI3.0 at 300 K and 1.013250·105 _{Pa. Left: Burning} velocity using full GRI3.0 kinetics (drawn line), mixture average Lewis numbers approximation (dashed) and Lei = 1 approximation (dash-dotted). Right: Flame gradient thickness, full GRI3.0 (drawn) and Lei= 1 approximation (dash-dotted).

diffusion effects.

2.4 Initial conditions

The initial conditions are a laminar spherical flame, calculated by CHEM1D43,44_,

su-perimposed on a turbulent field. There is no forcing in the simulation, so the turbulence will decay in time. In order to select a physical condition to be considered one normally refers to a certain region of the premixed combustion regime diagram given by Peters3_.

Given a certain chemical mechanism, the regime is given by the applied turbulence in terms of the amplitude of the velocity fluctuations and the mean turbulent coherence length given by the Taylor integral scale.

The procedure to initialize the turbulence starts with drawing random numbers for a stream function, ψ. This field is subsequently filtered multiple times by means of a top-hat filter. The discrete implementation of this top-top-hat filter over 2 grid spacings is done by a Simpson integration rule, with coefficients [1

4, 1 2,

1

4] and extended in a straightforward

way to three dimensions. This results in a smooth field in which derivatives can be approximated with sufficient accuracy and a certain Taylor scale depending on the number of filter operations is obtained. It must be remarked that the number of filter operations generally is of the order of 100, so that the effective filter is an accurate approximation of a Gaussian. As the number of iterations, ii, increases the Gaussian filter width, σi,

0 1 2 3 4 5 6 0.16 0.17 0.18 0.19 0.2 0.21 0.22 Time [ms] Burning velocity [m/s] dx=0.133 mm dx=0.1 mm dx=0.05 mm dx=0.025 mm 0 1 2 3 4 5 6 0.16 0.17 0.18 0.19 0.2 0.21 0.22 Time [ms] Burning velocity [m/s] dx=0.2 mm dx=0.133 mm dx=0.1 mm dx=0.05 mm dx=0.025 mm

Figure 3: Results for 1D flames with DNS and left: detailed chemistry, right: FGM with a single progress variable.

for the resulting filter, G,

G(r, σi) = à 1 6σ2 iπ !_3/2 exp à − r2 6σ2 i ! . (4)

In order to keep disturbances away from the domain boundaries, the resulting field is windowed with a tanh function, decreasing from 1 to 0 at a diameter of 0.8 times the domain size and with a width of 0.05 times the domain size. Note that the initial laminar flame is always contained within the turbulent region. Subsequently, derivatives are taken from the stream function ψ to obtain the velocity components of a solenoidal field (by construction),

ui = ²ijk

∂ψk

∂xj

, (5)

with ψk = ψ and the Levi-Civita tensor ²ijk.

3 RESULTS

3.1 Laminar test cases and acoustics

3.1.1 1D laminar case

For the compressible case a 1D version of the DNS code was used as described previously in Bastiaans et al.28_{. It is a time dependent solver, so the solutions are a function of time.}

This means that the burning velocity is not a direct result but must be assessed by tracking some characteristic point of a variable that is monotonically changing in the flame front. Complex chemistry is implemented by the TROT method, Evlampiev et al.45_{. The results}

of the burning velocity as function of time and resolution are given in figure 3.

criterion of 15 points/mm is met, being consistent with the incompressible calculations. At the very fine grid of spacing 0.025 mm an oscillation can be observed although the deviations of the amplitude stay within 1 mm/s, which is a half percent of the burning velocity. This probably has to do with the numerical setup with a fixed inlet and a partially non-reflecting outlet in which standing acoustic waves can emerge. An FGM of the GRI3.0 kinetics is constructed on the basis of the CHEM1D results. The outcome for the burning velocity as function of time is also given in figure 3. The converged burning velocity is somewhat higher then those obtained by detailed kinetics but the deviations are within reasonable limits. Convergence is obtained at about 10 points/mm, which is a little less restrictive compared to the cases with detailed kinetics. Again a deviation is observed at the finest grid, which must be due to the compressibility effects and boundary conditions. By using FGM the resolution requirements are marginally improved but the stiffness of the problem is totally gone. The time step could be increased with the grid size up to the most coarse grids and it was totally dictated by the acoustics. Therefore here the power of FGM is demonstrated. The number of species is lowered from 36 to only a single species and the time steps at a resolution of 10 points/mm is 4 times as large as the time steps at a resolution of 40 points/mm, which is two times as large as the time step for detailed chemistry. Therefore a speed-up of a factor of 50 can be achieved (including the decrease of the number of variables by a factor of 6), along with the large decrease in use of computer memory. Generally it can be seen that both the detailed chemistry and the FGM method give a convergence in the transient at the finest grids.

3.1.2 Acoustics in 1D laminar combustion

In order to keep the acoustics intact it was chosen not to take the temperature from the manifold. Instead the temperature equation was kept in the system and manifold was used for the source term of this equation. The acoustics were studied in Groot18 _{with one step}

Figure 4: The pressure distribution through a flat flame at t = 0.25, 0.30, 0.35, and 1.00 ms after the start of the simulations. Single-step reaction mechanism (dashed lines) and FGM method based on the single-step reaction mechanism (dashed lines).

3.1.3 3D expanding spherical flame

Another test case for the DNS program are perfectly spherical expanding laminar flames. These were simulated18_{, again using the single-step reaction mechanism and the}

FGM method for which two different manifolds were used based on the single-step reaction mechanism and on GRI-mech 3.0. These flames are computed in a cubic three-dimensional computational domain with a length of 5.0mm and 127 grid points in each direction. The initial flame radius was 0.85mm at 1500K.

A comparison between the direct numerical simulations performed with the DNS pro-gram and CHEM1D is shown in figure 5, where the flame propagation velocity is plotted as a function of the flame radius, at 1500K. figure 5 (left) shows that the behaviour of the flames computed with the two programs using the single step reaction mechanism is very similar. However, a small quantitative disagreement of less than 1.5% is present between the DNS program and CHEM1D, with the former yielding slightly lower values. Here the difference in treatment of compressibility, effects of partially reflecting outflow boundaries and the FGM being based on only one single progress variable are responsible for the differences.

3.2 3D turbulent flame kernels

3.2.1 Introduction

By having performed laminar test cases as described in the previous section we have obtained confidence in the method of DNS-FGM and the implementation. Results for turbulent flame kernels can not be validated that easily. However the validations already performed should be sufficient with the additional requirement that the phenomena we want to describe are in the flamelet regime. Here we include the thin reaction zones regime into the domain of flamelet regimes as stated in the general introduction. It has to be assessed however up to what Karlovitz numbers this is true for the application of DNS-FGM with sufficient accuracy.

In order to see whether DNS-FGM still gives reasonable results we will confront its outcome to findings based on the flamelet theory of De Goey and Ten Thije Boonkkamp11_,

which gives a prediction of mass burning rate as function of flame stretch field for flamelet combustion. If the local mass burning rate correlates with the theory we can conclude that we are still in the flamelet regime and that both the theory and DNS-FGM simulations are valid tools. In the theory11 _{an expression for the stretch rate is derived directly from}

its mass-based definition,

K = 1

M dM

dt , (6)

where M is the amount of mass in an arbitrary control volume moving with the flame velocity:

M = Z

On the basis of this definition, a model for the influence of stretch and curvature on the mass burning rate has been developed. In a numerical study by Groot and De Goey46_,

it was shown that this model, with a slight reformulation, shows good agreement with calculations for spherically expanding laminar flames. This formulation, for the ratio of the actual mass burning rate at the inner layer, min, relative to the unperturbed mass

burning rate at the inner layer, m0

in (for unity Lewis numbers), reads

min

m0 in

= 1 − Kain, (8)

with the integral Karlovitz integral being a function of flame stretch (6), flame surface area, σ, and a progress variable, Y,

Kain:= 1 σinm0in   sb Z su σρKYds − sb Z sin σρKds  _{. (9)}

The integrals have to be taken over paths normal to the flame and su, sb and sin are the

positions at the unburned side, the burned side and the inner layer, respectively. The flame surface area, σ, is related to the flame curvature, κ, which is related to the flame normals, ni on the basis of the progress variable, Y,

ni = − ∂Y/∂xi q ∂Y/∂xj∂Y/∂xj , (10) κ = ∂ni ∂xi = −1 σ ∂σ ∂s. (11)

In some studies28,30,33,34 _{we also looked at possible parameterizations of (9). Ideas}

covered the concept of i) the flame surface area being constant through the flame and assuming the inner layer value, ii) this also for the product of stretch and surface and iii) an infinite activation energy expression30,33_{, iv) constant curvature. Also v) cylindrical}

curvature and vi) spherical curvature were added to this set of possibilities47_{. It must be}

remarked that some of these assumptions are better than others (clearly the last three options are better) but they all seem to break down quite low in the thin reaction zones regime, whereas the theoretical value holds its correlation with DNS-FGM results. 3.2.2 Turbulent flame kernels

One of the first turbulent simulations we did was in the corrugated flamelet regime33_.

This was still on a small domain of 5 mm in each direction. Here the length scale ratio of the turbulent integral scale to the laminar flame thickness was lt/δf = 10 and the

fluctuating velocity scale over the laminar flame speed was u0_/s

L ≈ 3. Later simulations

-2 -1 0 1 2 -2 -1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2

Figure 6: Projection of flame paths (bold lines) on the plane z = 0 for turbulent expanding flames. The thin lines are isotherms corresponding to the position of the unburned side, the inner layer and the burned side of the flame. The vectors represent the gas velocity in the plane z = 0.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 y x -8 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 8 10 y x

Figure 7: Scaled mass burning rate at the inner layer as function of the Karlovitz integral. Left: case thin reaction zones regime. Right: case broken reaction zones regime.

with lt/δf = 0.9 and u0/sL ≈ 4 and u0/sL≈ 40, respectively. Results of these three cases

are given in figure 6.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 mil/m0ilDetailed mil / m 0 il 1 D F G M 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 mil/m0ilDetailed mil / m 0 il 1 D F G M

Figure 8: Scaled mass burning rate at the inner layer for the thin reaction zones regime case. Comparison of FGM against computations with detailed chemistry by back substitution of the stretch field in a 1D detailed calculation. Left: Computed with 1D FGM. Right: Computed with 2D FGM.

almost completely for a 2D FGM36_.

Flame kernels in larger domains of 12 mm and 254 points in each direction are also studied28,34,35 _{and an example is given in figure 9. Findings are in line with the studies}

mentioned above. Larger domains allow for larger initial radii, which on its turn allows for a broader turbulent spectrum to manipulate the flame sheet. The goal of these studies is to advance to a priori studies for the purpose of LES modelling. An important parameter in this area is the amount of flame surface and wrinkling. In figure 10 the dynamics of flame surface can be seen for three cases, C1, C2, C3, with increasing Karlovitz and

Reynolds numbers. Here the mean surfaces are calculated by averaging the entire field to a 1D profile and taking the radius of the mean inner layer to determine the average area. The actual turbulent flame surface areas are determined by integrating the actual surface at which the progress variable assumes the value of the inner layer value. The integration is carried out by the method of Geurts48_{. From these calculations the strong}

stretch theory of De Goey14 _{is confirmed}37_.

3.3 2D turbulent flame kernels

For direct validation purposes we perform 2D turbulent flame calculations with detailed chemistry (GRI3.0) on 4492_{grids. First results are given in figure 11. Again the agreement}

is very good, but quantitative analysis is subject of ongoing work.

4 Conclusions

It is shown that the method of FGM gives very good results when implemented in a DNS code for simulating turbulent premixed combustion. There is also a very good agree-ment with flamelet theory in which the mass burning rate is predicted as a function of flame stretch and curvature. Up to now we only worked with first implementations, gen-erally with only one single progress variable and not accounting for preferential diffusion. However it has already been shown that 2D manifolds can improve the results up to a level that is very accurate. There is a large speed-up associated to the DNS-FGM method, which is accompanied by an additional large decrease of required computer memory. We will proceed by including preferential diffusion effects and direct validations with detailed chemistry. The method can be employed as a very useful toolbox for a priori testing of LES sub grid models for reactive flows. In combination with flame surface density models we will develop LES formulations in which inclusion of the appropriate manifolds will provide us with necessary unclosed information.

0 0.2 0.4 0.6 0.8 1 x 10−3 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3x 10 −4 Time [s] <A>, A 0 0.2 0.4 0.6 0.8 1 x 10−3 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Time [s] A/<A>

Figure 11: Results from 2D calculations. Left: detailed Lei= 1, right: FGM Lei = 1.

Acknowledgement

The authors would like to thank the Dutch Technology Foundation (STW) under grant nos. EWO.5874 and EWO.6285 and the support of NCF, the Netherlands Computing Facilities Foundation.

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