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Implementation of a Steady Laminar Flamelet Model for nonpremixed combustion in LES and RANS simulations Conference Paper · June 2013 CITATION

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Implementation of a Steady Laminar Flamelet Model for non-premixed combustion in LES and RANS simulations Hagen M¨uller∗ 1 , Federica Ferraro1 and Michael Pfitzner1 1

Thermodynamics Institute, Department of Aerospace Engineering, Bundeswehr University Munich, 85577 Neubiberg, Abstract

As part of the development of a numerical tool to simulate non-premixed combustion in liquid rocket engines by means of LES, a steady laminar flamelet model has been implemented in OpenFOAM. In the current contribution, the implementation details are presented. Furthermore, the code is used to simulate non-premixed methane/air combustion with both LES and RANS methods. A comparison with available experimental data shows good agreement and proves the capability of the new implementation to model turbulent combustion at a reasonable computational effort.

1. INTRODUCTION The majority of energy conversion technologies, such as diesel engines and gas turbines, are based on turbulent non-premixed combustion. Naturally, there is great interest in modeling the prevailing physical phenomena by means of Computational Fluid Dynamics (CFD). Research in this field started off about sixty years ago and has undergone remarkable improvements. In this course many models of varying complexity and accuracy have been proposed. An excellent overview of state-of-the-art combustion modeling and new trends is given in the recent volume edited by Echekki and Mastorakos [1]. An established model for turbulent non-premixed combustion is the flamelet approach which was mainly advanced by Peters [2]. It is based on the view of a turbulent flame as an ensemble of many laminar diffusion flames, usually referred to as flamelets. Today several extensions of the flamelet approach exist, taking into effects which are neglected in the fundamental formulation. Details can be found in the books by Peters [3] as well as Poinsot and Veynante [4]. The version presented in the current contribution is referred to as the steady laminar flamelet model (SLFM) and takes a steady laminar counterflow diffusion flame as a basis for the flamelet library. Its structure is defined by the scalar dissipation rate which s for strain effects and is connected to the velocity gradients of the turbulent flame. In of computational efficiency, the main advantage of the flamelet approach is the decoupling of the chemistry structure from the turbulent flow. This is achieved by introducing the ive scalar mixture fraction which is transported in the CFD and denotes the mass fraction of fuel-generated species in a mixture. The influence of turbulence on combustion is taken into using a presumed probability density function (PDF). As commonly accepted, a β -distribution is used for the mixture fraction PDF

∗

Corresponding author: Hagen M¨uller ([email protected])

and the Dirac-function for the scalar dissipation rate. In this formulation the SLFM has been applied to several configurations with turbulent non-premixed combustion. The flamelet concept, however, has mainly been developed in the RANS context where it has shown its capability for instance in the studies by Zimmermann [5]. Recently, the approach has been extended to be used in Large-Eddy simulations (LES) as shown by Pitsch [6] and Kempf [7]. The flamelet approach is not yet among the chemistry models available in the OpenFOAM software package as provided by OpenCFD Ltd. Therefore, Cuoci et al. [8] have recently developed an extension that provides all steps necessary to use a SLFM with OpenFOAM in the RANS context. Their software package also contains various PDF shapes and the possibility to model soot formation as well as heat losses. Our long-term goal, however, is to develop LES methods to simulate combustion in high-thrust rocket engines which operate at high pressures and cryogenic injection temperatures. In that context, real-gas thermodynamic effects have to be taken into when calculating the flamelets as shown by Pohl et al. [9]. With the flamelet calculation being closed-source in the extension by Cuoci et al. [8] and the solver limited to RANS simulations, the need to perform real-gas LES motivated the implementation of a new SLFM OpenFOAM extension. The objective hereby is to consistently include the new libraries in the existing OpenFOAM structure in order to guarantee full flexibility. In the next section, the formulation of the implemented SLFM and a brief outline of the code structure is shown. In addition, the new implementation is validated for two test cases. A 2-dimensional laminar counter flow diffusion flame is simulated and the results are compared with the according flamelet solution. In a second step, the well-documented non-premixed methane/air Sandia Flame D, which has been investigated experimentally by Barlow et al. [10], is simulated using both RANS and LES methods.

2. MODEL FORMULATION The SLFM concept is based on the assumption that the flame in a turbulent flow can at any time be regarded as an ensemble of small laminar diffusion flames, generally referred to as flamelets. This assumption is justified when the Damk¨ohler number is large, i.e. when the chemical reaction zone is thin compared to the turbulent length scales. Within this limit the SLFM is a strong and widely-used approach that allows the to for finite-rate chemistry effects at a reasonable computational effort. Its main advantage is that the flamelets, which describe the local structure of the turbulent flame, are coupled to the turbulent flow by only a few parameters. In practice, this fact is used to calculate the flamelets independently from the turbulent flow in a pre-processing step and store them in a “flamelet library”. Thermodynamic properties and species mass fractions can then be extracted from these tables using representative parameters which are transported in the turbulent code. In this work, the flamelets are generated using a counterflow diffusion flame configuration (see figure 1). Oxidizer and fuel flow through opposed nozzles and form a diffusion flame in their midst which, along its centerline, can be reduced to a one-dimensional problem. The temperature, the species composition at the nozzles as well as the operating pressure is hereby chosen to match the configuration used in the turbulent code. This type of flame is typically described using the mixture fraction Z which is a measure for the amount of fuel atoms in a given mixture. By definition, it is 1 at the fuel and 0 at the oxidizer inlet. The governing equations for temperature and species mass fractions of the flamelet in mixture fraction space can then be obtained applying a coordinate transformation, as shown by Peters [2]. For uniform diffusion (Le = 1) the flamelet equations can be written as ∂Yk 1 ∂ 2Yk − ρχ − ω˙ k = 0 ∂t 2 ∂ Z2 2 ∂T 1 ∂ T 1 ∂ ∂ T 1 n − ρχ + + ρ ∑ hk ω˙ k = 0 ∂t 2 ∂ Z2 ∂ Z ∂ Z c p k=1 ρ

(1) (2)

where ρ is the thermodynamic density, Yk is the chemical species mass fraction, T denotes the temperature, c p and hk are the specific isobaric heat capacity and the specific enthalpy of species k, respectively.

Figure 1: Schematic view of a counterflow diffusion flame configuration. The chemical species source ω˙ k are calculated with a chemistry reaction mechanism. The scalar dissipation rate χ can be thought of as an inverse diffusion time scale and has the dimension 1/s.

∂Z χ = 2D ∂y

2 (3)

χ is a function of the mixture fraction, but can, as shown by Peters [3], be parametrized by its value at stoichiometric mixture χst . Therefore, χst acts as an external parameter that imposes strain on the flamelet and defines its structure. In the current work, the flamelets are calculated with the open-source chemistry software Cantera [11] where the scalar dissipation rate is adjusted by defining the mass flow at the two inlets. Starting with a low mass flow, several flamelets can be calculated by incrementally increasing the mass flow and also the imposed strain until the extinction limit is reached. Once the solution of the flamelet equations is known for a sufficient number of χst , the temperature as well as the species mass fractions can be tabulated as a function of mixture fraction and scalar dissipation rate at stoichiometry (T (Z, χst ), Yk (Z, χst )). The mean values for species mass fraction and temperature are then calculated using a Favre presumed probability density function (PDF) to integrate the flamelets. Z ∞Z 1

Yek =

0

0

Yk (Z, χst )P(Z, χst )dZdχst

(4)

T (Z, χst )P(Z, χst )dZdχst

(5)

Z ∞Z 1

Te =

0

0

This step is realized with the new OpenFOAM utility canteraToFoam which at large follows the procedure described by Echekki and Mastorakos [1]. Hereby, the t PDF P(Z, χst ) is decomposed assuming statistical independence. P(Z, χst ) = P(Z)P(χst ) (6) The shape of the PDF for the scalar dissipation rate is modeled with a simple Dirac function P(χst ) = fst ). For the mixture fraction, the shape of the PDF is approximated using a presumed β -PDF. δ (χst − χ P(Z) = Z α−1 (1 − Z)β −1

Γ(α + β ) Γ(α) Γ(β )

(7)

Γ is the gamma-function, α and β are the β -PDF parameters defining its shape depending on the mean 002 . mixture fraction Ze and its variance Zf ! e − Z) e Z(1 α = Ze −1 (8) 002 Zf

Cantera:

New OF-utility “canteraToFoam”:

Solve the Flamelet-equations

β-PDF integration

Y k (Z , χ st ) T (Z , χ st )

Ỹk ( Z̃ , χ̃st , Z̃ ' '2 ) h̃s( Z̃ , χ̃st , Z̃ ' ' 2)

Flamelet library:

PRE-PROCESSING

̃ Z

Ỹk , h̃s

χ̃st , Z̃ ' '2

New OF-solver “flameletFoam”: Pressure-Based RANS/LES CFD solver

Z̃ , χ̃st , Z̃ ' '2

CFDsolver

Figure 2: Schematic of the SLFM procedure as implemented in the current work.

β = 1 − Ze

e − Z) e Z(1 −1 002 Zf

! (9)

Large parts of the thermodynamic calculation in OpenFOAM are based on the sensible enthalpy hs , from which the temperature and other thermodynamic quantities are derived. Thus, in order to be consistent with the OpenFOAM standard and to be flexible in of thermodynamic modeling, the utility canteraToFoam calculates the enthalpy from the given temperature. The whole set of thermodynamic models of OpenFOAM is available for this step. After the PDF-integration and the calculation of hs , the mean values of species mass fraction Yek and enthalpy hes are stored in the “flamelet library” as function 002 and scalar dissipation rate at e mixture fraction variance Zf of the mean values of mixture fraction Z, fst . These three quantities have to be provided by the turbulent CFD code, such that stoichiometric mixture χ the species composition and enthalpy can be obtained from the library by interpolation. This functionality has been implemented in the new pressure-based OpenFOAM solver flameletFoam for both LES and

RANS simulations. The details of the modeling approach depends on the turbulence closure. In the RANS context two additional transport equations for the mean mixture fraction and its variance have to be solved. ! ∂ ∂ Ze ∂ ρ¯ Ze ∂ ρ¯ uei Ze = µe f f (10) + ∂t ∂ xi ∂ xi ∂ xi ! !2 002 002 002 ∂ Zf ∂ Ze ∂ ρ¯ Zf ∂ ∂ ρ¯ uei Zf = µe f f + 2µe f f − ρ¯ χe (11) + ∂t ∂ xi ∂ xi ∂ xi ∂ xi Here, the effective viscosity is composed of a laminar and a turbulent contribution (µe f f = µ + µt ). The turbulent or eddy viscosity is calculated with a turbulence model that can be chosen from the variety of models available in OpenFOAM. Using the assumption that the mixture fraction fluctuations decay in a manner proportional to the turbulent fluctuations, the scalar dissipation rate in equation 11 can be expressed as follows ε 002 χe = Cχ Zf (12) k where k and ε denote the turbulent kinetic energy and its dissipation, respectively. The constant Cχ is set to the value 2.0 as proposed by Janicka and Peters et al. [12]. In the LES context, the filtered transport equation for the mixture fraction formally looks identical to the Reynolds-averaged transport equation (see equation 10). However, the effective viscosity is now composed of a resolved contribution and a part which is due to the unresolved subgrid scale (SGS) fluctuations (µe f f = µ + µsgs ). The latter needs modeling and can be described by the SGS turbulence models available in OpenFOAM. Under the assumption of local equilibrium, the SGS mixture fraction 002 and the filtered scalar dissipation rate χ e are modeled according to the approach which has variance Zf been proposed by Pierce and Moin [13]. ∂ Ze 2 002 = C ∆2 Zf Z ∂ xi

(13)

∂ Ze 2 ∂ xi

(14)

µe f f χe = Cχ ρ¯

Moin et al. [14] have proposed a dynamic procedure to determine the constant CZ . However, for the current implementation a constant value of 1.0 has been found to be sufficient. A further simplification in the current implementation is made concerning the conditioning of the scalar dissipation rate on stoichiometric mixture. The flamelets in the library are characterized by χst , while only the unconditioned value can be calculated with equation 12 and equation 14. Among others, Pitsch et al. [15] demonstrated a model which makes use of a presumed shape PDF to connect the unconditioned scalar dissipation rate χ to its value at the point of stoichiometry χst . However, it is reported [16] that in practice the approximation χ = χst is acceptable since the error is small around the stoichiometric mixture, which is the focus of interest. Furthermore modeling the conditioned scalar dissipation rate introduces additional uncertainties. For these reasons, the unconditioned scalar dissipation rate is used in the current implementation to extract the species mass fractions and the sensible enthalpy from the “flamelet library”. The SLFM procedure as implemented in the current work is also illustrated in figure 2.

3. VERIFICATION AND VALIDATION Two test cases have been considered in order to the new implementation. In a first step, an axisymmetric laminar counterflow diffusion flame configuration (see figure 1) has been simulated on a 2-dimensional grid using the new solver flameletFoam. A comparison with the results from the chemical kinetic software Cantera for the same configuration, allows for a validation of the correct flamelet library

0.25

2500

T

2000

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1000

O2

CH4

0.05

0 0

Temperature [K]

mass fraction

0.2

500

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0.75

0 1

Z

Figure 3: Temperature and mass fraction profiles in mixture fraction space along the centerline of a counterflow diffusion flame: ◦ Cantera, —flameletFoam. setup and table interpolation. Furthermore the new solver has been used to simulate a turbulent diffusion flame (Sandia Flame D) which has been investigated experimentally by Barlow et al. [17]. Both RANS and LES turbulence closures have been used for the validation. 3.1 Laminar Counterflow Diffusion Flame As mentioned in the previous section, a laminar counterflow diffusion flame problem can be described with the one-dimensional flamelet equations (see equation 1 - 2). It is therefore expected that a CFD code which solves for the full compressible Navier-Stokes equations, reproduces the flamelet solution if the configuration and modeling is chosen properly. This is particularly true for the SLFM since the flamelet solution is used to construct the table library which is then used for the interpolation of species mass fraction and enthalpy. However, for a first verification of the new solver, a comparison of the flamelet solution as obtained from the chemical kinetics software Cantera with the simulation results of the new solver is necessary to validate the pre-processing and the interpolation at each time step. The boundary conditions for the fuel and oxidizer inlets of the tested counterflow diffusion flame have been chosen to match the boundary conditions of the diffusion flame which is investigated in the next section. This particular configuration therefore represents a single flamelet of the flamelet library which is needed to subsequently simulate the turbulent flame. Thus, temperature, pressure and species mass fractions at the nozzle boundaries have been chosen according to the experimental configuration by Barlow et al. [17] which is summarized in table 1. The coflow in the experiment corresponds to the oxidizer nozzle. Further details of the turbulent diffusion flame setup are discussed in the next section. For this first verification, the effect of strain has not been considered and the velocity at either side is therefore set to a value small enough to guarantee that strain is negligible (Uin j = 0.025 m/s). The computational domain is a 2-dimensional wedge with inlets at the front as well as at the rear end and an outlet at the upper boundary. The grid has a uniform distribution in both directions and contains 100 × 20 cells. In figure 3 the temperature and the methane (CH4 ) as well as oxygen (O2 ) mass fraction profiles are shown in mixture fraction space. The results show that the flamelet solution matches the results of the new solver exactly. This shows that the flamelet library is set up properly and the solver extracts the correct values at each time step.

Table 1: Boundary conditions for Sandia Flame D.

U [m/s] T [K] Z

Fuel 49.6 294 1

Pilot 11.4 1888 0.271

Coflow 0.9 294 0

3.2 Turbulent Diffusion Flame (Sandia Flame D) For further validation, a piloted diffusion flame has been chosen (Sandia Flame D). In this configuration, the fuel is a 25%/75% methane-air mixture by volume and is injected through an axisymmetric pipe with a diameter of D = 7.2 mm. It is enclosed by a pilot nozzle with an outer diameter of DP = 18.2 mm. The pilot flow is a pre-burned mixture which has been adjusted to match the species composition and temperature of the main fuel/oxidizer flow with the mixture fraction Z = 0.271. The pilot is surrounded by a slow (UCF = 0.9 m/s) air coflow. The bulk inflow velocities of fuel and pilot nozzle are U = 49.6 m/s and UP = 11.4 m/s, respectively. Table 1 summarizes the boundary conditions. The experiments for this configuration have mainly been performed by Barlow et al. [10, 17, 18]. They have measured the mixture composition by means of Raman and LIF spectroscopy, the temperature profiles have been obtained by Rayleigh measurements. The numerical setup for the configuration differs according to the turbulence closure. For the LES, a 3dimensional axisymmetric grid with 4.4 × 106 grid cells have been used which has been stretched in radial and axial direction. Turbulence on the subgrid scales has been modeled with the algebraic Smagorinsky model and a white-noise boundary condition has been used to for velocity fluctuations at the fuel inlet. Fewer cells are needed for the RANS-simulation, where a 2-dimensional wedge is sufficient to

Figure 4: Instantanious temperature (upper plot) and mixture fraction (lower plot) LES distribution as obtained from flameletFoam. The black isolines denote stoichiometric mixture Zst = 0.351

2500

1.2

2000

1

0.6

Ze

Te[K ]

0.8 1500 1000 0.4 500 0 0

0.2 20

40

x/D

60

80

0 0

20

40

60

80

x/D

Figure 5: Axial mean temperature (left) and mean mixture fraction (right) profiles: ◦ experiments Barlow et al., —flameletFoam LES, −− flameletFoam RANS reflect the axisymmetric configuration. In the current work, a grid with 3.8 × 104 cells has been used to discretize the domain. Turbulent fluctuations have been modeled with the unmodified κ-ε model. In figure 4, snapshots of the temperature and the mixture fraction in a plane perpendicular to the centerline are shown. Naturally, the results refer to the LES. The black isolines denote the location of stoichiometric mixture fraction (Zst = 0.351) and therefore mark the most reactive layer in the flow field. The snapshots show that in the region just downstream of the nozzle (x/D < 5), the fuel and pilot stream show hardly any turbulent fluctuations. This short laminar section is followed by a transitional area where vortical structures evolve on the outer side of the pilot stream and the shear layer between coflow and fuel starts to wrinkle. Depending on the realization, the flame becomes fully turbulent in the section between 25 < x/D < 45. It is interesting to observe the reaction zone around Zst in the temperature plot. In the transitional zone, the flame thickness clearly varies, i.e. narrow reaction zones are followed by broad ones and vice-versa. This is an effect, that has previously been observed by Pitsch and Steiner [19] and can be attributed to the local scalar dissipation rate. The narrow reaction zones correspond to spots of high local mixture fraction gradients and therefore scalar dissipation rate (see equation 14). Thus, the flamelets that are used to represent these regions correspond to counterflow diffusion flames which were generated with high inlet mass flows. This is also reflected in the decreased maximum temperature in this region. A quantitative comparison with the measurements is possible for the averaged temperature and mixture fraction profiles along the flame centerline as well as along the radial axis at different axial positions. Figure 5 compares the averaged LES results and the RANS simulations with the experiments on the symmetry axis. The LES results are in excellent agreement with the experiments. The mixture fraction profile is particularly well reflected in the fully turbulent region of the flame. In the transitional region, the deviation could be attributed to the velocity boundary condition where white-noise has been used to reflect the velocity fluctuations. It is, however, known that this method induces fluctuations which damp too quickly and lead to a delayed jet-breakup. Simulations with more sophisticated methods are planned. Although the delayed break-up can also be observed in the temperature profile, the transitional region is captured satisfactorily. Within the limits of the turbulence closure, the RANS results also match the measurements in good agreement. The shape of the temperature profile is, although slightly shifted downstream, well reproduced. The offset can be explained by the underestimated turbulence in the first section of the flame. Jet break-up is shifted downstream and the mixture fraction is therefore overpredicted. The same observations can be made in figure 6, where the radial temperature and mixture fraction profiles at three axial positions are compared. The radial simulation results at the location closest to the nozzle (x/D = 15) are much the same for the LES and the RANS closure. Both are in excellent agreement

2500

1 x / D = 15

0.6

1500 1000

0.4

500

0.2

0 0

1

2

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x / D = 30

x / D = 30 0.6

1500 0.4

Ze

Te[K ]

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x / D = 15

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Figure 6: Radial mean temperature (left) and mean mixture fraction (right) profiles at three different axial positions: ◦ experiments Barlow et al., —flameletFoam LES, −− flameletFoam RANS for the mixture fraction, but slightly overestimate the maximum temperature. This can be attributed to an inherent weakness in the SLFM formulation. It is reported that minor local extinction occurs in this area of the flame [10] - an effect which can not be reproduced sufficiently with the SLFM. Further downstream, the difference between RANS and LES becomes more apparent. The deviation between experiment and RANS simulation in the temperature profiles, can fully be attributed to the discrepancies in the transport of mixture fraction. The LES reproduces the experiments quite well. The discrepancy in the outer section of the LES profiles at the location x/D = 45 are very likely due to insufficient averaging time.

4. CONCLUSION A new OpenFOAM solver for reacting, pressure-based LES and RANS simulations using the steady laminar flamelet model has been implemented. The flamelets are generated with the open-source chemical kinetics software Cantera and are subsequently integrated with a new utility which uses a presumed β -shape PDF. The integrated flamelets are arranged in a flamelet library which is accessed by the new

solver to retrieve the species composition and the enthalpy at each time step. This procedure allows to include finite-rate chemistry in a turbulent CFD simulation at a reasonable computational effort. The new code has been tested for a laminar counterflow diffusion flame configuration as well as for a turbulent, jet flame (Sandia Flame D) and reproduced the reference data in excellent agreement. The code can be ed from our website (http://www.unibw.de/thermo/mitarbeiter-en/mueller) and comes with a manual as well as with a tutorial.

ACKNOWLEDGEMENTS Financial has been provided by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) in the framework of the Sonderforschungsbereich Transregio 40. Computational resources have been provided by the Leibniz-Rechenzentrum M¨unchen (LRZ).

REFERENCES [1] T. Echekki and E. Mastorakos. Turbulent Combustion Modeling. Springer, 2011. [2] N. Peters. Laminar diffusion flamelet models in non-premixed turbulent combustion. Progress in Energy and Combustion Science, 10(3):319–339, 1984. [3] N. Peters. Turbulent Combustion. Cambridge University Press, 2000. [4] T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. 3rd edition, 2012. [5] I. Zimmermann. Modeling and numerical simulation of partially premixed flames. PhD thesis, University of the Federal Armed Forces Munich, 2009. [6] H. Pitsch. Large-Eddy Simulation of Turbulent Combustion. Annu. Rev. Fluid Mech., 38:453–482, 2006. [7] A.M. Kempf. Large-Eddy Simulation of Non-Premixed Turbulent Flames. PhD thesis, Technische Universit¨at Darmstadt, 2003. [8] A. Cuoci, A. Frassoldati, T. Faravelli, and E. Ranzi. http://creckmodeling.chem.polimi.it/. [9] S. Pohl, M. Jarczyk, M. Pfitzner, and B. Rogg. Real gas effects in hydrogen/oxygen counterflow diffusion flames. 4th European Conference for Aerospace Sciences, 2011. [10] R.S. Barlow and J.H. Frank. Effects of turbulence on species mass fractions in methane/air jet flames. Proc. Combust. Inst., 27(1):1087–1095, 1998. [11] URL http://cantera.github.io/docs/sphinx/html/index.html. [12] J. Janicka and N. Peters. Prediction of Turbulent Jet Diffusion Flame Lift-Off Using a PDF Transport Equation. In Symposium (International) on Combustion, volume 19, Haifa, Israel, 1982. [13] C.D. Pierce and P. Moin. Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion. J. Fluid Mech, 504:73–97, 2004. [14] P. Moin, K. Squires, W. Cabot, and S. Lee. A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys. Fluids A, 3(11):2746–2757, 1991. [15] H. Pitsch, M. Chen, and N. Peters. Unsteady Flamelet Modeling of Turbulent Hydrogen-Air Diffusion Flames. 21st Symposium (International) on Combustion / The Combustion Institute, pages 1057–1064, 1998. [16] Ansys CFX-solver Theory Guide. Ansys CFX Release 11.0, 2006.

[17] R.S. Barlow, J.H. Frank, A.N. Karpetis, and J.-Y. Chen. Piloted methane/air jet flames: Transport effects and aspects of scalar structure. Combustion and Flame, 143:433–449, 2005. [18] R.S. Barlow and J.H. Frank. URL: http://www.sandia.gov/TNF/DataArch/FlameD.html. [19] H. Pitsch and H. Steiner. Large-Eddy Simulation of a Turbulent Piloted Methane/Air Diffusion Flame (Sandia Flame D). Physics of Fluids, 12(10):2541–2554, 2000.

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