A comparison between two different Flamelet reduced order manifolds for non-premixed turbulent flames

8 th U. S. National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, 2013 A comparison between two different Flamelet reduced order manifolds for non-premixed turbulent flames Hossam A. El-Asrag and Graham Golden Ansys, Inc. Lebanon, NH, 03766, USA To reduce the computational effort in 3D CFD simulations, a common practice is to parameterize and tabulate a priori the scalar evolution of a reactive turbulent environment by few variables that govern the scalar evolution in a laminar flame. Two famous methodologies that use this approach are the Flamelet Generated Manifold (FGM) and the Flamelet Progress Variable (FPV) models. Both FPV and FGM parameterize all species and temperature by a mixture-fraction (Z) and a progress variable or progress parameter (C). However, the two models treat the flamelet manifold between equilibrium and flame extinction in a different manner. The Stanford FPV model solves flamelet equations in the (unstable) middle and lower branches of the flamelet S-curve, while the FGM (Fluent approach) model solves an unsteady extinguishing flamelet from the last stable burned steady solution before extinction. The generated tables for FGM and FPV show similar behavior in the mixture fraction space but different behavior on the progress variable space. Where the FPV solutions, show an extinction curve where the progress variable and temperature are decreasing with increasing the mixture fraction. Both models are compared here with experimentally measured thermo-chemical states for Sandia turbulent jet diffusion flames C and F. The results show that under lean and stoichiometric conditions (Z< Zstoic) both models show similar behavior, with slightly better sample from the FPV model for ultra-lean mixture, and better behavior for the FGM model in the stoichiometric range. However, on the higher rich mixture fraction values FPV shows better predictions for the CO mass fraction and the temperature field than the FPV. 1. Introduction and Objectives Gas turbine engines rely heavily on complex chemical structure liquid fuels as a primary source of energy. The chemical oxidation mechanisms of such fuels always involve hundreds of species and thousand reactions. Simulating the details of such mechanisms is computationally prohibitive and can consume up to 75% of the simulation time. Chemistry tabulation is one well-known methodology to reduce the computational cost of reacting flow simulations. Many 1

models exist to achieve this goal by first pre-solving the stiff chemistry on a reduced model problem with the appropriate boundary conditions then tabulating the thermochemical solution on a reduced order manifold that can be represented be only few variables. In this paper two different methods are employed and compared, where counter-flow diffusion laminar flamelets are solved a priori as a model problem that represents both mixing and laminar chemistry. The Laminar Flamelet model postulates that a turbulent flame is an ensemble of laminar flames that have an internal structure not significantly altered by the turbulence. These laminar flamelets are embedded in the turbulent flame brush using statistical averaging. The Flamelet Generated Manifold (FGM) [1,2] and the Flamelet Progress Variable (FPV) [3,4,5] models assume that the scalar evolution (that is the realized trajectories on the thermochemical manifold) in a turbulent flame can be approximated by the scalar evolution in a laminar flame. Both FPV and FGM parameterize all species and temperature by a few variables, such as mixture-fraction, scalar-dissipation and/or reaction-progress, and solve transport equations for these parameters in a 3D CFD simulation. Note that the FGM and FPV models are fundamentally different from the Steady Laminar Flamelet (SLF) model [6]. For instance, since in SLF model the laminar flamelets are parameterized by strain, the thermochemistry always tends to chemical equilibrium as the strain rate decays towards the outlet of the combustor. In contrast, the FGM and FPV models are parameterized by reaction progress and the flame can be fully quenched, for example, by adding dilution air. The laminar flame thermo-chemistry is calculated in a simple configuration, such as a 0D plug-flow or perfectly-stirred reactor, or a 1D diffusion or premixed flame, and parameterized by mixture fraction (Z) and reaction progress (C). Prior research [x], however, has shown that premixed laminar flamelets generally perform better than diffusion flamelets for FGM application to turbulent premixed flames, and visa-versa. Diffusion flamelets are computed in opposed-flow strained laminar flamelet geometries. At zero strain, the flamelet is at chemical equilibrium (c=1) and as the strain is increased, the flamelet departs from equilibrium (C<1) until the extinction strain rate is reached, which corresponds to a particular reaction progress, C_extinction. Typically C_extinction is 0.8 or 0.9. There are different ways to model the flamelet manifold between C=0 and C=C_extinction: Solve the flamelet equations in the (unstable) middle and lower branches of the flamelet S-curve (Stanford FPV approach) [2,3,4]. Linearly interpolate the thermo-chemical state between C=C_extinction and C = 0 (Dutch approach) Solve an unsteady extinguishing flamelet (Fluent approach)[1,2] In this work we calculate the manifold by 1) and 3) and compare with experimentally measured thermo-chemical state for turbulent jet diffusion flame C-F [7,8,9]. 2

2. Laminar Flamelet Tabulation Any type of laminar flame can be used for parameterization. For example, ANSYS Fluent can either import an FGM calculated in a third-party flamelet code and written in Standard file format or calculate an FGM from 1D steady premixed flamelets or 1D diffusion flamelets. In general, premixed flamelets should be used for turbulent partially-premixed flames that are predominantly premixed. Similarly, diffusion FGMs should be used for turbulent partiallypremixed flames that are predominantly non-premixed. For turbulent partially-premixed flames that are predominantly non-premixed, similar to the C-F flames [7,8,9], diffusion flamelets are a better representation of the thermochemistry than premixed flameltes. An example of this is modeling CO emissions from a gas-turbine combustor where the primary combustion zone is quenched by rapid mixing with dilution air. If the outlet equivalence ratio is less than the flammability limit of a corresponding premixed flamelet, the premixed flamelets will predict sub-equilibrium CO, even if the combustor is quenched ( ). A diffusion FGM, however, will better predict super-equilibrium CO for. Similar to the FPV model [x], Diffusion FGMs are calculated in ANSYS Fluent using steady diffusion laminar flamelet. Steady diffusion flamelets are generated over a range of scalar dissipation rates by starting from a very small strain and increasing this in increments until the flamelet extinguishes. The diffusion FGM is calculated from the steady diffusion laminar flamelets by converting the flamelet species fields to reaction progress,. As the strain rate increases, the flamelet chemistry departs further from chemical equilibrium and decreases from unity towards the extinction reaction,. In FGM the thermochemical state between, and the unburnt state, is determined from the thermochemical states of the final, extinguishing diffusion flamelet. 3. FGM Flamelet Generation The laminar counterflow diffusion flame equations can be transformed from physical space (with as the independent variable) to mixture fraction space (with as the independent variable). In ANSYS Fluent, a simplified set of the mixture fraction space equations are solved. Here, equations are solved for the species mass fractions,, (1) and one equation for temperature: (2) 3

The notation in the above two equations is as follows:,,, and are the th species mass fraction, temperature, density, and mixture fraction, respectively. and are the th species specific heat and mixture-averaged specific heat, respectively. is the th species reaction rate, and is the specific enthalpy of the th species. The scalar dissipation,, must be modeled across the flamelet : (3) where is the density of the oxidizer stream. 4. Flamelet Progress variable model Pierce and Moin [3] proposed tabulation based on the mixture fraction f and a reaction progress parameter instead of the scalar dissipation rate (strain rate) in SLF model, which leads to the Flamelet/Progress Variable (FPV) formulation. The purpose of this parameter is to parameterize the other flame thermochemical states (i.e. fully burning state, transient solution between burned and unburned states, and the unburned state). The advantage is that the use of the reaction progress parameter as a parameter can allow description of local extinction and reignition. In the FPV the steady flamelet equation is solved for the thermochemical state in the mixture fraction space. Details of the definition of the reaction progress parameter can be found in Ihme and Pitsch [10]. The Flamemaster code by Heinz Pitsch [10] is used to generate the FPV library for the C-F Sandia flame conditions [7,8,9]. The upper branch of the S-shaped curve is computed by a Newton iterative solver form an initial unburned or burned solution. The middle branch is constructed using an Arclength continuation method, where the scalar dissipation rate is computed as an Eigen value of the problem that considers the arclength between the different solutions as a parameter [10]. 5. Results and Discussion In the current paper the laminar tables are generated using non-premixed flamelets independently for the Sandia Flames C-F operating conditions in Table [1] using Ansys-Fluent for FGM and FlameMaster [1] for FPV. A 100 flamelets are generated by FlameMaster to cover the fully burned stable and burned unstable solutions in the S-shaped curve, while 64 flamelets only are used with FGM. The GRI 2.11 for methane oxidation mechanism is used for both cases. Table [1]: Non-Premixed Laminar flamelets boundary conditions to mimic the Sandia Flames C-F. All flames are generated for pressure=0.993 atm. Temp YCH 4 YO 2 YN 2 Oxidizer Stream 291 K 0 0.233 0.767 Fuel Stream 294 K 0.156 0.196 0.648 4

For comparison, the definition of the mixture fraction and the progress variable are fixed for all simulations and for the data extracted from the measurements. The progress variable C is deifned as the sum of Y CO +Y CO2. The mixture fraction is defined using the Bilger formula [7]: =,.,,,.,, (4) Where M i is the molar mass of an element i, Z j is the mass fraction of an element and subscripts C, H, 1 and 2 refer to carbon, hydrogen and the fuel and oxidizer streams. The fuel and oxidizer stream composition is defined as Z H,1 =0.0393, Z C,1 =0.1170, Z H,2 =0.0007, and Z C,2 =0.0. Scatter plots for the mixture fraction and the progress variable from the measurements for Flame-C and Flame-F are shown in Figs [1] and [2], respectively. The C and F flames are partially premixed, where the jet composition is 75% by volume air and 25% CH4. Flame-F has higher inflow jet velocity than Flame-C and, therefore, shows flame local extinction. The experimental results show that the mixing rate is high enough that the flames burn in the non-premixed mode with no indication of premixed reaction in the fuel-rich regions [8]. This is indicated in Figs.[1] and [2] by the higher sample of lower value of progress variable C for the mixture fraction range. The reported stoichiometric value of the mixture fraction is 0.351. Similar scatter plots are shown for the generated tables from FGM (using Fluent) and FPV (using FlameMaster) models solution in Figs. [3] and [5], respectively. Hundred flamelets are generated for the FPV to cover the upper and middle branch of the S-shaped curve as shown in Fig. [5], while only 64 flamelets with 64 grid points along the computational domain in the Z and C direction are used for the FGMs generation. While experimental measurements show that the progress variable C decreases after the mixture fraction Z=0.5 nearly, Figures [3] shows a different behavior for FGM, where the progress variable C increases monotonically with the mixture fraction. FPV tabulation, however, shows that some C profiles have similar behavior to experiment while other profiles (flamelets) increase with Z then decrease in the far rich side sharply towards Z=1 at different turning points for each flamelet. To compare between the two different models the raw data are used directly from the generated tables and from the experimental measurements. The conditional (on mixture fraction) mean (averaged in the Z space) temperature and CO mass fraction are plotted against both the mixture fraction and the progress variable. These two variables are used in combustion modeling to extract the data form the tables generated a prior to the 3D CFD simulations. On each conditional mean value a numerical bar is added that corresponds to the possible variation in the raw data for each scalar in a given mixture fraction bin and progress variable. These numerical bars are not error bars; they are indication of the data availability from each table for a given Z and progress variable range. For example, if the mixture fraction space is divided between 0 and 1 to 5 bins each 0.2 range. The plot show the mean value of the table data inside this range and the variation of the scalar field inside this range is indicated by the numerical bars. The FPV and FGM databases are then compared with the experimental data for Flame-C and Flame-F for the same Z and C values. The forgoing strategy is first illustrated in Figs. [5], [6] and [7], which show the conditional mean temperature (T) and CO mass fraction (Y CO ) from the FGM database. Figure [5] shows the temperature variation in the mixture fraction space. The maximum temperature is between 5

Z=0.33 and Z=0.4. The reported experimental stoichiometric mixture fraction is 0.351. In Fig. [6], the same conditional mean temperature is plotted against the mean progress variable on the X-axis and the mean mixture fraction on the secondary Y-axis (on the right). This plot can be read as follows: for each database s mixture fraction on the right Y-axis (red open box) and the corresponding progress variable vertically on the X-axis, the conditional mean temperature is given by the black rhombus value on the same vertical line. The possible variance of the extracted scalar from the database at such mixture fraction and progress variable is given by the numerical bar. In other words, within the mixture fraction bin (range between two consecutive mean mixture fractions), the variation of the mixture fraction and progress variable within this bin will lead to different temperatures within the numerical bar above and below the mean value. This will give a straightforward and clear insight on how the database compare with the measurements for a given progress variable and mixture fraction. Figures [6] and [7] also show that the temperature increase with the progress variable then decrease at C =0.136. The CO mixture fraction however, increases monotonically with both the progress variable and the mixture fraction. Different definitions for the progress variable might lead to different behavior. Figures [8] and [9] show the same T-Z and T-C plots for the FPV database, respectively. The T-Z behavior in Fig. [8] for the FPV is similar to the FGM. The wider numerical bars above the mean value are indicative of a larger sample from FPV that includes extinct and burned solutions. The behavior is radically different than FGM, however, in the progress variable space as shown in Fig. [9], where the FPV tables show two solutions for the temperature: an upper higher temperature branch that corresponds to the burned stable solution (for Z range up to 0.55 and C < 0.095) and another low temperature branch (for Z >0.55 and C decreases from 0.095 to 0.05). The decrease if the C value indicates flame extinction. A similar behavior can be shown for the YCO-C distribution. It is important to stress here that the numerical bars indicate the range of validity and are not error bar. The next step is to compare the data extracted from FPV and FGM with the experimental measurements. Comparisons with Flame C measurements are shown in Figs. [10], [12], and [14] and with Flame F in Figs. [11], [13], and [15]. The following conclusions can be observed: The temperature field in the mixture fraction space is shown in Figs. [10] and [11] for both Flame C and Flame F, respectively. In the lean side (Z<0.22) FPV and FGM models show similar predictions (nearly overlap). FPV, however, shows closer predictions than the FGM for this range. At this region both models slight under-predicts with the experimental mean values can occur during CFD simulations. The FGM shows closer predictions with the experiment near stoichiometry for (0.22<Z<0.55), while the FPV shows better behavior on the rich side (Z>0.55) than FGM, which consistently over-predicts the temperature. At this range (Z>0.55, Z<0.22) flame extinction can occur. The conditional mean CO mass fraction in the mixture fraction space is shown in Figs. [12] and [13] for Flame C and Flame F, respectively. On the lean side, both FPV and FGM show typical behavior with possible over-predictions of the experimental values. Near stoichiometry both models show very good predictions with better behavior and range of validity for the FGM model. On the rich side, both FPV and FGM over predict the CO mass fraction. FPV, however, has better predictions for the CO mass fraction, especially for the C-flame. For the F-Flame, 6

both models over predict the CO mass fraction on the rich side. FGM is expected to over predict the CO mass fraction for ultra-lean and ultra-rich mixtures. Figures [14], and [15] show the T distribution in the progress variable space for flames C and F, respectively. Again, the FPV and the experimental database in the progress variable space show generally two branches, which indicates the burned and quenched states in the measurement s samples (black dots) and the modeling predictions. FPV in-general captures this trend as it solves for the middle unstable branch in the S-curve, while FGM shows nearly a middle value in-between the fully burned and quenched states. FGM, however, shows a wider range of C that agrees well with the experiment between 0.1 and 0.15. FPV is not capturing these high progress variable samples. This indicates better behavior for FGM near stoichiometric values. 6. Conclusions Two flamelet tabulation models for turbulent reactive flows are compared the Flamelet Generated Manifold (FGM) and the Flamelet Progress Variable (FPV) models. Both FPV and FGM parameterize all species and temperature by a mixture-fraction (Z) and a progress variable or progress parameter (C). The generated tables for FGM and FPV show similar behavior in the mixture fraction space but different behavior on the progress variable space. Where the FPV solutions, show an extinction curve where the progress variable and temperature are decreasing with increasing the mixture fraction. Both models are compared here with experimentally measured thermo-chemical states for Sandia turbulent jet diffusion flames C and F. The results show that under lean and stoichiometric conditions (Z< Zstoic) both models show similar behavior, with slightly better sample from the FPV model for ultra-lean mixture, and better behavior for the FGM model in the stoichiometric range. However, on the higher rich mixture fraction values FPV shows better predictions for the CO mass fraction and the temperature field than the FPV. From this study we conclude that FPV model will tend to have better predications for flame extinction under lean and rich conditions, while FGM will tend to have better predictions for stoichiometric burning conditions. 7

Figure [1] Scatter plot for Flame-C from the experimental measurements. The progress variable C =YCO+YCO2 and the mixture fraction Z is defined by Eq. [4]. Figure [2] Scatter plot for Flame-F from the experimental measurements. The progress variable C and the mixture fraction Z is defined by Eq. [4]. 8

Figure [3] Scatter plot for all the FGM tables generated for Flame-C-F boundary conditions. The progress variable C=YCO+YCO2 and the mixture fraction Z is defined using Eq. [4]. Figure [4] Scatter plot for all the FPV tables generated for Flame-C-F boundary conditions. The progress variable C=YCO+YCO2 and the mixture fraction Z is defined using Eq. [4]. 9

Figure 5 FGM database conditional mean temperature variations with mixture fraction. The numerical bars on each mean value corresponds to the variation in the extracted temperature from the FGM database for each mixture fraction. Figure 6 FGM database conditional mean temperature variations with mean mixture fraction (boxes) and the progress variable C. The bars on each mean temperature corresponds to the variation in the extracted temperature from the FGM database for each mixture fraction and progress variable. 10

Figure 7 FGM database conditional mean CO mass fraction variations with mean mixture fraction (boxes) and mean progress variable C. The bars on each mean CO mass fraction corresponds to the possible variation in the extracted values from the FGM database for all mixture fraction and progress variable variations inside the mixture fraction bin. Figure 8 FPV database conditional mean temperature variations with mixture fraction. The bars on each mean temperature corresponds to the possible variation in the extracted values from the FPV database for all mixture fraction and progress variable variations inside the mixture fraction bin. The wider numerical bars correspond to more available samples inside the mixture fraction range. 11

Figure 9 FPV database conditional mean temperature variations with mixture fraction (red boxes) and progress variable C. The bars on each mean temperature corresponds to the possible variation in the extracted values from the FPV database for all mixture fraction and progress variable variations inside the mixture fraction bin. Figure [10] Comparison of Flame-C conditional mean temperature with the FPV and FGM databases in the mixture fraction space. The wider bars indicate more possible solutions (more flamelets in FPV) 12

Figure [11] Comparison of Flame-F conditional mean temperature with the FPV and FGM databases in the mixture fraction space. The wider bars indicate more possible solutions (more flamelets in FPV) Figure [12] Comparison of Flame-C experimental mean CO mass fraction with the FPV and FGM databases in the mixture fraction space. 13

Figure [13] Comparison of Flame-F experimental conditional mean CO mass fraction with the FPV and FGM databases in the mixture fraction space. Figure [14] Comparison of Flame-C experimental conditional mean temperature with the FPV and FGM databases in the progress variable space. The corresponding mixture fractions for each progress variable is shown in Figs. [6] and [8]. 14

Figure [15] Comparison of Flame-F experimental conditional mean temperature with the FPV and FGM databases in the progress variable space. The corresponding mixture fractions for each progress variable are shown in Figs. [6] and [8]. References [1] A. van Oijen and L.P.H. de Goey. "Modelling of Premixed Laminar Flames Using Flamelet- Generated Manifolds". Combust. Sci. Tech.. 161. 113 137. 2000. [2] Ramaekers W.J.S., Oijen, J.A. and Goey. L.P.H. Flow Turb. Combust. (2010) 84:439-458 [3] C. D. Pierce, P. Moin, J. Fluid Mech. 504 (2004) 73 97. [4] Ihme, M. and Pitsch, H., Phys.Fluds, 20 (2008) 0055110-20 [5] Ihme, M. and Pitsch, H., Combust. Flame, 155 (2008) 70-89 [6] N. Peters, Turbulent combustion, Cambridge University Press UK, (2000) [7] Barlow, R. S. and Frank, J. H., Proc. Combust. Inst. 27:1087-1095 (1998) [8] Barlow, R. S., Frank, J. H., A. N. Karpetis, and Chen, J.-Y., "Piloted Methane/Air Jet Flames: Scalar Structure and Transport Effects," Combust. Flame 143:433-449 (2005). [9] Schneider, Ch., Dreizler, A., Janicka, J., "Flow Field Measurements of Stable and Locally Extinguishing Hydrocarbon-Fuelled Jet Flames," Combust. Flame 135:185-190 (2003). [10] FlameMaster, a C++ computer program for 0D combustion and 1D laminar flame calculations, H. Pitsch, 2000 15