The optimization procedure adopted has been developed in Python with a modified version of the Cantera library. This modification is necessary to ensure consistency between how the species transport, and their properties have been computed in the CFD solver and within Cantera. In this customized version of Cantera (adapted version of Cantera form CERFACS, 2017), the mass and thermal diffusion coefficients are computed considering constant Schmidt and Prandtl numbers, and the power law, reported in Eq. 1, is used for the molecular viscosity. μ = μ 0 T T r e f b , (1)

where the constants μ 0 and T r e f are calculated by fitting the molecular viscosity of the mixture from the unburned to the burned side of a freely propagating laminar flame.

The automated optimization procedure used to adapt the BFER* mechanism is schematically presented in Figure 1. First, the variables that control the kinetics and species diffusion, which have to be changed to obtain the target design requirements, must be chosen. In this work, the selected variables are all the Arrhenius constants (A_k, b_k, and Ea_k, with the subscript k referring to the reaction) and the two reaction orders ( n C H 4 a n d   n O 2 ) of the first reaction. Moreover, as described in Franzelli (2011), optimization has been performed to improve the laminar flame response with respect to the strain rate.

Once the variables have been defined, for a complete definition of the optimization problem, the cost function (CF) must be specified. In this work, CF is evaluated using Eq. 2, where TQ and PQ refer, respectively, to the target quantities and the predicted quantity on the laminar flame, while N q and N c are the numbers of different reference quantities and the number of tested operating conditions, respectively. It must be noted that the number of conditions can be different based on quantity q. Moreover, different weights w k i included in the CF formulation guide the choice set of variables that enhance the model prediction under the condition of major interest. C F = ∑ i = 1 N q ∑ k = 1 N c q w k i T Q k i − P Q k i T Q k i . (2)

The reference quantities chosen in this optimization process are LFS, computed from a 1D freely propagating flame, and the flame consumption speed S f u e l evaluated in the counterflow premixed flame configuration using Eq. 3. All TQs have been preliminarily computed for each condition ( N c using GRI3.0 (Smith et al., 2000) as a reference detailed reaction mechanism. This detailed mechanism, whose accuracy is well known for the description of CH₄/air combustion, has also shown optimal performance in predicting CO₂-diluted conditions (Halter et al., 2009; Galmiche et al., 2011; Chan et al., 2015; Chen et al., 2020; Duva et al., 2020; Xie et al., 2020). The considered conditions for each flame configuration are summarized in Table 1. S f u e l = 1 ρ f Y f ∫ − ∞ + ∞ − ω f ˙ d x . (3)

The optimization problem is now completely defined; thus, in principle, the minimization of CF can be afforded in multiple ways. In this work, because of the simplicity of the reaction mechanism and the low computational cost of each single-flame evaluation, the design of the experiment procedure has been chosen to explore the variable space and find the CF minima. The Latin hypercube sampling (lhs) method has been adopted to assemble the variable test matrix that has the dimension N v a r × N s a m p . This lhs criterion allows identifying the convenient set of variables spacing over the whole design space (variable space), the boundaries of this space, and the number of sampling points N s a m p that must be chosen. After the test matrix assembly, for each set of variables, the value of CF will be evaluated by solving a series of flames under all the conditions. It is worth noting that this process is very easily parallelizable, allowing a significant reduction in the computational cost if needed.

The set of variables for the new reaction mechanism, identified after the optimization process, is presented in Table 2.

The performances of the two-step BFER_CO₂* reaction mechanism in terms of LFS prediction are shown in Figure 2. This is in accordance with the reference GRI3.0 mechanism in the whole range of conditions included in the optimization. Focusing on the prediction of BFER_CO₂*, the accuracy gets better at 600 K and φ = 0.6, which are exactly the fresh mixture conditions for turbulent combustion investigations presented in the next paragraph. This was made possible by the proper choice of the w k i in Eq. 2. As mentioned previously, in the next turbulent combustion analysis, a perfectly premixed mixture at a well-specified temperature level will be considered. Despite that, the kinetics scheme is required to perform better in a wider range of conditions. The good LFS prediction at various unburnt temperatures shown in Figure 2 ensures the capacity of the mechanism to correctly characterize the wall heat loss impact on the flame.

The prediction of the BFER* mechanism always overestimates the reactivity, and this is expected mostly to be due to the kinetic effect of the CO₂ dissociation reactions. The BFER* was optimized for conditions in which these dissociation reactions play a marginal role, but as pointed out by Xie et al. (2020), these reactions should gain importance by increasing CO₂ enrichment, affecting the global fuel consumption capacity.

However, unlike BFER*, BFER_CO₂* does not include any pre-exponential adjustment (PEA). The PEA allows the pre-exponential factor A to be dependent on the local φ , allowing a better flame speed prediction in a wide range of φ . As a result, the validity range of the BFER_CO₂* mechanism is only within lean conditions, which are the conditions of interest in this study. According to the fact that the BFER* mechanism overestimates the reactivity, it can be observed in Figure 3 that this mechanism underestimates the flame thermal thickness (Eq. 4). On the other hand, the BFER_CO₂* prediction of D t is very close to the prediction of GRI3.0. D t = T b − T f max ∂ T ∂ x . (4)

As highlighted in Table 1, the reaction mechanism has been optimized considering counterflow premixed flames in the fresh-to-burnt configuration at various strain rate levels. The prediction of the BFER_CO₂* mechanism is not accurate in increasing the strain rate level, even though the Schmidt number has been included in the variables for optimization. In any case, the Schmidt number finally chosen allows the best reconstruction of the laminar consumption speed increasing the strain (Figure 4).

The experimental rig is described in De Persis et al. (2013). The facility architecture, shown in Figure 5, consists of an inlet plenum, an axial swirler, and a cylindrical Herasil quartz combustion chamber. The rig is operated at atmospheric pressure under fully premixed conditions. The CO₂/air/CH₄ mixture is prepared in the mixing volume upstream the swirler so that when it enters the combustion chambers, it can reasonably be assumed perfectly premixed. The fresh mixture can be heated up to 700 K to reproduce the gas turbine combustors’ inlet temperature levels. The objective of the experimental study was to investigate the effect of the CO₂ dilution on the flame stability and the corresponding change in the flame topology. The researchers have analyzed a wide range of operating conditions by varying the equivalence ratio and the CO₂ content in the fresh mixture. The observed flame shape changed from a lifted anchored condition toward the extinction, increasing the CO₂ dilution. It is worth noting that the CO₂ dilution adopted in this experimental study is not actually consistent with any EGR level. The relative N₂ content in EGR is expected to be higher than that in a pure CO₂ dilution.

The experimental apparatus consists of an ICCD camera for detecting CH* chemiluminescence and a gas analyzer for investigating the composition of the exhaust gases.

The details of the two high-temperature test points selected for the numerical models’ validation are reported in Table 3.

Reactive high-fidelity simulations have been carried out using the fully compressible AVBP code (CERFACS, 2023). The third order in time and space TTGC numerical scheme (Colin and Rudgyard, 2000) has been used to discretize the convective term of the Navier–Stokes equations. The time discretization is explicit, and the timestep is automatically chosen in order to ensure a constant maximum CFL = 0.7. The dynamic Smagorinsky–Lilly formulation (Lilly, 1992) is used to treat the sub-grid fluctuations.

The turbulence chemistry interaction has been modeled using ATFM (Colin et al., 2000) with a constant thickening factor selected to ensure a constant number of grid points in the unstretched laminar flame front equal to 5.

The computational domain is shown in Figure 6. At the inlet plenum, the mass flow condition is specified, whereas in the outlet plenum, the atmospheric pressure is prescribed on the dome and an auxiliary co-flow inlet has been assigned to the plenum ring. All the inlet and outlet boundaries are modeled with partially non-reflecting Navier–Stokes characteristic boundary conditions (NSCBCs) (Poinsot and Lelef, 1992). All the walls are modeled via the wall function approach. A fixed temperature has been imposed on the wall boundaries in the combustion chamber, and the estimated temperatures for the quartz, backplate, and the center body are 1000, 900, and 900 K, respectively. In terms of inlet temperature, mixture composition, and mass flow, the numerical boundary conditions are the same as described in Table 3 for the experimental test point.

The results discussion is organized as follows: first, the assessment of the flame prediction is focused with respect to the experimental measurements. Afterward, the impact of the heat loss on the flame shape is discussed, and finally a brief characterization of the numerically predicted flame dynamics is provided.