edcSMOKE: A new combustion solver for stiff chemistry based on OpenFOAM ®

. In the present work, two new OpenFOAM solvers for combustion problems requiring detailed kinetic mechanisms are presented. The Eddy Dissipation Concept (EDC) [1] is used for turbulence-chemistry interactions and for the integration of detailed chemistry. The solvers, called (cid:48) edcSimpleSMOKE (cid:48) for steady state problems and (cid:48) edcPimpleSMOKE (cid:48) for unsteady ones, were developed for a robust handling of large and detailed chemical mechanisms in the context of RANS simulations. The solver was validated using high-ﬁdelity experimental data from several Sandia ﬂames and Jet in Hot Co-ﬂow burner. In general, good agreement is observed between the simulations and the experimental results, for both temperature and species mass fraction proﬁles. What’s more, different formulations of EDC model are tested and the results are compared.


INTRODUCTION
Most combustion burners and devices operate in turbulent regime. Turbulent flames help improving mixing, thus enhancing combustion efficiency. Combination of chemical reaction and turbulence results in a multi-scale and multi-physics problem, whose solution is even more challenging, when the large kinetic mechanisms are used to understand complex phenomena such as pollutant formation.
The purpose of the present work is to implement a new OpenFOAM based solver, capable of incorporating detailed chemistry mechanisms. In the following sections, the structure of the edcSMOKE solver and the different formulations of the EDC model will be discussed. An extensive validation of the solver using experimental data from several model flames will be presented. Several large mechanisms will be used to demonstrate the robustness of the solver.

METHODOLOGY
Two versions of the solvers were developed, a steady one ( edcSimpleSMOKE ) based on the Simple algorithm, and a transient one ( edcPimpleSMOKE ), based on the Pimple algorithm. The solver is also coupled with the OpenSMOKE library [2], which is specially developed to manage large and detailed kinetic mechanisms for combustion chemical reactions. The integration of turbulence-chemical interactions is done with the model of Eddy Dissipation Concept (EDC).

Edddy Dissipation Concept for tubulent combustion
In turbulent flows, there are regions where molecular mixing and dissipation of turbulence energy into heat take place. These regions account for a small fraction of the total volume of the fluid and they are called fine structures. These fine structures are believed to be vortex tubes, sheets or slabs with the characteristic dimensions of the order of the Kolmogorov length scale [3]. According to Eddy Dissipation Concept, chemical reactions take place in the fine structures and the mass fraction of the fine structures, represented with γ λ , is defined as: (1) The mean residence time within the fine structures is expressed with: where ν is the kinematic viscosity and ε is the dissipation of turbulent kinetic energy, k. The fine structure volume and residence time constants C γ = 2.1377, C τ = 0.4803.
The mean reaction rate for the source term of the conservation equations has different formulations. Initially, in the first publication of EDC, the mean reaction rate for the k th species was modelled by Magnussen as [4]: where ρ represents the mean density of the mixture, y * k is the mass fraction of the k th species in the fine structures and y k denotes the mean mass faction of the k th species between the fine structures and the surrounding fluid y 0 k , with This version of EDC is called as EDC1981 in the present work.
In 1996, Gran et al. argued that the mass exchange between the fine structures and the surrounding fluid should be modelled as γ 2 λ /τ * [5], thus changing the formulation intȯ This formulation is adopted by the commercial software of Fluent. It is named as EDC1996 subsequently.
Later in 2005, EDC model was modified again. Magnussen suggested that the mean reaction rate should be formulated asω and the new mean mass fraction of the k th species between the fine strictures and the surrounding fluid now becomes [1]: This expression will be called EDC2005 .
In all three formulations mentioned above, the mean mass fraction y k for each species is obtained by solving the species transport equation whereas the mass fraction of each species inside the fine structure y * k is computed with the detailed chemistry approach.

The EDC detailed chemistry approach
In the detailed chemistry approach, the effects of chemical kinetics are taken into account by regarding the fine structures as constant-pressure adiabatic homogeneous reactors. One of these reactors is Perfectly Stirred Reactor (PSR), in which the development of y * k in time is expressed as: In Equation 8, y 0 k is the mass fraction of species k entering the reactor, which is also the mass fraction of species k in the surrounding fluid.

VALIDATION TEST CASES
Upon knowing the basics of the edcSMOKE solver, it is necessary to validate the robustness and accuracy of this solver. The following model flames were selected as validation test cases: Among these validation cases, the steady state solver edcSimpleSMOKE is used on most cases, except Adeleide JHC burner, with which the transient solver edcPimpleSMOKE is used. In the current scope, only the results for the JHC burner are presented and discussed. The other validation test cases will be demonstrated in the final submission of the full paper.

Adeleide Jet in Hot Co-flow (JHC) burner
The validation was conducted with MILD (Moderate or Intense Low Dilution) combustion to investigate the model performance in case of strong turbulence/chemistry interactions. MILD combustion has gained more attention in recent years due to its features of low pollution emission and high efficiency [6]. The results for mean temperature values of the Adeleide JHC burner [7] using a mixture of CH 4 and H 2 50/50 on molar basis are presented in Figure 1. The jet diameter is 4.25 mm and Re = 10, 000. The oxygen mass fraction is fixed at 3 % in the hot co-flow. In this simulation, the standard k − ε turbulence model is applied and the model constant C 1ε = 1.6. Radiation is turned off. Three detailed mechanisms, KEE (17 species, 58 reactions) [8], GRI3.0 (53 species, 325 reactions) [9] and San-Diego (50 species, 247 reactions) [10] are used. Very good agreement between the experiments and simulations is observed along the centerline. The performance of the solver at the axial position of 30 mm and 60 mm show an over-prediction of the temperature levels, but this is a known issue for this burner [6] and it can be addressed by changing the EDC model constants. What's more, different versions of EDC model are compared Figure 2, showing that with k − ε model constant C 1ε = 1.6, EDC1981 and EDC1996 have better performance.

CONCLUSIONS
Two solvers implementing the Eddy Dissipation Concept model are presented and tested on a number of validation test cases, including diffusion flames and a jet in hot co-flow burner, emulating MILD conditions. Results indicate the ability of the solver to efficiently handle detailed kinetic schemes, providing satisfactory predictions of temperature and species mass fraction profiles.