Computational Fluid Dynamics (CFD) of turbulent reacting flows is an essential tool to support the industrial sector transitioning to cleaner and energy efficient technologies. Reynolds-averaged Navier-Stokes (RANS) computations and large eddy simulations (LES) are widely used to accelerate innovation without the need of costly experimental campaigns. While simplified treatment of the chemical processes is often selected for computational cost saving, the integration of detailed chemical kinetics in CFD solvers improves the accuracy of complex phenomena, e.g., pollutant formation pathways, extinction and re-ignition processes. This level of fidelity is essential for designing and developing advanced combustion concepts and low-carbon fuel alternatives. This paper presents the FiReSMOKE solver suite, a collection of finite-rate chemistry solvers for RANS computations and LES of turbulent reacting flows. The suite is implemented in OpenFOAM and leverages OpenSMOKE++ to handle detailed chemistry. Along with the finite-rate models from literature and the availability of a wide range of ordinary differential equation (ODE) solvers, the FiReSMOKE suite includes the novel modal partially-stirred reactor (mPaSR) model, the sample-partitioning adaptive chemistry (SPARC) plug-in, a data-driven methodology for chemistry acceleration, and tabulated adaptive chemistry (TDAC). FiReSMOKE is, to the best of the authors' knowledge, the first OpenFOAM-based suite that offers integrated support for advanced chemistry solvers (SPARC, TDAC), tabulation techniques, and multiple combustion models in a unified and modular framework. This manuscript provides a summary of the theoretical background of the combustion models pertaining to the reactor-based models class and a methodology overview of the solver implementation. The modular design facilitates the integration of new combustion models and numerical techniques, making it adaptable to a wide range of research and engineering applications. Along with details on the numerical implementation of the code, test cases demonstrating the solver capabilities are presented.

Turbulent reacting flows are described as multi-scale processes with characteristic flow and chemical timescales spanning several orders of magnitude. Species source term closure models that rely on the description of such systems through a single scale make a strong assumption, failing to provide accurate estimations for chemical processes with significantly different characteristic timescales. The modal Partially Stirred Reactor (mPaSR) model overcomes this limitation by accounting for all chemical system dynamics through the modal decomposition of the Jacobian matrix of the species source terms. Following apriori testing on direct numerical simulation data and simulations using the Reynolds-averaged Navier-Stokes approach, this work details the first mPaSR model assessment in the context of Large Eddy Simulation (LES). Model validation is achieved through a series of LES of the Darmstadt Multi-Regime Burner (MRB). Attention is paid to the quality of temperature and carbon monoxide estimations in comparison to the measurements. Insights into the model are provided by assessing the resulting flow fields with tools from the Computational Singular Perturbation (CSP) theory. The study supports the use of the mPaSR model for the numerical investigation of complex turbulent reacting flows with the LES approach. Novelty and significance The novelty of this work lies in the first a posteriori testing, in the context of Large Eddy Simulation, of an innovative combustion model accounting for several timescales of dynamical chemical systems. This represents an important step towards developing well-suited approaches for modelling multi-regime combustion and multi-scale processes, such as pollutant formation in turbulent flames. The model demonstrates promising predictive capabilities in the investigated cases, motivating further studies across a broader range of combustion scenarios.

Accurately predicting turbulent combustion processes is fundamental for optimizing efficiency, reducing pollutant emissions, and ensuring operational safety in combustion systems. To this purpose, computational fluid dynamics (CFD) simulations are widely employed. In particular, large eddy simulations (LES) balance prediction accuracy with computational efficiency by resolving only the most energy-containing scales of turbulence and rely on modeling the turbulence-chemistry interactions (TCI) occurring at the smallest scales. Among the existing closures, the partially stirred reactor (PaSR) model incorporates finite-rate chemistry and estimates a cell reacting fraction based on the local Damköhler number to account for the subfilter-scale TCI. Although widely validated in CFD computations, the PaSR model was found limited by the way it computes the cell reacting fraction. To tackle this point, our study proposes a machine learning (ML) enhanced partially stirred reactor model for LES. A fully connected neural network is trained on direct numerical simulation (DNS) data of turbulent premixed jet flames to compute a correction coefficient for the cell reacting fraction. Maintaining the original model shape, this ML-enhanced closure aims at bridging the gap between physics-based models and advanced data-driven techniques. The proposed formulation not only improves the prediction accuracy of quantities of interest such as the heat release rate but also features computational feasibility and generalisation capabilities over a large range of LES grid refinement. This demonstrates the significant potential of ML-aided TCI closures in future applications of combustion engineering.

This paper investigates a turbulence-chemistry interaction model based on the Partially Stirred Reactor (PaSR) paradigm where the hypothesis of relying on an individual chemical timescale is relaxed to deal with multiscale problems. The modal Partially Stirred Reactor (mPaSR) model relies on the Computational Singular Perturbation (CSP) theory and performs an eigen-decomposition of the Jacobian matrix of the chemical source terms. The CSP manifold is then corrected by modal fractions that, similarly to the cell reacting fraction of the original PaSR model, account for the individual mode timescales. The vector of the chemical source terms, to be returned to the computational fluid dynamics solver, acts as an aggregated contribution of the corrected CSP modes. The predictive capabilities of the mPaSR model are demonstrated a posteriori through a series of Unsteady Reynolds-Averaged Navier-Stokes simulations of the well-documented Sandia flames. Promising results are observed at different turbulence levels making the mPaSR approach a valuable alternative to existing turbulence-chemistry interaction models. Particular attention is given to the formation of pollutants, and accurate predictions of nitric oxide NO are obtained. Novelty and significance statement The novelty of this work lies in the code development, integration and a posteriori testing of an innovative combustion model accounting for several timescales of dynamical chemical systems. This represents an important step towards well-suited approaches for the modelling of multiscale processes such as pollutant formation in turbulent flames. The model shows promising prediction capabilities with desirable computational efficiency on the investigated cases, motivating follow-up investigations in a larger range of combustion scenarios.

In the field of turbulent reacting flows, combustion phenomena, such as the mixing of cold fuel with hot products, propagation of flames, and auto-ignition, are profoundly affected by interactions between turbulence and chemistry, known as turbulence-chemistry interactions (TCI). Accurately modeling these intricate combustion processes requires a closure adept at capturing TCI behavior. Among the existing combustion models, the Partially Stirred Reactor (PaSR) model, as one of the finite-rate chemistry models, has shown significant suitability for modeling TCI within various combustion regimes. The modeling of chemical and mixing time scales is crucial to the performance of the PaSR model. Although numerous studies have extensively explored these aspects in separate efforts, there is a notable lack of a systematic study on employing the PaSR model to turbulent flames with multiple combustion regimes. In the present study, the Cabra flame, a vitiated coflow flame with multiple combustion regimes, is investigated by using large eddy simulations (LES) coupled with the PaSR model. Particular emphasis is placed on evaluating the combinations of the chemical and mixing time scales. Twelve combinations, involving three distinct chemical time scales and four different mixing time scales, are evaluated. The results reveal that both the chemical and mixing time scales significantly influence the model’s predictive accuracy, and various combinations exhibit varied predictive strengths in flame transition and diffusion regions. Based on the findings from these twelve combinations, a clustering model for Partially Stirred Reactor closure is first proposed. The model performance is then assessed, showing a better prediction in mean and root mean square values of temperature and species concentrations, as well as probability density functions of the reaction fraction, as compared to the traditional PaSR models.

Reactor-based models are well-suited Turbulence-Chemistry Interactions, Sub-Grid Scale closures for Large Eddy Simulation (LES) due to their ability to account for finite-rate kinetics. The Partially Stirred Reactor (PaSR) model relies on the estimation of characteristic time scales to define the reacting fraction of each computational cell. However, chemistry develops a spectrum of intrinsic chemical time scales, leaving no clear consensus on the definition of a single representative scale. Nevertheless, in numerical codes, a single chemical time scale formulation is used on the whole physical domain despite local and complex phenomena. Through an a priori assessment on Direct Numerical Simulation (DNS) data of turbulent non-premixed combustion, the present work proposes a numerical method to locally select an optimal chemical time scale formulation that minimises the model error. Data points are grouped into clusters via supervised partitioning algorithms where the optimal formulation is attributed to each cluster by means of distances minimisation. Using a combination of partitioning procedures can further improve the reconstruction quality of the clustered solutions, up to 35% global errors reductions with respect to standard solutions. Existing data partitions are then tested on unseen data points, yielding great prediction capabilities. DNS data of a turbulent premixed flame are used to demonstrate that the methodology is also robust across combustion regimes. The present proof of concept shows suitable features to introduce systematic improvements for the PaSR combustion closure in LES.

A generalized Partially-Stirred Reactor (PaSR) model is presented in this work based on the inclusion of multiple chemical times. The PaSR model has shown promising results at modelling turbulence-chemistry interaction in Large-Eddy Simulations (LES) and Reynolds-Averaged Navier-Stokes (RANS), providing an extension of the well-known Eddy Dissipation Concept (EDC). PaSR model divides the computational domain into reactive and non-reactive parts. The factor defining this partition is expressed as a function of the system characteristic chemical and mixing times. However, the estimation of these factors, particularly the chemical one, is often oversimplified. The approach proposed in this study seeks to include in the PaSR model the whole set of chemical times involved in the reactive system. Besides, the concept of fine structures, first introduced in the EDC and often adopted also in the PaSR model to characterize the evolution of chemistry in the reactive part of the fluid, is here abandoned in favour of direct manipulation of species production rates. The mean source term is formulated according to the new generalized model through a modal decomposition of the Jacobian matrix. The method is validated a priori with DNS data of a syngas non-premixed jet flame, whose filtered data represent the validation benchmark. A good agreement is found between the new PaSR model and the filtered data for all species at different filter widths. Comparison with the single time scale based model clearly shows the limitations of the old standard approach and the necessity of including the whole spectrum of chemical times for a more comprehensive description of turbulence-chemistry interaction. A thorough analysis with the time scale participation index reveals the complexity of reaction rates contributions to the development of a specific time scale, underlying the importance of developing a model able to inherit all kinetic pathways in the turbulent closure.

This article presents a joint numerical study on the Multi Regime Burner configuration. The burner design consists of three concentric inlet streams, which can be operated independently with different equivalence ratios, allowing the operation of stratified flames characterized by different combustion regimes, including premixed, non-premixed, and multi-regime flame zones. Simulations were performed on three LES solvers based on different numerical methods. Combustion kinetics were simplified by using tabulated or reduced chemistry methods. Finally, different turbulent combustion modeling strategies were employed, covering geometrical, statistical, and reactor based approaches. Due to this significant scattering of simulation parameters, a conclusion on specific combustion model performance is impossible. However, with ten numerical groups involved in the numerical simulations, a rough statistical analysis is conducted: the average and the standard deviation of the numerical simulation are computed and compared against experiments. This joint numerical study is therefore a partial illustration of the community's ability to model turbulent combustion. This exercise gives the average performance of current simulations and identifies physical phenomena not well captured today by most modeling strategies. Detailed comparisons between experimental and numerical data along radial profiles taken at different axial positions showed that the temperature field is fairly well captured up to 60 mm from the burner exit. The comparison reveals, however, significant discrepancies regarding CO mass fraction prediction. Three causes may explain this phenomenon. The first reason is the higher sensitivity of carbon monoxide to the simplification of detailed chemistry, especially when multiple combustion regimes are encountered. The second is the bias introduced by artificial thickening, which overestimates the species' mass production rate. This behavior has been illustrated by manufacturing mean thickened turbulent flame brush from a random displacement of 1-D laminar flame solutions. The last one is the influence of the subgrid-scale flame wrinkling on the filtered chemical flame structure, which may be challenging to model.

In Large Eddy Simulations (LES) of combustion, the accuracy of predictions might be heavily affected by deficiencies in traditional/simplified closure models, especially when employed to simulate non-conventional fuels and combustion regimes. The increasing availability of data from experiments and higher-fidelity numerical simulations offers attractive opportunities for improving combustion models with data-driven techniques. In this work, we focus on sub-grid turbulence-chemistry interactions with the Partially Stirred Reactor (PaSR) model and its associated cell reacting fraction sub-model. We combine machine learning and sparsity-promoting techniques to improve the predictive capabilities of PaSR by discovering new functional forms of the cell reacting fraction sub-model from data. The obtained models are parsimonious models that balance accuracy with model complexity to avoid over-fitting. We employ the proposed model identification approach on data from a Direct Numerical Simulation (DNS) of a three-dimensional non-premixed n-heptane/air jet flame. As a result, we single out the most plausible model form of the cell reacting fraction, expressed as a function of the local Damköhler number. Then, the capability of the model to generalize properly to new, previously unseen data is tested. The results demonstrate the ability of the machine learning approaches to infer robust corrections for turbulence-chemistry reactor-based combustion models.

In flames, turbulence can either limit or enhance combustion efficiency by means of strain and mixing. The interactions between turbulent motions and chemistry are crucial to the behaviour of combustion processes. In particular, it is essential to correctly capture non-equilibrium phenomena such as localised ignition and extinction to faithfully predict pollutant formation. Reactor-based combustion models - such as the Eddy Dissipation Concept (EDC) or Partially Stirred Reactor (PaSR) - may account for turbulence-chemistry interactions at an affordable computational cost by calculating combustion rates relying upon canonical reactors of small fluid size and timescale. The models may include multiscale mixing, detailed chemical kinetic schemes and high-fidelity multispecies diffusion treatments. Although originally derived for conventional, highly turbulent combustion, numerous recent efforts have sought to generalise beyond simple empirical correlations using more sophisticated relationships. More recent models incorporate the estimation of scales based on local variables such as turbulent Reynolds and Damkohler numbers, phenomenological descriptions of turbulence based on fractal theory or specific events such as extinction. These modifications significantly broaden the effective range of operating conditions and combustion regimes these models can be applied to, as in the particular case of Moderate or Intense Low-oxygen Dilution (MILD) combustion. MILD combustion is renown for its ability to deliver appealing features such as abated pollutant emissions, elevated thermal efficiency and fuel flexibility. This review describes the development and current state-of-the-art in finite -rate, reactor-based combustion approaches. Recently investigated model improvements and adaptations will be discussed, with specific focus on the MILD combustion regime. Finally, to bridge the gap between laboratory -scale canonical burners and industrial combustion systems, the current directions and the future outlook for development are discussed.

For their ability to account for finite-rate chemistry, reactor-based models are well suited Turbulence–Chemistry Interactions (TCI) Sub-Grid Scale (SGS) closures for Large Eddy Simulations (LES). The SGS closure in the Partially Stirred Reactor (PaSR) model relies on the determination of the reacting fraction of each computational cell, whose definition is based on estimates of the characteristic mixing and chemical time scales. Direct Numerical Simulations (DNS) of turbulent combustion can supply key information on TCI for the development, validation, and comparison of combustion models. In particular, a priori testing allows the direct validation of model assumptions. In the present work, an a priori assessment of the PaSR model is conducted. Its ability to reconstruct thermo-chemical quantities of interest is investigated along with model assumptions. Sub-grid quantities are extracted from the DNS to investigate the role of the cell reacting fraction. Various definitions are then proposed to estimate the characteristic chemical timescale in the PaSR model. Modeled chemical source terms and heat release rates are compared against the filtered quantities from DNS data of a two-dimensional, spatially developing, turbulent nonpremixed jet flame with detailed kinetics. The results demonstrate the importance of accounting for the fine structures quantities in the context of reactor-based models. A new formulation of the chemical timescale is proposed and provides improved overall predictions. Several issues are raised in the discussion, representing realistic prospects for further developments of the PaSR model as a SGS combustion closure for LES.

Moderate or intense low-oxygen dilution (MILD) combustion can deliver appealing features such as reduced pollutant emissions, elevated thermal efficiency, and fuel flexibility. Turbulence-chemistry interactions (TCI) in MILD combustion are enhanced due to the intense internal recirculation of combustion products, and complex unsteady phenomena may occur. Jet-in-hot-coflow burners emulate MILD conditions with highly diluted coflows, thus simplifying the fluid patterns. Industrial systems employ confined or reverse-flow configurations to enhance mixing and trigger MILD conditions in the combustion chamber, where the hot flue gases dilute the fresh reactant mixture. Despite successfully characterizing the flow field, Reynolds-averaged Navier-Stokes computations suffer at predicting turbulent mixing and nonequilibrium phenomena such as local extinction and reignition. Large eddy simulation (LES) can accurately capture turbulent mixing and unsteady flow structures. This chapter aims at presenting LES investigations in the context of MILD combustion along with the models employed for TCIs. General best-practice guidelines for MILD LES modeling are also discussed.

Moderate or Intense Low-oxygen Dilution (MILD) combustion has drawn increasing attention as it allows to avoid the thermo-chemical conditions prone to the formation of pollutant species while ensuring high energy efficiency and fuel flexibility. MILD combustion is characterized by a strong competition between turbulent mixing and chemical kinetics so that turbulence-chemistry interactions are naturally strengthened and unsteady phenomena such as local extinction and re-ignition may occur. The underlying physical mechanisms are not fully understood yet and the validation of combustion models featuring enhanced predictive capabilities is required. Within this context, high-fidelity data from Direct Numerical Simulation (DNS) represent a great opportunity for the assessment and the validation of combustion closure formulations. In this study, the performance of the Partially Stirred Reactor (PaSR) combustion model in MILD conditions is a priori assessed on Direct Numerical Simulations (DNS) of turbulent combustion of MILD mixtures in a cubical domain. Modeled quantities of interest, such as heat release rate and reaction rates of major and minor species, are compared to the corresponding filtered quantities extracted from the DNS. Different submodels for the key model parameters, i.e., the chemical time scale τc and the mixing time scale τmix, are considered and their influence on the results is evaluated. The results show that the mixing time scale is the leading scale in the investigated cases. The best agreement with the DNS data regarding the prediction of heat release rate and chemical source terms is achieved by the PaSR model that employs a local dynamic approach for the estimation of the mixing time scale. An overestimation of the OH species source terms occurs in limited zones of the computational domain, characterized by low heat release rates.

The Sandia piloted methane-air jet flames represent ideal test cases for the validation of combustion models, for the simple burner design and the high fidelity of the available measurements. In particular, the level of turbulence-chemistry interactions changes significantly from flame C to F: flame C has little to no extinction, while flame F is close to global extinction, representing a very challenging case for finite-rate combustion models. In the present work, simulations on Sandia flames C-F are carried out using two finite-rate chemistry (FRC) models, the Eddy Dissipation Concept (EDC) and the Partially-Stirred Reactor (PaSR) approach. Both EDC and PaSR models are reactor based models. In other words, they are based on the partition of a computational cell in a reacting and a non-reacting zone. However, the turbulence-chemistry interaction factor in EDC solely relies on a turbulent time scale, based on the hypothesis that the chemical time scale is negligible to the mixing one, while in the PaSR concept, the cell splitting factor compares both the chemical and mixing time scales. The objective of this work is to show the importance of explicitly accounting for a chemical time scale, when finite-rate chemistry effects become relevant (flames E and F) and extinction and re-ignition phenomena occur. Several approaches for mixing and chemical time scale evaluation will be benchmarked, and the comparison to the available experimental data will allow to make recommendations about the range of applicability of the investigated FRC models.