**1. Introduction**

The ever-stringent regulations on carbon dioxide and criteria pollutants (i.e., nitrogen oxides or particulate matter) for internal combustion engine vehicles (ICEV), as well as the original equipment manufacturers (OEM) needing to reduce the technology development times, are among the key drivers of modern computer aided engineering (CAE) for engine development toolchains. To thoroughly study and optimize engine fuel e fficiency and reduce pollutant formation, an experimentally driven campaign generally requires the deployment of expensive and highly complex techniques. Moreover, the ever-increasing hardware complexities being introduced in modern powertrains (i.e., pre-chamber or advanced multi-stage aftertreatment) make the experimental engine development process even more challenging. Each new technology introduces a new degree of freedom in the parameter range of a combustion engine development. In this scenario, numerical methods represent an attractive tool to aid the engine development process, more so if they are capable of accounting for both the chemical and physical phenomena occurring in an internal combustion engine-based powertrain. Provided that such numerical methods deliver an acceptable level of accuracy, engine development costs can be substantially reduced by running engine optimization campaigns within a virtual framework. However, modeling of the in-cylinder combustion process poses many challenges, such as (1) Turbulence-Chemistry Interaction (TCI), (2) fuel injection and mixture formation and (3) gaseous pollutants and particulates formation mechanisms. On top of the numerical complexity, computational cost is also among the major decisive factors of whether a certain modeling approach shall be deployed in the development stage or not. The methods need to be fast to deliver information in time during an engine or vehicle development process.

Among the di fferent reactive flow simulation frameworks currently adopted for engine development, 3-D Computational Fluid Dynamics (CFD) allows to model flow, turbulence and combustion chemistry processes with high level of detail. However, depending on the chosen model parameters such as computational grid/time-step size, numerical di fferencing scheme and the number of species/reactions in the chosen chemical kinetic model, 3-D CFD may require an unfeasible computational cost. This is particularly true when engine and fuel chemistry e ffects are to be considered across a large set of operating points or during transient operations. In this respect, lower order tool chains (i.e., 0-D and 1-D frameworks) require a small fraction of the computational times compared to advanced 3-D CFD analyses. The price for this benefit is the limited numerical accuracy and frequently a loss of chemical and physical information. The treatment of the combustion chemistry and the turbulence-chemistry interaction e ffects are among the main aspects to be addressed in order to achieve high accuracy and feasible simulation times. Both these phenomena become particularly important in the development and optimization of novel internal combustion engine concepts which may include, for instance, dual-fuel, highly premixed fuel/oxidizer mixtures or complex exhaust gas recirculation (EGR) strategies.

Numerous 0-D methods have been proposed to describe the Rate of Heat Release (RoHR) and the turbulence for both Spark Ignition (SI) and Compression Ignition (CI) engines to with di fferent level of complexity [1–6].

In addition to the turbulence/burn rate interaction, the computational treatment of in-homogeneities in the combustion chamber can strongly a ffect the predictive capability of a 0-D model especially under Diesel engine conditions. The most common approach is to discretize the trapped mass into several computational zones, which vary depending on the number of physical regions included in the model formulation (i.e., flame front, cylinder wall area, crevice [7–11]). While such models present a remarkable advantage against single or two zone models, a mean temperature and gas composition within each zone has still to be imposed by definition. This implies that the calculation of the chemical source terms is done assuming negligible variations in enthalpy and composition spaces within each zone and hence no TCI e ffects are considered. These simplifications, together with the lack of detailed chemistry sub-models, impact the quality of engine-out emission predictions.

An alternative method to consider turbulent mixing as well as detailed chemistry in 0-D is the Stochastic Reactor Model (SRM). The SRM discretizes the mixture in the combustion chamber in a given number of notional particles under the assumption of statistical homogeneity, as opposed to special homogeneity in multi-zone models. Each notional particle features a realization of possible temperature and mixture compositions. TCI is mimicked by stochastic mixing of particles, stochastic heat exchange with the walls and detailed chemistry evaluations. However, as it is the case for detailed chemistry-based 3-D CFD methods, depending on the size of the chemical mechanism considered, the chemistry step may take up to 99% of the simulation time. In addition to mechanism reduction techniques [12], chemistry storage and run-time retrieval methods are viable solutions to reduce computational costs.

Different tabulated chemistry-based methods have been proposed primarily for 3-D CFD applications. These methodologies are based on the decoupling of flow and chemistry. While the flow is computed during run-time, the chemistry solution, usually intended as auto-ignition and/or emission formation processes, is computed in a pre-processing step typically performed on a one-time basis for a given fuel. The two major combustion modeling concepts used to decouple flow and chemistry are various flamelet methods, including Conditional Moment Closure (CMC) and the Well-Stirred Reactor (WSR) methods. Remarkable efforts have been made towards formulation of predictive flamelet-based tabulated chemistry solvers for auto-ignition [13–17] as well as advanced soot and NOx emission formation modeling [18–21]. The present article is based on the WSR approach. This is the best choice for Probability Density Function (PDF) methods, such as the herein employed SRM. These methods use operator splitting loops to separate vaporization, mixing, compression, heat transfer and chemistry. In these processes, each particle is considered as a well stirred reactor. The chemistry storage is constructed by means of 0-D adiabatic constant pressure/volume reactors across wide ranges of initial pressure, temperature and equivalence ratio. Pires da Cruz et al. [22] proposed a method where both the high and low temperature ignition delay times are stored in the look-up table. The validation was done under 0-D adiabatic reactor conditions (assuming constant pressure/volume) as well as under diesel engine conditions in 3-D CFD. An improved version of such model was later proposed by Colin et al. [23] where a progress-variable-based approach was used rather than a tabulation of the ignition delays. In this configuration, an additional tabulation dimension is introduced as the tabulation is done across a predefined set of grid points, varying between unburned and fully burned mixture. During run-time, the progress variable source term was retrieved for each cell, and it was used to reconstruct the chemical state. Later improvements of such method proposed by Knop and co-authors [24] also incorporated a turbulence-chemistry interaction term during the tabulation process. Thanks to an additional tabulation dimension, TCI effects could be accounted for in 3-D CFD reactive flow simulations within the Extended Coherent Flame Model (ECFM) framework. The model (referred as ECFM-TKI in [24]) has been applied to predict the ignition process of Diesel and homogeneous charge compression ignition (HCCI) engines and is currently implemented in various commercial engine CFD software.

With respect to modeling of spray flames and Diesel engines, numerous models, such as the Partially Stirred Reactor (PaSR) concept [25], have been formulated and implemented in OpenFOAM [26,27].

When it comes 0-D/1-D frameworks, the number of studies featuring tabulated chemistry-based methods is rather limited. Its implementation, however, is potentially very useful as it allows to include detailed chemistry effects, as opposed to the commonly used empirically derived correlations for emission predictions. Leicher et al. [28] proposed a table look-up approach based on mixture fraction and reaction entropy as progress variable. Their methodology was implemented in an SRM and tested under constant pressure reactor conditions. Bernard and co-authors [10] simulated heat release and pollutant formation by means of a Flame Prolongation of Intrinsic Low Dimensional Manifold (FPI/ILDM initially proposed in [29]) as well as a timescale-based sub-model for NO formation. The tabulation method used constant volume reactors and a CO-CO2-based progress variable definition. Upon table generation, CO, CO2, H, H2 and the fuel molecule were stored as key species for the reconstruction of the thermodynamic state during engine simulation. The method was validated for a wide range of Diesel engine conditions. Within the SI engine simulation framework, Bougrine et al. [30] proposed a two-zone 0-D model (referenced as One-Dimensional Coherent Flame Model-Tabulated Chemistry (CFM1D-TC)) where the chemical part of the combustion process was tabulated using laminar 1-D premixed flame solutions. In addition, a time-scale model was formulated to better represent the relaxation towards equilibrium of CO and NO species with the help of homogeneous

reactor calculations. Bozza et al. [31] implemented the previously mentioned ECFM-TKI [24] approach within a 0-D/1-D phenomenological combustion model for better knock prediction in spark ignited engines compared to the traditionally used Livengood-Wu [32] approach. Validation under both 0-D reactors and knocking SI engine simulations under stoichiometric conditions showed promising results when compared to the online chemistry solutions.
