## 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 efficiency 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 different 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 differencing 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 effects 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 effects 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 different level of complexity [1; 2; 3; 4; 5; 6]. In addition to the turbulence/burn rate interaction, the computational treatment of in-homogeneities in the combustion chamber can strongly affect 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; 8; 9; 10; 11]).