## 2 Numerical Methodology
### CFD Models  
CONVERGE 3.0 CFD commercial software is used to solve numerically, mass, density-based Navier-Stokes equations, energy, and species equations for conventional diesel engine CDC and RCCI engine [34]. The in-cylinder turbulent flow is predicted using the Reynolds-averaged Navier-Stokes (RANS) equations in conjunction with the average generalized re-normalization group (RNG) turbulence model [35]. The automated cut-cell Cartesian grid approach automatically generates meshes and orthogonal mesh with hexahedrons. The mesh is updated at each timestep for movable boundaries. Adaptive mesh refinement (AMR) is employed to adjust the mesh resolution according to the varying conditions of the moving fluid, including factors such as velocity, temperature, species, scalars, passive variables, and boundary conditions, ensuring convergence. Furthermore, fixed embedding is applied to refine the mesh at specific critical locations, including cylinders, nozzles, and the fuel spray region, thereby allowing the coarse grid to accelerate the numerical solution in other regions [36]. The modified Pressure Implicit with the Splitting of the Operator (PISO) is utilized to solve the transport equations [37]. In this study, the Euler-Lagrangian method was employed. The Lagrangian method was employed to trace the movement of each droplet within the flow field. Meanwhile, the Eulerian method was applied to model the gaseous phase. The fuel parcels are introduced into the cylinder at a specific rate. The parcels are used to predict an identical group of drops. During the injection period, these droplets undergo various processes that promote their vaporization. The fuel particles may become unstable due to the forces exerted on their surfaces as they move through the gaseous medium within the cylinder. A breakup model is employed to solve this problem. The hybrid Kelvin-Helmholtz-Rayleigh-Taylor (KH-RT) model is employed to forecast the behavior of droplet breakup [38], and it is preferable since it is used in many other similar studies [39; 40; 41]. A detailed chemical model, SAGE,is utilized to simulate the combustion process [42]. The reduced primary reference fuel (PRF) mechanism is employed to predict the n-heptane and methane chemical reactions, composed of 464 reactions and 76 species [43]. For simplicity, n-heptane (C7H16) is used as a substitute for diesel fuel because of its analogous cetane number. The Frossling model was selected to simulate the droplet evaporation process [44]. For spray drop coalescence, the post-collision outcome model is used [45], and the No Time Counter (NTC) method is applied to model drop collision [46]. The Rebound/slide model is used to simulate the spray droplets' interaction with solid surfaces [47]. The extended Zeldovich mechanism is utilized to find the amount of NO formation [48]. The empirical model of Hiroyasu-NSC is used to predict soot oxidation [49]. CONVERGE 3.0 CFD calculated the species, including CO, CO2, methane slip (CH4), and UHC, by solving transport equations with the help of the SAGE combustion model and reaction mechanism. Table 3 provides a concise overview of the sub-models employed in the simulation. These models are extensively used in other studies [50; 51; 52].