## 2 Numerical Analysis
### Computational Mesh, Boundary, and Initial Conditions  
The 3D meshes of the engine combustion chamber for CFD analysis have been created by the ESE-diesel platform. Because of the axial symmetry characteristics of the engine combustion chamber, the injector nozzle has 10 identical holes, and for the computation time saving purpose, only one-tenth of the whole engine combustion chamber has been meshed. The simulation began at the IVC and ended at the EVO of the engine. It was carried out in parallel by a twelve-core AMD Ryzen 7 5800H processor (Advanced Micro Devices, Inc., Santa Clara, California and Austin, Texas, U.S.A.) and took approximately 25 h of CPU time. The computational meshes at the TDC of the three combustion chamber types in this study are presented in Figure 2.  
The surfaces of the piston, cylinder head, and cylinder liner of the engine were defined as impermeable walls. The cylinder liner and cylinder head surfaces were fixed walls, while the piston surface was a movable wall. Due to the axial symmetry characteristics of the cylinder geometry, the two cutting surfaces on the CFD computational domains have been assigned as cyclic boundary conditions (BCs). Diesel oil was injected directly into the engine cylinder at 12 crank angle degrees (CADs) before the top dead center (BTDC) with an injection duration of 32 CADs. During the simulation process, the computational mesh was re-meshed, corresponding to the piston movement by using the dynamic layering re-meshing method. In this method, the boundary layers of the meshes will be collapsed or added depending on the piston movement direction (upward or downward movement). Therefore, the number of layers of the computational dynamic mesh and the number of elements will consequently decrease or increase according to the moving up or down of the piston. In this way, the aspect ratio, skewness, orthogonal qualities, and resolution of the mesh are kept almost constant during the computations. To elimin  
Figure 2: 3D computational meshes of (**a**) U-type, (**b**) \(\omega\)-type, (**c**) re-entrant type combustion chambers at the TDC of the engine. of the cylinder liner and piston surfaces, as well as to accurately capture all the behaviors of the fluid flow near the walls, three inflation layers with a 1.2 mm thickness were applied on these surfaces. In engine modeling, the time-step size for the simulation is calculated based on the piston speed. In this case, the time-step size was 5.656 \(\times\) 10\({}^{-5}\) s. However, owing to the complexity of the combustion process, the time-step size of the simulations during the combustion period of fuel (12 CADs BTDC to 20 CADs ATDC) was reduced to 10 times to ensure the convergence of the solution. In this research, the simulations of the combustion had been performed according to the assumption that the simulated engine was working in a stable thermal situation. The thermal situation on the engine combustion chamber walls was almost stable. With this assumption, the temperature level on the engine cylinder liner surface was set to 197 \({}^{\circ}\)C, while the temperature levels on the piston and cylinder head surfaces were set to 297 \({}^{\circ}\)C, based on the technical suggestions in the literature [41]. The temperature of the cylinder liner surface was lower than that of the piston and cylinder head surfaces, owing to the cooling effect of the freshwater cooling system of the engine. The engine was charged with the fresh air by an exhaust gas turbocharger. The temperature and pressure in the engine cylinder at the start of the simulations (i.e., at the IVC) were 37 \({}^{\circ}\)C and 2.5 bar, respectively, the same as reported in the engine experiments. Additionally, the properties of the fresh air, such as density, thermal conductivity, viscosity, specific heat capacity, etc. had been set as functions of temperature in order to ensure the right responses of the air to any change in temperature.