## 2 Numerical Analysis
### Computational Mesh, Initial and Boundary Conditions  
The AVL FIRE R2018a ESE-diesel platform was employed to build a three-dimensional (3D) model of the combustion chamber of the engine and a computational mesh for the CFD analysis. Because the combustion chamber of the engine is axially symmetric; the pilot fuel nozzle has twelve identical holes; and for the calculation time reduction purpose, only one-twelfth of the entire combustion chamber was used to create the computational domain. The calculation began from the IVC (intake valve closing) to the EVO (exhaust valve opening) and was carried out in series using a twelve-core Intel Xeon processor. It took approximately 36 h of CPU time. The 3D movable computational mesh was generated using the dynamic layering method. The movement of the piston surface was controlled by Equation (1). \[\mathrm{P_{1}}=\sqrt{\left(\mathrm{r_{c}}+\mathrm{L}\right)^{2}-\mathrm{x_{ offset}^{2}}}-\mathrm{r_{c}}\cos(\theta_{\mathrm{c}})-\sqrt{\mathrm{L}^{2}- \left(\mathrm{r_{c}}\sin(\theta_{\mathrm{c}})+\mathrm{x_{offset}}\right)^{2}} \tag{1}\]  
where, \(\mathrm{P_{1}}\) is the piston location, \(\mathrm{r_{c}}\) is the crank radius, \(\mathrm{L}\) is the length of the connecting rod, \(\mathrm{x_{offset}}\) is the piston pin offset, while \(\theta_{\mathrm{c}}\) is the current CAD. The computational mesh with the boundary conditions (BCs) when the piston is at 40 CADs (crank angle degrees) ATDC (after the top dead center) is presented in Figure 2. The combustion chamber of the engine is built by the cylinder head, cylinder liner, and the piston surfaces. This engine is designed to run smoothly in both the diesel and DF modes without the compression ratio (CR) adjusting requirements. That is, the CR was the same in both the diesel and DF combustion simulations. To achieve the correct calculated CR in the CFD simulation, which is the same as the actual CR in the experiments, the \"compensation volume\" option was active to create a compensation volume on the top of the piston, to compensate for the lacking volume part of the swept volume when building the computational mesh for the simulations. For the CFD analysis, the BCs at the cylinder liner, cylinder head (cover), and piston surfaces were assigned as impermeable walls. Owing to the axially symmetric characteristic of the combustion chamber geometry, the cutting surfaces of the computational domain were assigned as periodic (cyclic) BCs. The simulation started from the IVC of 35 CADs after the bottom dead center (ABDC) to the EVO of 62 CADs before the bottom dead center (BBDC) of the engine. In the diesel mode, the injection of the diesel fuel started at 12 CADs BTDC and took place in 32 CADs (corresponding to 7.5 milliseconds). In the DF mode, the diesel pilot fuel was injected into the engine cylinder at 12 CADs BTDC as well, but took place in 12 CADs only (corresponding to 2.35 milliseconds). In addition, the combustion simulations had been carried out, based on the assumption that the engine was running in stable thermal conditions. In which, the thermal condition on the combustion chamber wall was stable. With this assumption, the temperatures of the cylinder head and piston surfaces were set to 297 \({}^{\circ}\)C while the temperature on the cylinder liner was set to 197 \({}^{\circ}\)C, based on the reference to similar studies [8; 9] and the suggestion in the literature [26]. The temperatures on the cylinder head and piston surfaces were higher than that on the cylinder liner surface because of the cooling effect of the cylinder liner's freshwater cooling system of the engine. This engine is supplied with the charge air by a turbocharger. The pressure and temperature inside the engine cylinder at the start of the simulation (i.e., at the IVC) are 3.5 bar and 47 \({}^{\circ}\)C, respectively, the same as those recorded in the experiments. In addition, all of the properties of the air were set as temperature-dependent to ensure the response of air to the change of temperature.