**2. Numerical Modeling**

In this study, numerical simulations were implemented using CONVERGE Version 2.2 [24], which was used to create constant volume models and set the turbulence model, spray model, and sub-model, as shown in Table 1. This model was created to predict the diesel spray performance by simulating the spray shape and the spray penetration distance in both a liquid and a vapor state, as well as the mixing behavior. The shape of the model is defined as a constant volume combustion chamber with a diameter of 105 mm and a length of 105 mm to reduce the grid number and increase computational e fficiency, with input from the case study boundary conditions under the experimental conditions [23]. The injector was placed at the top center of the cylinder, as shown in Figure 1. The spray was designed using the Spray A condition (detailed information is shown in the ECN [23]). The C12 H26 reaction mechanism was used as a diesel fuel agen<sup>t</sup> like the experimental considerations since the current model estimation of the trends and evolution of vapor penetration are independent of the fuel type [25].


**Table 1.** Modeling and Numerical Parameters.

**Figure 1.** Model characteristics of cylinders.

The grid size has a large influence on the penetration distance [26]. When the grid size is too small, a long simulation time will be required. In this study set up, a fixed grid embedded with a minimum grid size for the spray flow field is shown in Equation (1) [24], where the "Size of cells in scaled grid" represents the sizes of the cells in the scaled grid in each axial "Size of cells in base grid" indicates the sizes of the cells in the base grid setting of each axial, and "amr\_embed\_vel\_scale" is the level of embedding. In this study, CONVERGE can easily change the overall grid resolution prior to executing a simulation to assess grid sensitivity, which is useful for reducing the working time for constant stimulation. For simulations in areas or periods that are not of grea<sup>t</sup> importance, a rough grid can be used and then adjusted to consider only the important periods by setting the level of embedding for each section to obtain results that are accurate and not unnecessarily time intensive for calculations:

$$\text{Size of cells in scaled grid} = \text{Size of cells in base grid} \ast 2^{-\text{ann\\_embed\\_val\\_scale}}.\tag{1}$$

The computational fluid dynamics simulation of the in-cylinder process has many modifications in the spray modeling. The interaction of spray turbulence modeling has an important influence on the spray penetration and mixture formation prediction, which emerges at the end of the entire combustion process. A recent literature review on fuel spray modeling showed that the use of RANS guidelines is of fundamental importance to ensure sufficient performance of the spray mixing and combustion processes. In this study, we used the RANS method to describe the spray development process. The RANS model was used in conjunction with the RNG k-ε turbulence model to determine the effects of smaller movements. A drop evaporation model base on the Frossling model was also used. The liquid penetration length is defined as the maximum distance from the axial position with 99% injected fuel mass at the injection location, while the vapor penetration length is defined as the distance from the farthest location with a 0.01% fuel mass fraction to the nozzle exit. For a fast and accurate collision calculation response, the NTC model with an O'Rourke outcome was used in this study. To increase the efficiency of our droplet collision calculations, we used a dynamically simulated gas technique for spraying. For this technique to work effectively, it must be able to handle a common case, where the number of droplets in each particle varies. The configuration can work effectively with general cases under different conditions.

For the spray breakup model in this study, the Kelvin–Helmholtz instability (KH) and Rayleigh–Taylor instability (RT) models were used. The KH model size constant is defined as proposed by Reitz [27]. The wave breakup formulation was used to model the liquid breakup process. The wavelength (ΛKH) and the breakup size constant (B0) determined the child droplet sizes. The drop radius equation is calculated as follows:

$$
\mathbf{r}\_{\rm KH} = \mathbf{B}\_0 \boldsymbol{\Lambda}\_{\rm KH}.\tag{2}
$$

By requiring that B0 equal 0.61 for the droplet breakup regime that has the characteristics of 25 < We < 50, under a higher injection velocity, the B0 will be in accordance with the applications offered by Hwang et al. [28]. The KH model velocity constant was determined to be 0.188, which is the basic value commonly used by researchers. During the breakup, the parent droplet parcel radius (r) is continuously reduced until it reaches a stable droplet radius (rKH) according to the following equation:

$$\frac{\text{dr}}{\text{dt}} = \frac{\text{r} - \text{r}\_{\text{KH}}}{\text{\textdegree\textsuperscript{\text{r}\_{\text{KH}}}}}, \text{ \texttt{r}\_{\text{KH}} \le \text{r}}\tag{3}$$

where τKH is the breakup time, given by:

$$\pi\_{\rm KH} = \frac{3.276 \text{B}\_1 \text{r}}{\Lambda\_{\rm KH} \Omega\_{\rm KH}} \tag{4}$$

where ΩKH is the KH wave with the maximum growth rate. The KH model breakup time constant (B1) determines the primary breakup time. In this study, B1 equals 21, which is more accurate than the recommended value (recommended value equals 7) in other references. The RT breakup length equation is given by:

$$\mathbf{L}\_{\rm b} = \mathbf{C}\_{\rm bl} \sqrt{\frac{\rho\_{\rm l}}{\rho\_{\rm g}}} \mathbf{d}\_{\rm 0} \tag{5}$$

where Cbl is the RT model breakup length constant, ρl is the fuel density, ρg is the ambient gas density, and d0 is the orifice diameter. The RT model breakup length constant (Cbl) generally equals 1.0.

It is necessary to switch the liquid breakup parameter from the KH model to the RT model. Using this approach, the breakup time can be determined by the RT model breakup time constant (Cτ). The value of Cτ should be less to reduce the breakup delay, as shown in the RT breakup time equation below:

$$
\tau\_{\rm RT} = \mathcal{C}\_{\tau} \frac{1}{\Omega\_{\rm RT}} \tag{6}
$$

where ΩRT is the RT wave with the maximum growth rate. The RT model size constant (CRT) used to determine the scaled wavelengths and the radius of the RT breakup with a higher value increases the predicted RT breakup radius size:

$$\mathbf{r} = \pi \frac{\mathbf{C}\_{\text{RT}}}{\mathbf{K}\_{\text{RT}}} \tag{7}$$

where KRT is the wavenumber. In this study, CRT equals 0.1, in which the parent radius decreases continuously until it reaches the constant value in Equation (3), as in the KH model. The setting of the spray breakup model provides an additional configuration for the breakup model constants. Notably, different breakup model constants are used depending on the test conditions and the numerical calculation tools.

The spray dispersion angle is another important input variable. The spray distribution angle defines the fluctuation of air in the injected fuel due to the impulse exchange between gases and liquids. The spray distribution angle will change under different injection intervals (start of injection, the transient regime, the stable regime, and the EOI) due to changes in the nozzle sac flow, momentum, and the interactions between air and the fuel. The actual spray distribution angle detected by each method is different and cannot be compared. The differences in the spray distribution angles measured in the far-field show a higher angle variance than when measuring the angles using the near field method. Generally, the spray distribution angle can be represented in two ways: the spray angle and the spray cone angle. This study analyzed the shapes of different injection rates that have different injection pressures, making the spray distribution angle a very important variable because it is influenced by pressure. Here, the spray distribution angle is measured by taking an image of the cut-plane from the

direction of the spray by specifying two lines to determine the area of the spray and then measuring the distribution angle. The spray angle is measured from the nozzle outlet to the spray penetration distance/2 and the spray cone angle is measured from the nozzle outlet to the spray penetration distance equal to the 100×hole diameter. The spray cone angles were used in this study as the input in the spray model to analyze the spray behavior. In this study, the spray cone angles changes depend on the operating conditions of the di fference in rail pressure. The spray cone angle will increase noticeably when the injection pressure is higher because the pressure of the fuel mixture inside the nozzle increases and the distribution is higher. The relationship between spray cone angle and injection pressure is discussed in the fourth section.
