# Study on Spray Characteristics and Breakup Mechanism of an SCR Injector
## 2 Numerical Models and Methods
### Model Description and Methods of Primary Breakup
#### 2.2.3 Secondary Breakup Model  
With the movement of the droplet, the droplet may deform or even breakup under the action of drag force, gravity and additional force, and the fragment of the droplet is resisted by surface tension and viscosity. For inviscid fluids, the gas Weber number (\(W\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!  
Figure 4 shows the development and breakup of the liquid jet at different times when the inlet pressure is 0.6 MPa, where the volume fraction of the phase interface is 0.5. With the opening of the needle valve, the liquid flows obliquely from the pressure chamber into the nozzle and finally sprays into the static air. The jet is subjected to the interaction of aerodynamic force, viscous force, and surface tension, resulting in an impact of the jet front, and then, surface waves are generated and transmitted to the upstream of the liquid jet. The length of the liquid column gradually develops until the aerodynamic force overcomes the resistance of the liquid surface tension and viscous force, and then, the liquid column becomes unstable and fragments, forming ligaments and small droplets. The breakup process of the liquid jet can be divided into two stages. In the first stage, the liquid column shows a periodic fluctuation with an intense amplitude, as shown in Figure 4a-c. The periodic fluctuation of the liquid jet tends to be gentle, and the breakup process tends to be stable in the second stage, as shown in Figure 4d-f.  
Figure 4: _Cont._  
Figure 3: Displacement curve of the needle valve.  
Figure 5 explains the cause of the difference in the primary breakup. The pressure field inside the nozzle is constantly changing, due to the movement of the needle valve. The variation of the pressure field leads to a periodic fluctuation of the initial radial velocity of the jet (the radial velocity at the nozzle exit), and then causes the periodic radial disturbance of the liquid jet in the first stage. When the needle is stationary (at top dead center), the  
Figure 4: Development of the primary breakup at different times: (**a**) 0.5 ms; (**b**) 1 ms; (**c**) 1.5 ms; (**d**) 2 ms; (**e**) 2.5 ms; (**f**) 3 ms.  
pressure field in the injector as well as the liquid distribution in the nozzle tend to be relatively stable, the radial velocity fluctuation weakens and tends to fluctuate in a small range, and the primary breakup is dominated by asymmetric surface waves.  
There are two basic types of surface waves: dilational wave and sinuous wave. At low jet velocities (usually corresponding to low \(Re\) numbers), the jet periodically exhibits a wave pattern of increasing and decreasing diameter, which is dominated by a dilational wave. With the increase in the \(Re\) number, the jet axis mainly presents asymmetric disturbance, which is the sinuous wave [30; 31].  
In the study of primary breakup of the liquid jet, three important dimensionless numbers, Reynolds number (\(Re\)), Weber number (\(We\)) and Ohnesorge number (\(Oh\)), are often used to classify the breakup mode of the jet. They are defined as:  
\[Re=\frac{\rho_{l}\mathbf{u_{0}}d_{0}}{\sigma_{l}} \tag{16}\]  
\[We=\frac{\rho_{l}\mathbf{u_{0}^{2}}d_{l}}{\sigma_{l}} \tag{17}\]  
\[Oh=\frac{\mu_{l}}{\sqrt{\rho_{l}\sigma_{l}d_{0}}} \tag{18}\]  
where \(d_{0}\) and \(\mathbf{u_{0}}\) denote the diameter and average axial velocity of the jet at the nozzle exit, respectively. As shown in Figure 6, [PERSON] et al. [32] divided the breakup modes of the liquid jets into four categories based on the \(Re\) number and \(Oh\) number, which are Rayleigh breakup, first wind-induced breakup, second wind-induced breakup and atomization. When a criterion of the \(Re\) and \(Oh\) is considered, the cases of primary breakup in this study should be in the first wind-induced breakup regime.