# Drift Potential Characteristics of a Flat Fan Nozzle: A Numerical and Experimental Study
## 2 Model Construction and Numerical Simulation
### Numerical Calculation Model
#### 2.2.2 Discrete-Phase Model Selection  
The discrete-phase-droplet transport equation is solved using the Euler-Lagrange method proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (1953). Equation (4) is the discrete-phase particle motion equation, expressed as:  
\[\frac{d\overrightarrow{u}_{p}}{dt}=\frac{18\mu}{\rho_{p}d_{p}^{2}}\frac{C_{D} R_{e}}{24}\left(\overrightarrow{u}-\overrightarrow{u}_{p}\right)+\frac{g_{x} \left(\rho_{p}-\rho\right)}{\rho_{p}}+\frac{1\rho d}{2\rho_{p}dt}\left( \overrightarrow{u}-\overrightarrow{u}_{p}\right) \tag{4}\]  
where \(\overrightarrow{u}\) is the continuous phase velocity (m/s), \(\overrightarrow{u}_{p}\) is the particle velocity (m/s), \(\rho_{p}\) is the particle density (kg/m\({}^{3}\)), \(d_{p}\) is the particle diameter (m), \(g_{x}\) is the acceleration due to gravity (m/s\({}^{2}\)), and \(C_{D}\) is the coefficient of drag. \(R_{e}\) is the relative Reynolds number calculated as follows:  
\[R_{e}=\rho vd/\mu \tag{5}\]  
where \(v\) is the velocity (m/s), \(\rho\) is the density (kg/m\({}^{3}\)), \(\mu\) is the viscosity coefficient of the fluid (Pa-s), and \(d\) is a characteristic length.  
The flat fan atomizer model was adopted for the nozzle model, and the linearized unstable liquid film atomization model proposed by [PERSON] et al., was adopted for the atomization fragmentation model. The spray simulation considers droplet collision and aggregation in the discrete-phase to determine droplet merging and rebound, mainly based on the critical value obtained by [PERSON] (1953). The critical value is a function of the collision Weber number, radii of the droplets, and the collection droplets:  
\[b_{crit}=(r_{1}+r_{2})\sqrt{min\left(1.0,\ \frac{2.4f}{W_{e}}\right)}, \tag{6}\]  
where \(b_{crit}\) is the critical value for judging droplet collision, merging, or rebound (m). \(r_{1}\) and \(r_{2}\) are the small droplet radii (m), and \(f\) is a function of \(r_{1}\)/\(r_{2}\). \(W_{e}\) is the collision Weber number:  
\[W_{e}=\rho v2l/\sigma \tag{7}\]  
where \(\rho\) is the fluid density (kg/m\({}^{3}\)), \(v\) is the characteristic flow rate, \(l\) is the characteristic length, and \(\sigma\) is the surface tension coefficient of the fluid.  
The Taylor Analogy Breakup (TAB) model was selected as the droplet collision secondary fragmentation model. ANSYS Fluent offers four droplet breakup models: the TAB, Wave, KHRT, and Stochastic Secondary Droplet (SSD) models. The TAB model is recommended for low-Weber-number injections and is well suited for low-speed sprays into a standard atmosphere. For Weber numbers greater than 100, the Wave model is more applicable. The KHRT offers a transition between both application areas because it considers the competing effects of aerodynamic breakup and instabilities due to droplet acceleration. It is combined with the Levich model for the liquid core length. The Wave and KHRT models are popular for use in high-speed fuel-injection applications. The SSD model treats breakup at high Weber numbers as a discrete random event resulting in a distribution of diameter scales over a range. The choice of model depends on the relative importance of momentum and surface tension in the droplet breakup.