# Modeling the Formation of Urea-Water Sprays from an Air-Assisted Nozzle
## 2 Experimental and Numerical Methods
### Computational Method
#### 2.2.2 Treatment of the Turbulent Gas Jet  
The air flow escaping from the twin-fluid atomizer forms a single turbulent gas jet. This gas jet is accelerated to supersonic velocity, due to the under-expanded flow condition in the outflow channel of the nozzle. An under-expanded flow occurs if the pressure ratio of the compressed air in the mixing chamber to the surrounding pressure in the test section exceeds the critical pressure ratio [23]. Before the turbulent gas jet is initialized, the hot air flow in the test rig is precalculated until it achieves convergence. The gas jet is then initialized by means of a mass flow boundary in the outflow channel of the nozzle in the computational domain. The temperature of the gas flow inside of the atomizer has been experimentally determined by means of a thermocouple, which is situated directly upstream of the mixing chamber. On this basis, the temperature of the final mixture of the gas and liquid phase within the mixing chamber is used as an estimate of the initial temperature of the gas jet in the computational domain. In contrast to the initial temperature, no information about the initial turbulence level of the gas jet is available. Hence, a standard turbulence level of 5% is initialized as a rough estimate. A compressible solver, able to take into account supersonic flows [24], is applied for the calculations. Second-order accuracy differencing schemes are chosen for spatial discretization, and a first-order accuracy scheme is used for temporal discretization. The timestep is set to 1 \(\upmu\)s to achieve an appropriate temporal resolution. The turbulent flow is calculated using the Reynolds-averaged Navier-Stokes equations with the k-\(\varepsilon\) turbulence model. Equations (1)-(5) illustrate the transport equations of turbulent kinetic energy \(k\) and turbulent dissipation \(\varepsilon\) in dependence of the coordinate \(x\), the time \(t\), and the velocity \(U\). The viscosity and the turbulent viscosity are represented by \(\mu\) and \(\mu_{t}\). \(S\) represents the stress tensor in the production term of the turbulence \(P\). \[\rho\frac{\partial k}{\partial t}+\rho U_{j}\frac{\partial k}{\partial x_{j}}=P +G-\varepsilon+\frac{\partial}{\partial x_{j}}\bigg{(}\mu+\frac{\mu_{t}}{ \sigma_{k}}\frac{\partial k}{\partial x_{j}}\bigg{)} \tag{1}\]  
\[\rho\frac{D\varepsilon}{Dt}=\bigg{(}C_{\varepsilon 1}P+C_{\varepsilon 3}G+C_{ \varepsilon 4}k\frac{\partial U_{k}}{\partial x_{k}}-C_{\varepsilon 2} \varepsilon\bigg{)}\frac{\varepsilon}{k}+\frac{\partial}{\partial x_{j}} \bigg{(}\frac{\mu_{t}}{\sigma_{k}}\frac{\partial\varepsilon}{\partial x_{j}} \bigg{)} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cell Size (mm)** & **Distance from Nozzle (mm)** & **Radius (mm)** \\ \hline
0.125 & 25 & 3 \\
0.25 & 70 & 11 \\
0.5 & 125 & 15 \\
1 & 297 & 35.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mesh refinements with dimensions related to the nozzle exit. \[P=2\mu_{l}S*S-\left(\frac{2}{3}\right)\left[\mu_{t}(trS)+k\right](trS) \tag{3}\]  
\[G=-\frac{\mu_{t}}{\rho\sigma_{p}}\
abla\rho \tag{4}\]  
\[\mu_{t}=C_{\mu}\rho\frac{k^{2}}{\varepsilon} \tag{5}\]  
The model contains eight constants, which are available for calibration. In the present implementation, the standard values listed in Table 2 are utilized. It is known from the literature that one set of fixed parameters is not suitable to describe different turbulent flow phenomena. In the case of round jets, this results in a too-high spreading rate using the standard values [25]. To overcome this problem, the parameter \(C_{\varepsilon 1}\), which is used to control the production term in the transport equation of the dissipation rate in Equation (2), can be replaced by a function of the velocity decay rate [26], or an additional source term, known as Pope correction, can be added to the transport equation for the dissipation rate [27].