# A Heterogeneous Parallel Algorithm for Euler-Lagrange Simulations of Liquid in Supersonic Flow
## 3 Results and Discussion  
### Code Validation  
Simulations of a liquid jet in supersonic flow are performed to numerically replicate the experiments of [PERSON] et al. [20]. Figure 8 shows a schematic diagram of the computational domain and experimental conditions. The Cartesian coordinate system employed is also demonstrated in Figure 8; \(x\), \(y\), and \(z\) denote the streamwise, transverse, and spanwise directions, respectively. The width of the computational domain is 40 mm, while the dimensions of the streamwise and transverse directions are 200 mm and 40 mm, respectively. The nozzle with a diameter of 0.5 mm is installed in the center of the bottom floor and 50 mm downstream of the leading edge. The number of the grid points is 401 \(\times\) 21 \(\times\) 41 in \(x\), \(y\), and \(z\) directions, respectively, and the grid near the nozzle and the wall are refined. The effect of the grid has been investigated in our previous work, and the grid used in this  
Figure 7: Code structure of the CPU/GPU two-phase computational mode.  
paper fulfills the requirement of independence [42; 43; 44]. Water is used as the test substance, whose physical properties are illustrated in Table 1, the initial droplet diameter is 100 \(\upmu\)m. The freestream crossflow air has a Mach number of 1.94, a static pressure of 29 kPa, and a static temperature of 304.1 K. Details of the simulation are listed in Table 2.  
For the gas phase, a supersonic inflow condition is used at the inlet, and the 1/7 th power-law velocity profile is adopted. The extrapolation method is used at the outlet. Symmetry boundaries are employed at the spanwise boundaries. All walls are in non-slip, non-penetration, and adiabatic condition. For the droplet-wall collision, the particle rebound correlation proposed by [PERSON] [45] is adopted.  
Figure 9 demonstrates the pressure contour at the bottom wall and center plane perpendicular to the \(z\)-axis, the streamwise velocity contour at the plane of \(x\) = 0.08 m, and the simulated droplets, whose color represents their streamwise velocity. When injected into the supersonic crossflow, the water jet acts as an obstruction to the supersonic flow. As a result, a bow shock is formed in front of the jet, where the pressure changes sharply, manifesting as a high-pressure zone upstream of the jet. Near the boundary layer, the reverse pressure gradient transports air upstream and forms a separation zone. The droplets are rapidly broken down into finer sizes by interacting with the crossflow. As the droplets are swept down in the direction of the crossflow, the span of their distribution increases in width. The droplet streamwise velocity in the periphery of the atomization zone is relatively high, while in the core and the near-wall region, the velocity is lower. Moreover, due to the spray, the streamwise velocity of air of the core region is lowered even down to subsonic levels. These qualitative phenomena are consistent with the experimental and numerical results given in reference [13].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Density** & **Viscosity** & **Surface Tension** \\ \hline
988 kg/m\({}^{3}\) & \(2.67\times 10^{-3}\) kg/(m\(\cdot\)s) & 0.075 N/m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical properties of water.  
Figure 8: Schematic of the computation domains and conditions.  
Figure 10 compares the penetration height of our model with that observed in the experiment of [PERSON] et al. [20]. The black dots represent the instantaneous water droplets. The solid and dashed lines represent the average jet boundary of numerical results based on the CPU/GPU model and that of the experiment, respectively. It can be seen that the droplet distribution obtained is in good agreement with that of the experiment [20]. In addition, the penetration height obtained by [PERSON] et al. [46] using conservation-element and solution-element (CE/SR) is also shown in Figure 10. These results prove that our model can simulate a liquid jet in supersonic crossflow well.