# Combined CFD-Stochastic Analysis of an Active Fluid Injection System for Jet Noise Reduction
## 2 Model Description and Validation
### Model Validation  
The SNGR approach has been firstly validated against experimental and LES results of a subsonic jet with diameter \(D_{j}=0.050\) m, Mach number \(M=0.75\) and the Reynolds number, \(Re_{D}=5.0\times 10^{4}\), described by [PERSON] [7]. The two-dimensional geometry of the domain, shown in Figure 3, has been built using the CAD software Workbench by ANSYS. A CFD RANS simulation based on the nozzle geometry used for the LES simulation [7] has been performed. The external profile of the nozzle has been reproduced by scanning from the original work [7]. The RANS simulation of the subsonic jet has been performed in the cold conditions, for which \(T_{j}/T_{\infty}=1\), where \(T_{j}\) is the static temperature of the jet at the nozzle outlet section, and \(T_{\infty}\) is the ambient temperature. In this study, the static temperature and pressure of the jet at the nozzle outlet  
Figure 2: Two-point space correlation of a jet plume. (**a**) First approach based on blobs structure; (**b**) Second approach based on an adaptive Cartesian mesh. section are respectively \(T_{j}=288\) K and \(P_{j}=101\),300 Pa. The temperature and pressure at the nozzle inlet section, respectively \(T_{0}\) and \(P_{0}\), were calculated using Rankine Hugoniot equations:  
\[ \begin{array}{l}\frac{P_{0}}{P_{j}}=\left(1+\frac{k-1}{2}M^{2}\right)^{\frac{ k}{k-1}}\\ \frac{T_{0}}{T_{j}}=1+\frac{k-1}{2}M^{2}\end{array}  \tag{9}\]  
where \(M=0.75\) is the Mach Number while \(k=c_{p}/c_{v}=1.4\) is the specific heat ratio of air, temperature and pressure at the nozzle inlet section, obtained by solving these two equations, are \(P_{0}=147\),100 Pa and \(T_{0}=320.4\) K.  
The jet flow simulation has been carried out using the CFD software Fluent by ANSYS. The mesh extends from 0 to 50 nozzle diameters, \(D_{j}\), in the axial direction. The radial extension is \(10D_{j}\) at the nozzle outlet position and it increases up to \(20D_{j}\) at the far-field outlet position. Mesh cells are refined in the region near the nozzle exit and in the jet shear layer. A 2D axisymmetric transient pressure based second-order upwind scheme has been employed to converge fully coupled RANS equations with turbulence accounted for through \(K\)-\(\epsilon\) and \(K\)-\(\omega\)\(SST\) models. A view of the mesh made up of \(1.85\times 10^{5}\) cells and contour plots of the axial velocity and turbulent kinetic energy are shown in Figure 4.  
The aerodynamic field results were extracted along the centerline and along radial lines at three axial positions downstream of the nozzle exit, according to Figure 5.  
Figure 4: (a) Mesh used for the Reynolds Averaged Navier Stokes (RANS) jet flow simulation; (b) Contour plots of the RANS solution. Mean velocity on the top and turbulent kinetic energy on the bottom. Figure 3: Computational domain (Adapted from [PERSON] (2016)). The laminar core and the decay of the centerline velocity are in good agreement with the experimental results. The predicted laminar core length of \(8D_{j}\) is slight higher than the experimental value as usual for RANS computations (Figure 6a). The maximum of centerline turbulent kinetic velocity occurs at about \(10D_{j}\), in accordance with experimental results but its amplitude is slightly underestimated (Figure 6b). Moreover, the radial profile of the axial velocity is well predicted by both RANS and LES computations (Figure 7a), whereas, the \(uv\) correlation levels of the radial velocity profile are underestimated by the RANS analysis due to lower mixing predicted and overestimated by the LES (Figure 7b). Figure 5: The vertical dashed lines indicate lines along which profiles of time-averaged quantities were extracted (Adapted from [PERSON] [7]). Figure 6: (a) Centerline profile of the axial velocity; (b) Axial profiles of turbulence intensity. (Dotted black lines: experimental results; continuous red line: \(K\)-\(\varepsilon\) model; dashed green line: \(K\)-\(\omega\) SST model; dotted-dashed blue line: LES).