# Combined CFD-Stochastic Analysis of an Active Fluid Injection System for Jet Noise Reduction
## 2 Model Description and Validation
### Model Validation
It can be argued that the RANS analysis results are in fairly good agreement with experimental data and LES results although they underestimate the value of \(uv\) correlation and do not predict the gradual increase of turbulent energy in the region of the laminar core. The acoustic field is evaluated by processing the RANS data by means the SNGR approach. To compute the stochastic noise sources, the axisymmetric CFD solution has been initially projected on a 3D domain, i.e., a uniform grid that consists of \(1.74\times 10^{6}\) cubic cells with \(\Delta x=5\) mm (Figure 8). The noise has been computed directly from the Fourier transform of the Lamb vector by using the integral solution of Powell\({}^{\prime}\)s equation. Acoustic results are computed on a total of 25 microphones on two microphones arcs in the jet far-field region (Figure 9) and compared against LES and experimental results. Then, 14 microphones are arranged on a first arc of radius \(30D_{j}\) from the center of the nozzle outlet section and positioned by 20 degrees to 150 degrees with an angular step of 10 degrees. The remaining 11 microphones were placed on a second arc of radius \(50D_{j}\) from the center of the nozzle outlet section and positioned by 50 degrees to 150 degrees with an angular step of 10 degrees. Figure 8: (a) 3D acoustic domain; (b) 3D contour plot of turbulent kinetic energy. Figure 7: (a) Radial profiles of axial velocity; (b) Radial profiles of \(uv\) correlation. The profiles have been staggered according to their axial location. (Dotted black lines: experimental results; continuous red line: \(K\)-\(\varepsilon\) model; dashed green line: \(K\)-\(\omega\) SST model; dotted-dashed blue line: LES). The acoustic results are expressed in terms of the PSD (Power Spectral Density) and OASPL (Over-All Sound Pressure Level). To simplify the comparison with experimental data, the PSD spectra were filtered in third-octave bands. To represent all on the same graph, the PSD spectra have been staggered by multiplying the amplitude by a factor \(10^{2n}\), where \(n=\left(\frac{\theta-20}{40}\right)\) and \(\theta\) being the angle from the jet axis. SNGR method does not take account for either convective effects or refractions by the shear layer. Therefore, SNGR results have been corrected, as suggested by [PERSON] [15], to account for convective effects by using a Doppler factor of \(1/(1-M_{c}cos\theta)^{\alpha}\), where \(\alpha\) is a proper exponential coefficient and \(M_{c}\) is the convective Mach number. Furthermore, in a first approximation, mean flow refraction effects have been neglected. Figures 10 and 11 depict acoustic results for both arcs in terms of PSD spectra and OASPL levels showing that SNGR results are in very good agreement with the experimental results and LES analyses, with a matching within 2 dB except at the smaller angles where the maximum discrepancy is of about 5 dB (Figure 10b). In particular, the SNGR method allows predicting the trend of PSD spectra over a large part of the frequency range and up to \(St=1.5\div 2\). Instead, the LES method allows a good prediction only up to \(St=1\); after that, the PSD spectra decay abruptly because of sub-grid filter that does not resolve the smallest scales, i.e., at the high frequencies. There are not great differences between the two models of turbulence, \(K\)-\(\epsilon\) and \(K\)-\(\upomega\) SST. Figure 10: (**a**) Power spectra of far-field pressure signal for a few observer locations on the inner arc, \(30D_{j}\). The Power Spectral Density (PSD) spectra have been staggered by multiplying the amplitude by a factor \(10^{2n}\), where \(n=\left(\frac{\theta-20}{40}\right)\) and \(\theta\) being the angle from the jet axis; (**b**) Overall Sound Pressure Level Directivity. Figure 9: Position of the microphones (Adapted from [PERSON] [7]). In conclusion, the SNGR model, appropriately corrected by the Doppler factor, allows predicting the acoustic field of a subsonic jet both in terms of PSD spectra and in terms of directivity.