# Noise Reduction Using Synthetic Microjet Excitation in Supersonic Rectangular Jets
## 2 Methodology
### Computational Setup
Generally, maintaining an end cap region size that is about 5% to 10% of the overall permeable surface length in the primary flow direction proves effective [40]. Within this region, using between 5 to 8 end caps for averaging is optimal, helping to smooth out the noise signal without excessive computational cost. The study also showed that the spacing between these end caps has a minimal impact on the effectiveness of noise reduction, as doubling the cap spacing does not significantly change the results. However, it is essential to recognize that strong streamwise vortices may still affect noise levels even with a large end-cap region. In this investigation, five end caps are placed at equal distances from each other, from \(60D\) to \(64D\). Their noise contribution is averaged at each observer to eliminate the low-frequency spurious noise compared to the approach with no end caps. The averaging method reduces the artificial non-zero contribution [39]. The placement of the FW-H surface must be studied further because this surface should be properly placed to account for all (or most) noise sources, avoiding regions with coarse mesh where the solution is not properly resolved. The simulations are run using 896 processors, each operating at 2.3 GHz. Considering a sampling frequency of 200 kHz, 2048 samples would require 120,000 and 180,000 processors for the cases with 43 and 112 million cells, respectively. The small difference between both cases compared to the number of grid cells is due to the CFL number being limited by the ratio between cell length and flow velocity. The finer grid has more points because its growth ratio from the nozzle exit is smaller, which improves the LES simulation accuracy. Still, the cells become larger as they approach the jet centerline, which improves the simulation speed. Figure 2: Mesh structure. Post-processing involves sampling surfaces to collect flow data at every cell in the mid-span cuts in both the X-Y and X-Z planes and on the FWH surface. Once gathered, these data are processed separately using an in-house code, where they are subjected to either a Spectral Proper Orthogonal Decomposition (SPOD) or permeable-surface FW-H integration using the Farassat 1A formulation [41]. The obtained time histories of the unsteady pressure fluctuations at the far-field observer locations are transformed into frequency spectra by applying the Fast Fourier Transform (FFT) method [42]. Upon gathering the FWH surface data, the results at the far-field observer points can be analyzed as required. The nomenclature for the coordinates of these observer locations can be discerned by referring the diagram illustrated in Figure 5.  
For both data processing tools, a single-period sample is divided into three segments, each containing 1024 samples with a Hanning window with 50% overlap. FFT is then applied to each segment, and the resulting spectra are averaged. This procedure may help avoid significant'spectral leakage' and reduce the amplitude of the discontinuities at the boundaries of each finite segment. Although using more window blocks reduces the frequency resolution, this method smooths out the signal and filters it without causing phase distortion, thereby providing a reasonable spectrum estimate. In this research, SPOD enables some valuable insights into the fundamental physical processes of noise generation and improves our understanding of energy transfer between structures of different frequencies. Before carrying out a 2-dimensional SPOD, the DES data must be interpolated onto a structured Cartesian grid of equally stretched cells. This approach does not raise concerns regarding the accuracy of the analysis, as the main emphasis lies in decomposing the coherent structures, which are considerably larger than the maximum cell length of the interpolation grid. Figure 4: FW-H surface (in black) and refinement zones (in white) with maximum spacing specified. Figure 3: Mesh details near the nozzle exit.