# Noise Reduction Using Synthetic Microjet Excitation in Supersonic Rectangular Jets
## 2 Methodology
### Computational Setup
The core of the computational domain extends up to 64\(D\) downstream, 3\(D\) upstream, and 20\(D\) radially. Surrounding this core, a buffer zone is implemented to dissipate fluctuations before they reach the domain boundaries, extending up to 125\(D\) downstream, 16\(D\) upstream, and 40\(D\) radially. The grid spacing close to the nozzle walls is selected to achieve a \(y^{+}\) value of 50 on both coarse and fine grids. The recommended values from the literature range between 30 and 100. This is because the wall models for \(\omega\) in the \(k-\omega\) SST model do not perform well when they are either too close to or too distant from the wall [37]. The value for \(y^{+}\) is calculated based on flow results from the DES simulation itself. Figure 2 offers a general view of the fine mesh structure, while Figure 3 zooms in on the nozzle exit area. The cell size varies the grid spacing from \(D/65\) at the nozzle wall to \(D/25\) at 32\(D\) away along the jet axis to resolve the jet potential core and turbulent mixing, thus utilizing a growth ratio of 1.01. It then gradually increases to \(D/6.475\) to capture the jet decay and growth of large-scale structures until it reaches a distance of 64\(D\) downstream. The computational domain's outer region extends from the nozzle exit to 12\(D\) in the radial direction and to 64\(D\) downstream, and maintains a maximum grid spacing of \(\Delta=D/6.475\). This grid has a buffer zone from 64\(D\) to 125\(D\) downstream of the nozzle. This is necessary to dissipate the fluctuations before they reach the outflow boundaries, and thus ensures that fewer numerical reflections bounce back into the domain to adversely affect the predicted results. The FWH method is used to capture the far-field acoustics [21]. The external surface of the outer region serves as the FWH surface to capture the flow information for acoustic calculations (see Figure 4). The maximum grid spacing ensures a grid cut-off frequency with the consideration of \(N=20\) grid points per wave chosen based on the second-order central upwind scheme adopted in this simulation. The cut-off frequency is estimated using the line-of-sight (LOS) method [38]. Instead of temporal sampling, the maximum resolved frequency is limited by the local grid spacing on the acoustic data surfaces for the FWH method. The LOS method assumes that the noise source is at the end of the potential core, and the grid cut-off is determined by the largest grid spacing where the LOS intersects the FWH surface. The grid spacings in every direction are considered. The cut-off frequency is calculated using Equation (3), where \(\Delta\) is the maximum value between \(\Delta_{1}\), \(\Delta_{2}\) and \(\Delta_{3}\) defined in Equation (4):  
\[St_{max}=\frac{1}{N\Delta M_{d}}\sqrt{\frac{T_{\infty}}{T_{j}}} \tag{3}\]  
\[\Delta_{1}=\frac{\Delta x}{D}\cos\phi\quad\Delta_{2}=\frac{\Delta y}{D}\cos \phi\quad\Delta_{3}=\frac{\Delta z}{D}\sin\phi, \tag{4}\]where \(M_{d}\) is the jet exit Mach number, \(T_{j}\) is the jet exit static temperature, and \(T_{\infty}\) is the freestream temperature. The plane wave propagation angle (\(\phi\)) is between the jet axis and the observer. The worst-case scenario occurs when the observer is positioned at either \(\phi=0^{\circ}\) or \(\phi=90^{\circ}\), where the grid cut-off frequency is \(St_{\max}=0.30\). In contrast, the cut-off frequency reaches its peak value of \(St_{\max}=0.42\) when the observer is at \(\phi=45^{\circ}\). The end cap of the FWH surface might create spurious signals as strong eddies pass through it [39]. For reducing spurious low-frequency noise generated by vorticity traveling near the caps of the FWH surface, it is recommended to carefully configure both the extent and number of end caps used for the permeable surface. A recent study by [PERSON] et al. [40] has shown that the distance between the first and last end cap, or the \"end cap region\", should be sufficiently large to allow for the substantial decorrelation of coherent flow structures as they pass through the caps.