# Noise Reduction Using Synthetic Microjet Excitation in Supersonic Rectangular Jets
## 2 Methodology
### Computational Setup  
The computational setup is similar to that employed in our previous numerical simulations [33]. The previous computations are now repeated on a finer grid to achieve more accurate results. A density-based compressible OpenFOAM v1912 solver [35] using the total-variation-diminishing (TVD) scheme is employed to simulate the flow field of a supersonic, fully expanded heated jet. The current computational study adopts the \(k-\omega\) Shear Stress Transport (SST) turbulence model to conduct DES. This reduces the computational costs compared to the full LES, which would require extensive near-wall treatment. In the current simulations, a statistically steady-state solution is first achieved with the \(k-\omega\) SST RANS model. Then, DES simulations are marched in time using the second-order backward time scheme and the second-order central upwind scheme in space, with the RANS results applied as an initial solution. The initial RANS solution is obtained using the OpenFOAM pseudo-transient first-order accurate time marching scheme \"localEuler\" based on the Euler implicit time marching scheme, with the cell-based time step set by the Local Time Stepping (LTS) solver. To analyze the simulation data, 3072 samples are collected with a sampling rate of 200 kHz (\(dt=5\times 10^{-6}\) s). The minimum resolved frequency from Equation (1) is calculated based on the maximum simulation time of approximately 593 \"flow times\", with the \"flow time\" \(\tau\) defined as the physical time \(t\) non-dimensionalized by the jet exit velocity (\(U_{j}\)) and the characteristic length scale \(D\), so that \(\tau=t\frac{U_{j}}{D}\). This translates to a simulation physical time of 0.01024 s. Considering the utilization of the second-order backward time scheme, a resolution of approximately \(N=12\) points per cycle should suffice, ensuring that the minimum resolvable frequency corresponds to \(St=0.0135\). If the signal is divided into windowed blocks, the minimum resolvable frequency increases as the number of windows increases because the time duration of each window shortens. It is desired to utilize three windows of 1024 samples; thus, the minimum resolvable frequency corresponds to \(St=0.0404\). \[St_{min}=\frac{ND}{\tau U_{j}} \tag{1}\]  
The inlet conditions are prescribed for the desired \(NPR\) and \(NTR\). The advective far-field condition is imposed at the rest of the domain boundaries. This boundary condition, also called \"waveTransmissive\", is implemented using the Navier-Stokes Characteristic Boundary Condition (NSCBC) [36]. The non-reflective boundary condition is based on the characteristic wave relationship derived from the Euler equations and reformulated in the orthogonal coordinate system such that one of the coordinates is normal to the boundary [36]. Finally, the nozzle inner walls are prescribed as an adiabatic no-slip condition so that RANS simulations near the wall can resolve the boundary layer with a specified \(y^{+}\) value. Figure 1: Nozzle geometry. Units in meters. It is important to note that the mean-flow velocity at the exit is \(U_{j}=750\) m/s and the characteristic length \(D=0.01295\) m. The Reynolds number is defined by Equation (2) and matches the one from the experiments [2,3]. \[Re=\frac{\rho U_{j}D}{\mu}=1.73\times 10^{5} \tag{2}\]  
The current computational study utilizes a coarse and a fine grid of 43 and 112 million cells, respectively. These are composed predominantly of hexahedral cells. The main reason for this is that it offers superior accuracy when solving acoustic waves that travel away from the sources. By using hexahedral cells, one can ensure that the code does not lose accuracy and provides precise results even when dealing with complex acoustic wave phenomena. Additionally, it offers better mesh quality and a higher aspect ratio, which results in a better representation of the geometry and fluid flow physics. Therefore, it is essential for obtaining accurate and reliable simulation results.