# Noise Reduction Using Synthetic Microjet Excitation in Supersonic Rectangular Jets
## 3 Results
### Time-Averaged Flow  
Figure 8 presents a comparison of the non-dimensional time-averaged velocity at the center line and upper lip line for the baseline case against previous LES simulations under the same operating conditions as is [PERSON] et al. [45] using the jet noise-reduction (JENRE) code developed at the U.S. Naval Research Laboratory. The LES grid utilized  
Figure 7: Nozzle lip pressure spectra in minor and major planes.  
65 million points. The results show agreement at the center line up to 15\(D\) from the nozzle exit (see Figure 7(a)). However, the fine grid improves the prediction when considering amplitude at the lip line (see Figure 7(b),c).  
As shown in Figure 7(a), the current simulation predicts a potential core length of 9.5\(D\) for the rectangular jet. Very weak diamond shock cells are visible through the potential core in the center line plot, and Figure 7(b) reveals that remnants of the throat shock are visible in the minor axis plane lip-line plot. However, the major plane, with its straight walls, displays only the lip shock. The near-field shock cell locations, size, and strength are similar for the rectangular planes, and Figure 9 serves as a complementary comparison to the previous analysis as it provides contour plots of the time-averaged streamwise velocity at different stations. This figure displays how the jet plume develops further downstream, changing its transverse section shape from rectangular to near-circular. The excitation increases the jet spread along the minor and major axes, resulting in cross-shaped sections.  
Figure 10 compares the jet decay, a ratio between the jet exit and centerline velocities, and the jet half-width, a ratio between the jet exit length and the length where the jet plume reaches half of the centerline velocity at the specific station. It is evident that the excitation shortens the jet potential core length and increases the momentum thickness further downstream. Also, the jet spread is higher for the excited case. The jet decay lines are also a good indicator of whether the simulation reaches convergence to initiate acoustic data collection. These lines must be as straight as possible from the end of the jet potential core to the end of the grid or until the buffer zone is reached. A commonly used term to describe this condition is the flow-through time. It represents the average time taken for the flow to travel from the nozzle exit to the end of the computational domain, and can be determined using the grid spacing along the jet axis and the mean flow velocity  
Figure 8: Axial velocity distributions [45].  
Figure 9: Distribution of Streamwise Mean Velocity Downstream.  
at the centerline. The flow-through time is calculated as \(T_{FT}=0.0048\) s based on the given jet decay. Therefore, the DES simulation of the jet should run for approximately 10 flow-through times before analyzing the acoustic data or time-averaged quantities. In this simulation, 8 flow-through times are completed before the acoustic data collection is conducted.  
Figure 11 displays the contours of the time-averaged turbulence kinetic energy (TKE) normalized by the jet exit velocity for every case. This quantity is calculated from the first Reynolds stress tensor component \(u_{\infty}^{\prime 2}\). The square root of this component is known as RMS velocity fluctuation, and isnormalized by the jet exit velocity \(U_{j}\). The excitation modifies the development of the jet plume by increasing the turbulent mixing. The TKE peak occurs at the lip line, close to the nozzle exit.