## 4 Results  
### Shadowgraphy Results  
Figure 4 shows the results from the shadowgraphy acquisitions during a 0.8 s pulse. Each frame is an average of 90 raw images (10 ms), and the figure displays one frame every 0.1 s from 0.05 to 0.85 ms after the gas pulse starts. The figure also shows the pressure measured in the showerhead volume upstream of the nozzle during the pulse (top). From the pressure time history, three regimes can be identified during one pulse. First, a transient is visible in the first 0.1 s as the pressure builds up in the showerhead. The pressure is then nearly steady between 0.12 and 0.24 s; this is the quasi-steady phase of the jet. Finally, the jet declines after 0.3 s as the pressure decreases exponentially.  
At the entrance of the straight nozzle, a vena contracta develops, which allows the jet to transition to supersonic. Because the throat is not controlled, this results in an underexpanded supersonic jet, whose topology is recognizable in Figure 7. The three phases of the jet identified from the pressure history are confirmed by the shadowgraphy visualizations. Figure 7 presents additional shadowgraphy frames at the beginning of the pulse with shorter temporal increments. The bottom frames are for time 0.14 to 0.22 s and are nearly identical. There, the upstream pressure is high and the flow is highly underexpanded, leading to a sharp first Mach disk at \(x=6.8\) mm followed by additional Mach disks with reduced intensity and size. Typical features of an underexpanded jet are observed, including barrel shocks and reflected shocks. The pressure in the showerhead peaks at 97 psig (7.7 bar). As the upstream pressure decreases after 0.3 s, the width of the first Mach disk and its distance from the nozzle also decrease. We have developed an intensity-based single-pixel-accuracy algorithm to track the lateral extent (width) and \(X\)-location of the main Mach disk on the shadowgraphy frames. Figure 8 shows the discrete results as grey circles, along with further smoothing using a moving average. The three regimes are distinctly observable in the figure, with the first Mach disk reaching a maximum depth of 6.8 mm and a maximum width of 2.4 mm.  
Figure 6: Mach number in the portion of the simulation domain closest to the nozzle for three different grid sizes.  
Figure 8: Temporal evolution of the lateral extent (width) and \(X\)-location of the main Mach disk. The grey circles are obtained at single-pixel accuracy from the shadowgraphy images and the solid dark line smooths the data using a moving average.  
Figure 7: Time evolution of shadowgraphy results in 10 ms increments from time 0.05 to 0.22 s; each frame is an average of 90 raw images captured during the 10 ms time window. The figure shows the evolution and quasi-steady nature of the Mach disks below the nozzle.  
A comparison with CFD results is shown in Figure 9 for the quasi-steady phase of the jet. In the shadowgraphy images, the pixel intensity changes as the rays from the light source are deflected by local variations of the refractive index through the jet. Two quantities from the CFD results are displayed in the figure: the variable \(f_{S}(x,y)\) that relates to the divergence of ray deflection and the simplified function \(f_{L}\left(x,y\right)\), which consists of the Laplacian of the refractive index (see definitions in Appendix A).  
The CFD result predicts the approximate axial position of the first Mach disk within 0.5 mm, as well as its approximate size. The modeled Mach disk appears to be narrower compared to the experimental one, while the reflected shocks appear to have comparable sizes. The modeled \(f_{S}(x,y)\) quantity in Figure 9 exhibits signs of numerical oscillation that can be addressed in future work by testing advanced numerical schemes such as those in the ROUND family [16; 17].