## 4 Results and Discussion
### Experimental Result from Schlieren Method  
The gas flow visualized during the Schlieren experiment using a high-speed camera reveals the subtle unsteadiness in the Mach shock wave generation during the high-pressure nozzle flow. A stable Schlieren image was chosen among a series of images captured during the experiment and compared with the density gradient map obtained from the simulation results which is shown in Figure 11. The similarities between the Schlieren experiment result and the simulation results for the \(k\)-\(\varepsilon\) (realizable) and \(k\)-\(\omega\) turbulence models can be seen during comparison. However, the results from RNG turbulent model shows higher fluctuation in the flow which was not visible in the experimental result whereas the standard \(k\)-\(\varepsilon\) turbulent model was not capable of showing all the expansions and contraction in flows due to the shock wave as seen in the Schlieren experiment result. Figure 12 shows the velocity map at the stand-off distance predicted by four turbulent models after the refinement of the model using mesh adaption. It can be seen that the standard \(k\)-\(\epsilon\) turbulent model and the \(k\)-\(\epsilon\) (realizable) turbulent model were not able to predict the low-velocity spot behind the Mach shock disk where usually the flow velocity can reach almost zero. The image from the Schlieren experiment result was analyzed with an image processing technique, using the image processing toolbox in MATLAB. Figure 13 shows the image processing algorithm applied to the Schlieren image for image analysis in this study. Figure 11: Density gradient from CFD simulation and Schlieren image. (**a**) \(k\)-\(\epsilon\) (realizable) turbulence model, (**b**) \(k\)-\(\epsilon\) (standard) turbulence model, (**c**) \(k\)-\(\epsilon\) (RNG) turbulence model, (**d**) \(k\)-\(\omega\) turbulence model, and (**e**) Schlieren experiment result. Figure 12: Velocity map at the stand-off distance predicted by numerical simulation after mesh adaption compared with Schlieren image. (**a**) \(k\)-\(\epsilon\) (realizable) turbulence model, (**b**) \(k\)-\(\epsilon\) (standard) turbulence model, (**c**) \(k\)-\(\epsilon\) (RNG) turbulence model, (**d**) \(k\)-\(\omega\) turbulence model, and (**e**) Schlieren experiment result. The outliers were removed to extract the region of interest (ROI) from the images obtained from the Schlieren experiment. Figure 14a,b shows the extracted ROI images obtained before and after gas injection during the Schlieren experiment, respectively. Gas movement data can be obtained from the differences between these images, by subtracting the former one from the latter [23]. Figure 14c shows the subtracted image. However, the output image from the subtraction operation has some noise. A morphological operation was applied to this image to suppress unwanted noise, and the gas flow image was obtained as shown in Figure 14d. Figure 15 shows a side-by-side comparison of the images of the flow pattern predicted by the numerical simulation in the mesh model after mesh adaption using the \(k\)-\(\omega\) turbulence model; the image obtained from the Schlieren experiment; and the result obtained after image processing, respectively. Similarities found in these images are marked by the dotted regions in Figure 15. In the dotted region 1, when the gas is emitted from the nozzle, it shows an under-expansion in flow. In the dotted region 2, it can be seen that a stagnation region has formed at the point where the under-expanded gas comes into contact with the top surface at the end of the cut kerf slot. The flow is then deflected along the top surface. In the dotted region 3, the Mach shock disk pattern inside the kerf can be seen. The comparison of these three regions in images from the Schlieren experiments and those obtained from numerical simulations show a good agreement in terms of flow structure. Figure 14: Sequence of image processing.