# Influence of Incident Shock on Fuel Mixing in Scramjet
## 4 Results and Discussion
### Jet Penetration  
In the study of fuel mixing, penetration depth stands as a crucial measure. To further investigate the effect of varying isolator lengths on the jet penetration with non-uniform inflow conditions, the penetration depth is calculated using Equation (29). Figure 24: Mass-weighted average pressure in the x-direction for all the cases. Figure 23: Y \(=-0.207\)m pressure gradient distribution (red dashed line represented the leading edge of the cavity). In Figure 25, the black line represents the isoline where the mass fraction of gaseous kerosene is 0.7. It can be observed that in case 5, the separation zone before the jet at the z = 0 plane is minimal. Consequently, the bow shock exhibits a sharper incline when compared with case 4 and case 6, which enhanced fuel plume transfer towards the core flow. In case 4 and case 6, the slanted bow shocks limit the fuel's upward dispersal, directing the areas of higher mass fraction towards the lower wall. Therefore, as depicted in Figure 26, with A representing the leading edge of the cavity and B the trailing edge, part (b) offered a detailed enlargement of the area shown in part (a). It was observed that the penetration depth for both case 4 and case 6 diminished after the injection site. In these cases, the jet's wake merged with the cavity's shear layer. Owing to the recirculating flow, the fuel is conveyed to areas located further away from the lower wall surface. Influenced by the recirculating flow, the fuel is transported to regions further from the lower wall surface. As a result, the penetration depth commenced its ascent at the leading edge of the cavity. In case 5, the boundary layer separation at the leading edge of the cavity induced a separation shock wave. Subsequently, the fuel plume is redirected upon interacting with this shock wave, persisting its upward transport towards the upper wall surface. As a result, the inception of penetration depth's ascent precedes that observed in case 4 and case 6. Interestingly, in case 2, the absence of side injections resulted in a weaker bow shock compression effect, leading to the formation of a less intense separation shock wave at the leading edge of the cavity. The fuel plume, upon passing through this separation shock wave, was also redirected and transported upwards towards the upper wall surface. However, as shown in Figure 25, there were differences in the extent of separation and the intensity of the separation shock wave compared to case 5. Conversely, in the corresponding cases 1 and 4, as well as case 3 and case 6, the interaction of shock waves did not significantly affect the surge in cavity pressure due to the period of the reflected shock; hence, these cases exhibited similar behaviors. Figure 27 illustrates an iso-surface representation with a mass fraction value of 0.0677, where the coloring corresponds to the distance from the lower wall surface. Given the similarity in flow field structures between case 4 and case 6, Figure 27 exclusively shows a comparison of case 4 with case 5 and case 1 with case 2. It can be observed that case 4, characterized by a larger separation zone, results in increased spanwise fuel entrainment due to the interaction between the boundary layers of the side jets and the central jet. This effect enhances the spanwise penetration depth of the jet when compared to case 1. Therefore, from the jet to the leading edge of the cavity, the difference in penetration depth between case 4 and case 1 gradually increased, as in case 3 and case 6. In case 5, the smaller separation zone resulted in a relatively thinner boundary layer, so there was almost no interference between the boundary layers of the side jets and the central jet. This meant  
Figure 25: Shock wave and fuel iso-line for different cases. that before being affected by the separation shock wave at the leading edge of the cavity, the penetration depth in case 2 and case 5 was nearly identical.