5.5 Shock Reflection at a Solid Boundary

What happens when an oblique shock reaches a solid wall? To sort this out we will analyze the schematic flow situation illustrated in Figure Figure 5.16. The figure shows a supersonic flow through a channel where there is a sudden bend of the lower wall leading to the generation of an oblique shock in order to deflect the flow such that it follows the wall downstream of the corner. The shock angle $\beta_a$ is a function of the upstream Mach number $M_1$ and the flow deflection $\theta$. A bit further downstream the shock will reach the upper wall of the channel. The question now is what will happen at the point reaches the upper wall. As indicated in the figure the shock will deflect but in what way will it deflect. Will it be a specular reflection, i.e. will the angle of the shock be the same but in the other direction? To answer this question let’s sort out why the shock will reflect. After the first shock, the flow will be deflected the angle $\theta$, which means that, at the upper wall, the flow must be deflected again such that it follows the direction of the upper wall. Hence, the deflection angle will again be $\theta$ but in the opposite direction. So, the deflection angle is the same, does that mean that the shock angle will be the same? The answer is no, but why? Passing the first shock, the Mach number is reduced and thus the $\theta$-$\beta$-Mach relation will give us another shock angle $\beta_2$ for the second shock. Hence, the shock is not reflected specularly since $\beta_1\ne\beta_2$.

The situation discussed in the previous paragraph assumed that the flow deflection at the upper wall was less than the maximum flow deflection possible for the Mach number ahead of the reflection (downstream of the first shock). If, however, the flow deflection that must take place exceeds the maximum possible flow deflection angle, it will not be possible to generate an oblique shock that fulfills the requirements. Instead, a normal shock will be generated at the upper wall (a so-called Mach reflection) that will be gradually converted into an oblique shock. This situation is depicted in Figure Figure 5.17. The slip line indicated in the figure is a consequence of the fact that the flow the goes through the shock system experiences different entropy increases depending on if the flow passes a single stronger shock that generate higher losses or a set of weaker oblique shocks that will generate less losses.