Detached Shock

When the mathematical quantity $D$ becomes positive, for large deflection angle, there isn't a physical solution to an oblique shock. Since the flow ``sees'' the obstacle, the only possible reaction is by a normal shock which occurs at some distance from the body. This shock is referred to as the detach shock. The detached shock's distance from the body is a complex analysis and should be left to graduate class and researchers in this area. Nevertheless, a graph and a general explanation to engineers is provided. Even though this topic has few applications, some might be used in certain situations which the author isn't aware of.

Figure: $\;$ The schematic for a round-tip bullet in a supersonic flow.

Analysis of the detached shock can be carried out by looking at a body with a round section moving in a supersonic flow (the absolute velocity isn't important for this discussion). Figure 13.12 exhibits a round-tip bullet with a detached shock. The distance of the detachment is determined to a large degree by the upstream Mach number. The zone A is zone where the flow must be subsonic because at the body the velocity must be zero (the no-slip condition). In such a case, the gas must go through a shock. While at zone C the flow must be supersonic. The weak oblique shock is predicted to flow around the cone. The flow in zone A has to go through some acceleration to became supersonic flow. The explanation to such a phenomenon is above the level of this book (where is the ``throat'' area question<sup>13.25</sup>. Yet, it can be explained as the subsonic is ``sucked'' into gas in zone C. Regardless of the explanation, these calculations can be summarized by the flowing equation

$${\hbox{detachment distance}\over \hbox{body thickness}} = \hbox{constant} \times \left(\theta - f(M_{\infty}) \right)$$ (13.64)

where $f(M_{\infty})$ is a function of the upstream Mach number which tabulated in the literature.

The constant and the function are different for different geometries. As a general rule, the increase in the upstream Mach results in a decrease of the detachment distance. Larger shock results in a smaller detachment distance, or, alternatively, the flow becomes ``blinder'' to obstacles. Thus, this phenomenon has a larger impact for a relatively smaller supersonic flow.