Detached Shocks

Let us return to the problem of supersonic flow, of Mach number ${\rm Ma}_1$ , over a symmetric wedge of nose angle $2\,\theta$ , that was previously discussed in Section 15.3. What happens when the wedge angle, $\theta$ , is greater than $\theta_{\rm max}$ ?

In fact, there is no rigorous analytical treatment for cases where $\theta$ exceeds $\theta_{\rm max}$ . Experimentally, it is observed that the flow configurations are like those sketched in Figure 15.9(a). The flow is compressed through a curved shock front, known as a bow shock, that is detached from, and stands some distance upstream of, the wedge apex. The shape of the bow shock, as well as its detachment distance, depend on the geometry of the wedge, as well as the upstream Mach number, ${\rm Ma}_1$ .

On the central streamline, where the shock is normal, as well as on the nearby ones, where it is nearly normal, the flow is compressed to subsonic speeds. Farther out, as the shock becomes weaker, its inclination becomes less steep, approaching the upstream Mach angle asymptotically. Thus, conditions along the bow shock run the whole range of oblique shock solutions for the given Mach number. This is illustrated in Figure 15.9(b), which shows one curve from Figure 15.2. The complete description of the bow shock is complicated by the fact that its location cannot be found explicitly, because of the subsonic influence upstream, which implies that the entire flow field must be solved as one. There is an absolute limit of $\theta$ , just greater than $45^\circ$ , beyond which the shock will always detach, no matter how large the value of the upstream Mach number, ${\rm Ma}_1$ . Hence, if the wedge is replaced by a blunt-nosed obstacle then the shock will always detach (because $\theta=90^\circ$ at the nose of such an obstacle).

For a given wedge angle, $\theta$ , the sequence of events with decreasing upstream Mach number, ${\rm Ma}_1$ , is as follows. When ${\rm Ma}_1$ is sufficiently high, the shock front is attached to the wedge apex, and its straight portion is independent of both the shoulder and the afterbody. [See Figure 15.9(a).] The straight portion of the shock front lies between the wedge apex and the point where the first Mach line, emanating from the shoulder, intersects it. As ${\rm Ma}_1$ decreases, the wave angle, $\beta$ , increases. With a further decrease in the Mach number, a value is reached at which the fluid behind the shock becomes subsonic. The shoulder now has an effect on the whole shock front, which may become curved, although still remaining attached to the wedge apex. These conditions correspond to the region between the lines ${\rm Ma}_2=1$ and $\theta =\theta _{\rm max}$ in Figure 15.2. At the Mach number corresponding to $\theta =\theta _{\rm max}$ , the shock front starts to detach from the apex. This Mach number is called the detachment Mach number. With a further decrease of ${\rm Ma}_1$ , the detached shock moves upstream of the wedge apex.