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next up previous index
Next: Oblique Shock Examples Up: Detached Shock Previous: Detached Shock   Index

Issues Related to the Maximum Deflection Angle

The issue of maximum deflection has a practical application aside from the obvious configuration used as a typical simple example. In the typical example, a wedge or a cone moves into a still medium or gas flows into it. If the deflection angle exceeds the maximum possible, a detached shock occurs. However, there are configurations in which a detached shock occurs in design and engineers need to take it into consideration. Such configurations seem sometimes at first glance not related to the detached shock issue. Consider, for example, a symmetrical suction section in which the deflection angle is just between the maximum deflection angle and above half of the maximum deflection angle. In this situation, at least two oblique shocks occur and after their interaction is shown in Figure (13.13). No detached shock issues are raised when only the first oblique shock is considered. However, the second oblique shock complicates the situation and the second oblique shock can cause a detached shock. This situation is referred to in the scientific literature as the Mach reflection.

Figure: The schematic for a symmetrical suction section with Mach reflection.
\begin{figure}\centerline{\includegraphics{cont/oblique/LargeSymmetrialAngle}}
\end{figure}
It can be observed that the maximum of the oblique shock for the perfect gas model depends only on the upstream Mach number i.e., for every upstream Mach number there is only one maximum deflection angle.

Figure: The ``detached'' shock in a complicated configuration
\begin{figure}\centerline{\includegraphics {cont/oblique/MachReflection}}
\end{figure}
Additionally, it can be observed for a maximum oblique shock that a constant deflection angle decrease of the Mach number results in an increase of Mach angle (weak shock only) $ M_1 > M_2
\Longrightarrow \theta_1 < \theta_2$ . The Mach number decreases after every shock. Therefore, the maximum deflection angle decreases with a decrease the Mach number. Additionally, due to the symmetry a slip plane angle can be guessed to be parallel to original flow, hence $ \delta_1 = \delta_2$ . Thus, this situation causes the detached shock to appear in the second oblique shock. This detached shock manifested itself in a form of curved shock (see Figure 13.14).

The analysis of this situation is logically very simple, yet the mathematics is somewhat complicated. The maximum deflection angle in this case is, as before, only a function of the upstream Mach number. The calculations for such a case can be carried out by several approaches. It seems that the most straightforward method is the following:

(a)
Calculate $ {M_1}_B$ ;
(b)
Calculate the maximum deflection angle, $ \theta_2$ , utilizing (13.36) equation
(c)
Calculate the deflection angle, $ \delta_2$ utilizing equation (13.12)
(d)
$\textstyle \parbox{0.92\textwidth}{
Use the deflection angle, $\delta_2=\delta_...
...le is achieved in
this shock.
Potto-GDC can be used to calculate this ratio.
}$
This procedure can be extended to calculate the maximum incoming Mach number, $ M_1$ by checking the relationship between the intermediate Mach number to $ M_1$ .

In discussing these issues, one must be aware that there are zones of dual solutions in which sharp shock line coexists with a curved line. In general, this zone increases as Mach number increases. For example, at Mach 5 this zone is $ 8.5^\circ$ . For engineering purposes when the Mach number reaches this value, it can be ignored.


next up previous index
Next: Oblique Shock Examples Up: Detached Shock Previous: Detached Shock   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21