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Appendix: Oblique Shock Stability Analysis

**Figure:** Typical examples of unstable and stable situations.
$\begin{figure}\centerline{\includegraphics {cont/oblique/ball}} \end{figure}$

The stability analysis is an analysis which answers the question of what happens if for some reason, the situation moves away from the expected solution. If the answer turns out to be that the situation will return to its original state then it is referred to as the stable situation. On the other hand, if the answer is negative, then the situation is referred to as unstable. An example to this situation, is a ball shown in the Figure (13.22). Instinctively, the stable and unstable can be recognized. There is also the situation where the ball is between the stable and unstable situations when the ball is on a plane field which is referred to as the neutrally stable. In the same manner, the analysis for the oblique shock wave is carried out. The only difference is that here, there are more than one parameter that can be changed, for example, the shock angle, deflection angle, and upstream Mach number. In this example only the weak solution is explained. The similar analysis can be applied to strong shock. Yet, in that analysis it has to be remembered that when the flow becomes subsonic the equation changes from hyperbolic to an elliptic equation. This change complicates the explanation and is omitted in this section. Of course, in the analysis the strong shock results in an elliptic solution (or region) as opposed to a hyperbolic in weak shock. As results, the discussion is more complicated but similar analysis can be applied to the strong shock.

**Figure 13.23:** The schematic of stability analysis for oblique shock.
$\begin{figure}\centerline{\includegraphics {cont/oblique/stablityShock}} \end{figure}$

The change in the inclination angle results in a different upstream Mach number and a different pressure. On the other hand, to maintain the same direction stream lines, the virtual change in the deflection angle has to be in the opposite direction of the change of the shock angle. The change is determined from the solution provided before or from the approximation (13.62).

$\displaystyle \Delta \theta = {k + 1 \over 2} \Delta \delta$

(13.66)

Equation (13.66) can be applied for either positive, $\Delta \theta ^{+}$ or negative $\Delta \theta ^{-}$ values. The pressure difference at the wall becomes a negative increment which tends to pull the shock angle to the opposite direction. The opposite happens when the deflection increment becomes negative, the deflection angle becomes positive which increases the pressure at the wall. Thus, the weak shock is stable.

Please note that this analysis doesn't apply to the case of the close proximity of the $\delta = 0$ . In fact, the shock wave is unstable according to this analysis to one direction but stable to the other direction. Yet, it must be pointed out that it doesn't mean that the flow is unstable but rather that the model is incorrect. There isn't any known experimental evidence to show that flow is unstable for $\delta = 0$ .

Next: Prandtl-Meyer Function Up: Oblique Shock Previous: Summary Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21