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Appendix: Oblique Shock Stability Analysis
Figure:
Typical examples of unstable and stable situations.

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.

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).
Equation (
13.66) can be applied for either
positive,
or negative
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
.
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
.
Next: PrandtlMeyer Function
Up: Oblique Shock
Previous: Summary
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Created by:Genick BarMeir, Ph.D.
On:
20071121