Page 281 - 48Fundamentals of Compressible Fluid Mechanics

13.6. APPENDIX: OBLIQUE SHOCK STABILITY ANALYSIS 243

original state then it 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 13.21. Instinctively, the stable and unsta-
ble can be recognized. There is also the situation where the ball is between the
stable and unstable situations when the ball is on plan field which referred 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 changed, for example, the shock angle, deflection angle, 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 remember that when
the flow became subsonic the equation change from hyperbolic to elliptic equation.
This change complicates the explanation and omitted in this section. Of course,
in the analysis the strong shock results in elliptic solution (or region) as oppose
to hyperbolic in weak shock. As results, the discussion is more complicated but
similar analysis can be applied to the strong shock.
The change in
∆θ +
the inclination angel
results in a different ∆θ −
upstream Mach num-
∆δ −
ber and a different
pressure. On the
other hand, to maintain ∆δ +
same direction stream
lines the virtual change
in the deflection angle Fig. 13.22: The schematic of stability analysis for oblique shock
has to be opposite di-
rection of the change of shock angle. The change is determined from the solution
provided before or from the approximation (13.51).

(13.57)

c
values.

The pressure difference at the wall becomes negative increment which tends to
or negative
Equation (13.57) can be applied either to positive,
pull the shock angle to opposite direction. The opposite when the deflection incre-
ment became negative the deflection angle becomes positive which increase the
pressure at the wall. Thus, the weak shock is stable.

Please note this analysis doesn’t applied to the case in the close proximity
. In fact, the shock wave is unstable according to this analysis to one
direction but stable to the other direction. Yet, it must be point out that doesn’t
of the c
mean that flow is unstable but rather that the model are incorrect. There isn’t
.

known experimental evidence showing that flow is unstable for c