# 12.2.2.5: Flow in a Semi–2D Shape

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Example 12.2

In Figure 12.9 exhibits wedge in a supersonic flow with unknown Mach number. Examination of the Figure reveals that it is in angle of attack. 1) Calculate the Mach number assuming that the lower and the upper Mach angles are identical and equal to $$\sim 30^\circ$$ each (no angle of attack). 2) Calculate the Mach number and angle of attack assuming that the pressure after the shock for the two oblique shocks is equal. 3) What kind are the shocks exhibits in the image? (strong, weak, unsteady) 4) (Open question) Is there possibility to estimate the air stagnation temperature from the information provided in the image. You can assume that specific heats, $$k$$ is a monotonic increasing function of the temperature.

Solution 12.2

Part (1)
The Mach angle and deflection angle can be obtained from the Figure 12.9. With this data and either using equation (59) or potto-GDC results in

 Oblique Shock Input: $$\theta_w$$ and $$\delta$$ k = 1.4 $$M_1$$ $$M_x$$ $${{M_y}_s}$$ $${{M_y}_w}$$ $$\theta_{s}$$ $$\theta_{w}$$ $$\delta$$ $$\dfrac{{P_0}_y}{{P_0}_x}$$ 2.6810 2.3218 0 2.24 0 30 10 0.97172

The actual Mach number after the shock is then
\begin{align*}
M_2 = \dfrac{{M_2}_n}{\sin\left(\theta-\delta\right)} = \dfrac{0.76617}{\sin(30-10)} = 0.839
\end{align*}
The flow after the shock is subsonic flow.

Part (2)
For the lower part shock angle of $$\sim 28^\circ$$ the results are

 Oblique Shock Input: $$\theta_w$$ and $$\delta$$ k = 1.4 $$M_1$$ $$M_x$$ $${{M_y}_s}$$ $${{M_y}_w}$$ $$\theta_{s}$$ $$\theta_{w}$$ $$\delta$$ $$\dfrac{{P_0}_y}{{P_0}_x}$$ 2.9168 2.5754 0 2.437 0 28 10 0.96549

From the last table, it is clear that Mach number is between the two values of 2.9168 and 2.6810 and the pressure ratio is between 0.96549 and 0.97172. One of procedure to calculate the attack angle is such that pressure has to match by "guessing'' the Mach number between the extreme values.

Part (3)
The shock must be weak shock because the shock angle is less than $$60^\circ$$.