Compressible Flow Assignment 6 — Oblique Shock Waves  
Instructions
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
05.21.14
Question #1
Starting from the equations of motion governing a normal shock: $$ {\rm d}\left(\rho v\right)=0$$ $$ {\rm d}\left(\rho v^2 +P\right)=0$$ $$ {\rm d}\left(h+\frac{1}{2}v^2\right)=0$$ Show that the following relationship is correct for an oblique shock: $$ \frac{{\rm tan}(\sigma)}{{\rm tan}(\sigma-\delta)}= \frac{(\gamma+1)M_1^2{\rm sin}^2\sigma}{2+(\gamma-1)M_1^2 {\rm sin}^2\sigma} $$ where $\delta$ is the flow turning angle, where $\sigma$ is the angle between the oblique shock and the incoming flow vector, and where the subscripts 1 and 2 denote the properties upstream and downstream of the oblique shock, respectively.
Question #2
Air flows in the following passage with an initial Mach number of $2$:
diffuser2.png ./download/file.php?id=1709&sid=da11f407df106af2b104683fd6d4df02 ./download/file.php?id=1709&t=1&sid=da11f407df106af2b104683fd6d4df02
Determine the maximum turning angle $\delta$ for which three regular reflections of the original shock are possible.
Question #3
Consider the following supersonic intake diffuser, in which air is decelerated by two weak oblique shock waves followed by a normal shock wave:
intake.png ./download/file.php?id=1710&sid=da11f407df106af2b104683fd6d4df02 ./download/file.php?id=1710&t=1&sid=da11f407df106af2b104683fd6d4df02
The effective flow area at the inlet is 0.1 $\rm m^2$ and the inlet area at the normal shock is 0.12 $\rm m^2$. The flight Mach number is $2.8$ and the ambient conditions are 70 kPa and 200 K. Determine the Mach number downstream of the normal shock, the stagnation pressure loss and the mass flux through the diffuser.
Question #4
A two-dimensional, spike-type supersonic intake is operating in the supercritical mode (i.e. normal shock behind the inlet cowl):
spikeintake.png ./download/file.php?id=1711&sid=da11f407df106af2b104683fd6d4df02 ./download/file.php?id=1711&t=1&sid=da11f407df106af2b104683fd6d4df02
In operational flight configuration, as illustrated above, the flight Mach number is 2.0 and the ambient air conditions are 0.2 bar and 200 K. The cross-sectional area at the cowl of the inlet is $A_1=1$ $\rm m^2$. The normal shock is at the 1.2 m$^2$ position. The half angle of the inlet spike is $10^\circ$. Determine the mass flow and the loss of stagnation pressure through the engine inlet.
Question #5
The wedge-shaped probe shown below may be used for estimating the Mach number of a supersonic wind tunnel.
wedge3.png ./download/file.php?id=1712&sid=da11f407df106af2b104683fd6d4df02 ./download/file.php?id=1712&t=1&sid=da11f407df106af2b104683fd6d4df02
(a)  In a particular instance, a wedge with a $10^\circ$ included angle is used. Schlieren photographs show that the included angle of the shock front is $98^\circ$. Estimate the Mach number of the tunnel.
(b)  What is the minimum Mach number for which a $10^\circ$ probe can be used in this way as a Mach number indicator?
Question #6
Consider the following triple-shock configuration:
tripleshock.png ./download/file.php?id=1713&sid=da11f407df106af2b104683fd6d4df02 ./download/file.php?id=1713&t=1&sid=da11f407df106af2b104683fd6d4df02
Find the values of $M^\star$ and the flow direction $\theta$ in fields 2, 3, and 4 for $M_1^\star=1.50$ and $\theta_1=0$.
(a)  $\theta_2=+4^\circ$ (Ans. $M_2^\star=1.421$, $M_3^\star=0.759$, $M_4^\star=0.679$, $\theta_3=\theta_4=-6^\circ$)
(b)  $\theta_2=+12^\circ$ (Ans. $M_2^\star=1.231$, $M_3^\star=0.823$, $M_4^\star=0.696$, $\theta_3=\theta_4=+9.2^\circ$)
Sketch the shock directions to scale.
Answers
2.  $9.7^\circ$.
3.  0.5344, 0.56, 193 kg/s.
4.  256 kg/s, 0.748.
5.  1.47, 1.24.
11.22.17
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