2008 Compressible Flow Final Exam  
December 16th 2008
15:00 — 18:00
 
 
ANSWER ALL 6 PROBLEMS; ALL PROBLEMS HAVE EQUAL VALUE; NO NOTES OR BOOKS; TAKE $\gamma=1.4$ IN ALL CASES; USE GAS TABLES THAT WERE DISTRIBUTED.
05.01.14
Question #1
A 2-D converging-diverging nozzle is driven from a reservoir at 800 kPa and 300 K. The exit-to-throat area ratio is 2.005. Once the flow exiting the nozzle reaches steady state, a Schlieren photograph is taken:
nozzle.png
Determine quantitatively the range of back pressure for which the shock configuration, illustrated in the Schlieren photograph above, will be observed. The flow within the nozzle can be assumed essentially frictionless.
Question #2
Consider an essentially frictionless converging-diverging nozzle connected to a pipe in which friction effects take place:
pipe.png  ./download/file.php?id=1347&sid=036585c28da2b10b4e0e2a324f0753e5  ./download/file.php?id=1347&t=1&sid=036585c28da2b10b4e0e2a324f0753e5
Then, what is the Mach number distribution along the pipe if the nozzle exit Mach number is $M_{\rm e}=1.65$ and the discharge tank pressure is 220 kN/m$^2$?
Question #3
A two-dimensional intake diffuser for an air-breathing engine for supersonic aircraft consists of a wedge forebody leading into a cowl, as shown in the following sketch:
inlet.png  ./download/file.php?id=1348&sid=036585c28da2b10b4e0e2a324f0753e5  ./download/file.php?id=1348&t=1&sid=036585c28da2b10b4e0e2a324f0753e5
The purpose of the forebody is to generate an oblique shock (or shock waves) for external compression. This is followed by a series of internal oblique shock waves and a single normal shock wave to complete the compression to subsonic levels compatible with engine combustor requirements. The external portion of the compression process can be done in a one-step deflection or multi-step deflections as shown in the (a) and (b) versions of the sketch. For $M_\infty=3$, determine quantitatively the relative merits (compression ratio, stagnation pressure loss, Mach number reduction) of each type of configuration for the external portion of the compression process only.
Question #4
As shown below, a cambered supersonic aerofoil is simulated by an articulated flat plate where the articulated deflection is $4^\circ$:
thinairfoil2.png  ./download/file.php?id=1349&sid=036585c28da2b10b4e0e2a324f0753e5  ./download/file.php?id=1349&t=1&sid=036585c28da2b10b4e0e2a324f0753e5
If the angle of attack for the aerofoil is $\alpha=2^\circ$, determine the lift and drag forces for the aerofoil per unit span using exact shock-expansion theory.
Question #5
For the airfoil in the previous problem, calculate the lift and drag forces per unit span using first order linearized theory. Recall that linearized theory yields a pressure coefficient of: $$ {C_P}_{f,g}= \mp \frac{2 \theta_{\rm defl}}{\sqrt{M_\infty^2-1}} $$
Question #6
After graduation, you are working at a petro-chemical plant. Your office is in a building, a few hundred meters away but facing process equipment that handles highly volatile materials (i.e. hydrogen, ethylene, propane, etc). During the course of the day a pressure vessel ruptures and releases some hot volatile gases which then mix with air to form an explosive cloud of significant size ($R_{\rm init}\sim 5$ m), which, as sometimes happens, then detonates. You are located 200 m away and just happen to be looking out the window facing the explosion when the latter occurs. As the detonation transmits from the cloud at $R_{\rm init}\approx 5$ m into the surrounding air, the initial shock strength characterized by the dimensionless shock over-pressure $\Delta P / P_x = (P_y-P_x)/P_x$ corresponds to a shock Mach number $M_s \simeq 7$.

What will be the damage to your window and office building caused by the explosion?

Note that the spherical blast (shock wave) strength will decay roughly as the inverse cube of the distance, i.e.
$$ \frac{\Delta P}{P_x} \propto \frac{1}{R^3} $$ and that the window and building will “feel” the reflected shock pressure.
Damage data
$\Delta P / P_x \approx 0.001$ breaks windows
$\Delta P / P_x \approx 0.01$ causes minor structural damage
$\Delta P / P_x \approx 0.1$ causes major structural damage
$\Delta P / P_x \approx 1$ is total destruction (rubble)
Answers
1.  $179.5~{\rm kPa}$, $314.2~{\rm kPa}$.
2.  $252.2~{\rm kPa}$.
3a.  $P_2/P_\infty=3.37$, $P_2^\circ/P^\circ_\infty=0.84$, $M_2=2.1$.
3b.  $M_2=2.18$, $P_2/P_\infty=3.46$, $P_2^\circ/P^\circ_\infty=0.979$.
4.  $F_{\rm L}=3.89 L~{\rm kN/m^2}$, $F_{\rm D}=0.252 L~{\rm kN/m^2}$.
5.  $F_{\rm L}=3.94 L~{\rm kN/m^2}$, $F_{\rm D}=0.256 L~{\rm kN/m^2}$.
6.  $\Delta P/P_x=0.00175$.
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