2013 Compressible Flow Midterm Exam  
Compressible Fluid Flow
Midterm Quiz
November 15th, 2013
14:00 — 16:20

Question #1
Starting from the conservation of mass and momentum for a quasi-1D flow: $$ {\rm d} (\rho v A)=0\\[0.5em] \rho v {\rm d} v+ {\rm d} P =0 $$ Show that the force that a fluid exerts on a structure (in the opposite direction of the fluid motion) is: $$ F=\left( \rho v^2 A + PA\right)_2-\left( \rho v^2 A + PA\right)_1 $$
Question #2
To obtain hypervelocity flow in a laboratory, a gun tunnel is commonly used, as illustrated below.
question2.png  ./download/file.php?id=2291&sid=bc13d753abcb2e8cc9991558b0b155c1  ./download/file.php?id=2291&t=1&sid=bc13d753abcb2e8cc9991558b0b155c1
Initially, the pressure and temperature are 1 atm and 300 K and are uniform throughout the whole assembly. A normal shock with the velocity $v_{\rm s}$ travels towards the nozzle. If $A_0 \gg A_1$, the nozzle section is effectively like a solid wall, from which the incident shock reflects. The idea of the gun tunnel is to use the high pressure behind the reflected shock as a reservoir to drive the hypersonic nozzle. After shock reflection, the gas is effectively stationary.
(a)  Determine the throat area as a function of the test area to obtain $M_2=6$.
(b)  Determine the effective nozzle stagnation temperature $T_0$ that is needed to obtain hypersonic flow in the test section at $M_2=6$ and $T_2=220$ K.
(c)  Determine the shock speed in the lab frame $v_{\rm s}$ that yields the stagnation temperature obtained in part (b).
(d)  Determine the static pressure on the left of the piston (denoted as $P_{\rm piston}$ in the figure above) that yields $v_{\rm s}$ obtained in part (c).
Question #3
Consider the following shock tube followed by a nozzle and a test section, just prior to the bursting of the diaphragm:
question3.png  ./download/file.php?id=2292&sid=bc13d753abcb2e8cc9991558b0b155c1  ./download/file.php?id=2292&t=1&sid=bc13d753abcb2e8cc9991558b0b155c1
After the diaphram bursts, a shock forms and reflects off the nozzle converging section. Knowing that the area ratio between the test section and the throat is $A_2/A_1=25$, do the following:
(a)  Find the resulting Mach number, pressure, and temperature in the test section
(b)  Determine as accurately as possible the maximum testing time that can be obtained with such a shock tube
Question #4
After obtaining your Masters degree from Pusan National University, you are hired by the Korea Aerospace Research Institute in Daejeon in the satellite control department. The first task given to you is to estimate the importance of the stagnation enthalpy on the thrust of a small rocket used for altitude control. To save weight, it is decided not to use a converging-diverging de Laval nozzle but rather to limit the nozzle to a converging duct with the flow exiting from the throat directly to a vacuum. Find an expression for the thrust of the rocket, simplify it as much as possible, and show that it depends only on the mass flow rate, the ratio of specific heats $\gamma$, and the stagnation enthalpy ($h_\circ=c_p T_\circ$).
2.  53.2, 1804 K, 1194 m/s, 13.64 atm.
3.  5, 0.23 atm, 423 K, 3.8 ms.
4.  $F=\dot{m}\sqrt{h_\circ}\sqrt{2(\gamma-1)(\gamma+1)}/\gamma$.
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