2008 Compressible Flow Midterm Exam  
November 1st 2008
15:00 — 17:00
 
 
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.
05.22.14
Question #1
Consider the following nozzle which is designed to yield an efficiency essentially of 1.0:
supersonictunnel2.png  ./download/file.php?id=1747&sid=67334ffe70a7c84130b7536ab6317448  ./download/file.php?id=1747&t=1&sid=67334ffe70a7c84130b7536ab6317448
Air flowing in the nozzle is driven by a large reservoir in which the pressure is 30 bar and the temperature $75^\circ$C. A pitot tube measurement in the exit plane gives a pressure of 23.3 bar. Determine the exit Mach number, the exit pressure and sketch out the Mach number distribution along the length of the nozzle from the throat to the exit.
Question #2
A continuous-running supersonic wind tunnel has the usual configuration of a converging-diverging nozzle, followed by a short, straight test section, then by a converging-diverging diffuser culminating in mechanical recompression beyond the diffuser exit. The wind tunnel is intended to be run at close to “ideal” operating conditions with an intended diffuser exit Mach number of about $M_{\rm e}\approx 0.3$ before mechanical recompression. The wind tunnel segment itself is instrumented in the following way: at the nozzle throat, which is 5 $\rm cm^2$, there is a pressure tap (i.e, a tiny hole perpendicular to the nozzle throat wall and connected to a pressure sensor). The test section itself is equipped with a pitot tube. In a typical operational run at effectively “ideal” operating conditions, the following pressures were recorded:

pressure tap at nozzle throat: 483.4 kPa

pitot tube pressure in test section: 616.9 kPa

Based on these measurements determine:
(a)  the wind tunnel dimensions in terms of nozzle throat, test section, diffuser throat and exit areas
(b)  the pressure and Mach number distributions from the nozzle entrance to diffuser exit (including the pressure and Mach number at the nozzle entrance and diffuser exit)
Question #3
To obtain hypersonic flow in the test section of a wind tunnel, a shock tunnel followed by an hypersonic nozzle is generally used, as shown below.
hypersonictunnel.png  ./download/file.php?id=1748&sid=67334ffe70a7c84130b7536ab6317448  ./download/file.php?id=1748&t=1&sid=67334ffe70a7c84130b7536ab6317448
Initially, air is at a pressure and temperature of 1 atm and 305 K throughout the whole assembly. After the shock tunnel is started, a normal shock is formed and travels towards the nozzle section. The speed of the shock with respect to the laboratory frame is of 1747 m/s. Since $A_1\gg A_2$, the nozzle section effectively acts like a solid wall, from which the incident shock reflects. The idea of the shock 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 essentially stationary. Calculate the pressure and temperature after the reflected shock assuming normal reflection. What are the resulting stagnation pressure and temperature?
Question #4
A fixed geometry converging-diverging intake diffuser is designed for shock-free operation at $M_\infty=1.6$. Determine the minimum flight Mach number $M_\infty$ to first achieve “choking” at the throat in the take-off sequence. Further, calculate the % mass spill at $M_\infty=0.8,$ 1.0, 1.4, and 1.6 (i.e. before the shock is swallowed). Determine the “overspeed” Mach number necessary to “swallow” the shock.
Answers
1.  ${M}_3=0.52$, $P_3=19.4~{\rm bar}$.
2.  ${M}_{\rm N}=1$, ${M}_{\rm T}=2.1$, $A_{\rm T}= 9.18~{\rm cm^2}$, $A_{\rm D}=7.43~{\rm cm^2}$, ${M}_{{\rm D}x}=1.84$, ${M}_{{\rm D}y}= 0.608$, $A_{\rm e}=12.85~{\rm cm^2}$
3.  $T_y= 3626~{\rm K}$, $P_y=192~{\rm atm}$, $P_\circ= 192~{\rm atm}$, $T_\circ= 3626~{\rm K}$.
4.  ${M}_1= 0.55$, ${\rm spill} = 17\%$, ${\rm spill} = 20\%$, ${\rm spill} = 14.6\%$, ${\rm spill} = 10.6\%$, ${M}_{1x}= 2.17$.
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