2009 Compressible Flow Midterm Exam  
November 7th 2009
16:00 — 18:20
 
 
NO NOTES OR BOOKS; USE GAS TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
05.23.14
Question #1
A ramjet is to fly at 20,000 m altitude with a speed of $2000$ km/hr.
(a)  Design the best fixed-geometry, convergent-divergent diffuser for this aircraft, and compute for it the least percent loss in stagnation pressure.
(b)  Suppose that it were possible to overspeed the aircraft to $2600$ km/hr. Design the best convergent-divergent diffuser which could then be used, and find for it the least percent loss in stagnation pressure at the operating speed of $2000$ km/hr.
Question #2
A fixed-geometry, convergent-divergent wind tunnel diffuser is to be designed for Mach number $3$. Assuming no friction, compare the minimum possible percent loss in stagnation pressure during operation for the following cases:
(a)  The best possible design is employed
(b)  The design is conservative, with a throat area 5 percent larger than that required for starting, and with the shock located during operation at an area 5 percent greater than the throat area.
(c)  The converging portion is eliminated, and the process comprises a normal shock followed by reversible subsonic compression.
Question #3
A converging-diverging nozzle has an exit-to-throat area ratio of about 1.6. The nozzle is supplied by a reservoir of air at a pressure of 600 kPa and a temperature of 300 K. A pitot tube has been installed in the exit plane of the nozzle and in some preliminary tests it is recorded to measure a pressure of about 594 kPa. Using this test information, and assuming that the nozzle is essentially frictionless, plot the Mach number and the pressure distributions within the nozzle.
Question #4
A gun tunnel typically consists of a propellant-driven piston which drives a shock ahead of it towards the nozzle. The shock on reflection creates the reservoir of hot, high pressure gas necessary to create high enthalpy supersonic flow in the converging-diverging nozzle typically located at the end of the driver section. The piston and the shock effectively reach their steady terminal velocities by the time the nozzle entry is reached by the shock. The shock can be essentially considered to reflect from a closed end before any significant efflux through the nozzle can occur. Initially, the driver tube is filled with air at 1 bar (100 kPa) and 305 K. If the terminal piston-driven shock velocity is observed to be about 3500 m/s, determine the initial nozzle reservoir conditions (i.e. the conditions created by the first shock reflection at end of driver section).
Answers
1.  $1.189$; $-12\%$; $1.305$; $-7\%$
2.  $-56.2\%$; $-59.7\%$; $-67.2\%$
3.  $M=1.0$, $1.22$, $0.83$, $0.40$; $P=317$, $241$, $378$, $532~{\rm kPa}$;
4.  $885.3~{\rm bar}$; $13792~{\rm K}$;
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