2010 Compressible Flow Midterm Exam (Including Solutions)
November 6th 2010
9:40 — 12:00

NO NOTES OR BOOKS; USE COMPRESSIBLE FLOW TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
 05.23.14
 Question #1
Design a continuous-flow supersonic wind tunnel to simulate the conditions $M=2.6$, $P=30$ kPa and $T=230$ K with the best possible performance. The test section must have a cross-section area of 180 cm$^2$ and is followed by a fixed-geometry diffuser. Specifically, find the diffuser and nozzle throat areas and the mass flux through the system necessary to start and operate the system. At operating conditions, sketch the Mach number distribution along the wind tunnel from the nozzle inlet to just past the diffuser throat.
 Question #2
A supersonic aircraft is equipped with a two-dimensional, converging-diverging, variable throat area, intake diffuser. The diffuser is designed for a cruise Mach number of $2.0$. What percent increase in throat area is required to “swallow” the shock? If in the takeoff sequence the aircraft has to “loiter” at Mach 1.8 due to tactical reasons, what percent of mass spill of air occurs, with the rest, of course, passing through the engine, if the diffuser configuration happens to be set for cruise at $M_\infty=2.0$?
 Question #3
Consider the following nozzle which is designed to yield an efficiency essentially of 1.0: Air flowing in the nozzle is driven by a large reservoir in which the pressure is 1.048 atm and the temperature $75^\circ$C. The pressure of the environment (back pressure) can be taken as 1.0 atm. Determine the exit Mach number, the exit pressure and sketch out the Mach number distribution along the length of the nozzle from the entrance to the exit.
 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).
 1. $6.47 ~{\rm kg/s}$; $62.2 ~{\rm cm^2}$; $135.2 ~{\rm cm^2}$; $M= 1.0$, $2.6$, $2.29$, $0.54$. 2. $38.7\%$; $30.7\%$; 3. $1~{\rm atm}$; $0.26$; $0.48$; $0.895~{\rm atm}$; $0.32$; $0.976~{\rm atm}$; 4. $885~{\rm bar}$; $13792~{\rm K}$;      $\pi$