2011 Compressible Flow Midterm Exam  
May 7th, 2011
14:00 — 16:10
Question #1
In order to provide thrust-vector control for a space vehicle, air at a stagnation pressure of 2.7 MPa and a stagnation temperature of 295 K is expanded through a nozzle. The ambient surrounding pressure is $50$ kPa and the flow rate is 0.05 kg/s. Determine the throat and exit areas of the nozzle resulting in maximal thrust, as well as the thrust developed.
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
A flow is referred to as “hypersonic” when the Mach number is 4 or above. In order to achieve hypersonic level Mach numbers, a gun tunnel is commonly used, as illustrated below.
question03.png  ./download/file.php?id=1993&sid=f6dce1b69881a226c3e34f6c5235a356  ./download/file.php?id=1993&t=1&sid=f6dce1b69881a226c3e34f6c5235a356
Initially, the pressure and temperature are 1 atm and 300 K respectively, 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 #4
Consider a continuous supersonic wind tunnel as illustrated below:
question04.png  ./download/file.php?id=1994&sid=f6dce1b69881a226c3e34f6c5235a356  ./download/file.php?id=1994&t=1&sid=f6dce1b69881a226c3e34f6c5235a356
The test section has a cross-sectional area of 5 $\rm m^2$, and the wind tunnel should be designed such that the pressure in the test section is 0.1 atm and the Mach number in the test section is 2.5. Perform the following tasks:
(a)  For a fixed-geometry nozzle and a fixed-geometry diffuser, find the nozzle throat area ($A_2$) and the diffuser throat area ($A_5$), and sketch the Mach number distribution between stations 1 and 6.
(b)  For a fixed-geometry nozzle and a variable-geometry diffuser, find the nozzle throat area and the minimum and maximum diffuser throat area. As well, sketch the Mach number and pressure distribution between stations 1 and 6 when the wind tunnel operates at maximum efficiency.
(c)  What role does the compressor play for a fixed-geometry diffuser? What role does the compressor play for a variable geometry diffuser?
1.  $42~{\rm mm^2}$; $7.7~{\rm mm^2}$; $31.7~{\rm N}$;
2.  $38.7\%$; $30.7\%$;
3.  $53.2$; $1804{\rm K}$; $1194~{\rm m/s}$; $13.64~{\rm atm}$;
4.  $1.89~{\rm m^2}$; $3.79~{\rm m^2}$; $1.89~{\rm m^2}$; $0.9~{\rm atm}$; $M=1.0$, $2.5$, $2.2$; $P=0.9$, $0.1$, $0.9~{\rm atm}$;
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