2012 Compressible Flow Midterm Exam  
Compressible Fluid Flow
Midterm Quiz
May 11th, 2012

Question #1
A variable geometry, 2D supersonic inlet diffuser is designed to operate at $M_\infty=2.8$. What is the inlet to throat area ratio? In the take-off sequence, the shock is swallowed at $M_\infty=1.9$ by enlarging the throat area. Subsequently, due to hydraulic line failure the variable geometry device jams at this shock swallowing configuration. What is the inlet to throat area ratio? By playing with the fuel throttle, the pilot tries to achieve the best possible engine operating conditions under the circumstances it still being necessary to go to $M_\infty=2.8$. Describe how he would do this and plot the Mach number distribution within the diffuser.
You are working for a satellite company. Your first design project consists of optimizing the rocket nozzle used for altitude control on board the satellite. Your idea consists of replacing the converging-diverging “De Laval” nozzle by a simple converging nozzle. Determine how much thrust would be lost in doing so, and on what flow and geometric parameters this depends (find the ratio between the thrust of the De Laval nozzle and the thrust of the converging-only nozzle and simplify the expression as much as possible). For a fair comparison, assume that the stagnation properties and the mass flow rate are the same for both configurations. Also assume that the flow exiting the De Laval nozzle reaches terminal velocities.
Question #3
Consider that you are faced with designing the following variable geometry supersonic wind tunnel:

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The nozzle should be designed such that the test section area $A_2=A_3=1$ m$^2$, the pressure at the beginning of the test section is $P_2=0.1$ atm, and the temperature at the beginning of the test section is $T_2=250$ K. Assuming that the flow within the nozzle is frictionless, determine the nozzle throat area $A_1$, as well as the stagnation pressure $P_\circ$ and stagnation temperature $T_\circ$ upstream of the nozzle that will yield these desired flow conditions at the entrance of the test section. After starting the wind tunnel, you measure the pitot and static pressure at stations 2 and 3 and find the following: $$ \begin{array}{ll} (P_{\rm pitot})_2=0.564~{\rm atm} & (P_{\rm pitot})_3=0.569~{\rm atm}\\ (P_{\rm static})_2=0.1~{\rm atm} & (P_{\rm static})_3=0.135~{\rm atm}\\ \end{array} $$ Taking the latter data into consideration, find the minimum diffuser throat area $A_4$ necessary to start the wind tunnel. Also find the diffuser throat area $A_4$ that will yield optimal performance once the wind tunnel has been started. As well, provide the best explanation of why the measured static and pitot pressures at station 3 are different from those at station 2.
Question #4
A calorically and thermally perfect gas flows through a stationary normal shock and experiences a drop in velocity from 450 to 150 m/s. If the pressure and the density upstream of the shock are 62 kPa and 1.5 kg/m$^3$, calculate:
(a)  The specific heat ratio $\gamma$
(b)  The pressure and density downstream of the shock
1.  $3.5$; $1.193$; $2.61$; $0.50$;
2.  $F_{\rm b}/F_{\rm a}=\gamma/{\sqrt{(\gamma+1)(\gamma-1)}}$
3.  $0.782~{\rm atm}$; $450{\rm K}$; $0.593~{\rm m^2}$; $\gt 0.873~{\rm m^2}$; $0.747~{\rm m^2}$;
4.  $1.24$, $264.6~{\rm kPa}$, $4.504~{\rm kg/m^3}$.
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