Compressible Flow Assignment 4 — Normal Shock Waves II
 Instructions
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
 05.08.14
 Question #1
Consider a fixed geometry axi-symmetric supersonic intake diffuser for a high performance aircraft designed for optimal operational conditions of Mach number $M_\infty=2.2$ at 12,000 m altitude:
The intake opening has a cross-sectional diameter of 60 cm and the combustor-turbine requirements are such that the diffuser Mach number at the inlet face of the axial compressor is to be 0.4. During acceleration to its flight envelope from takeoff, the turbojet passes through the shock stand-off regime before the shock wave is “swallowed” by overspeeding. Consider a particular point in this shock stand-off phase namely that corresponding to a flight Mach number of $M_\infty=1.8$. Determine under these conditions the mass spillage around the outside of the diffuser intake and the associated loss of total pressure at the axial compressor face. The ambient conditions correspond to the 12,000 m altitude.
 Question #2
A variable geometry, 2D supersonic inlet diffuser is designed to operate at $M_\infty=2.2$. What is the inlet to throat area ratio? In the take-off sequence, the shock is swallowed at $M_\infty=1.5$ 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 $M_\infty=2.2$. Describe how he would do this and plot the Mach number distribution within the diffuser.
 Question #3
A continuous flow wind tunnel for supersonic flow has a nozzle throat area of 200 $\rm cm^2$ and test section area of 337.5 $\rm cm^2$. The tunnel is driven by air at an initial pressure of 5 atm, $20^\circ$C, and negligible flow velocity at entry. Determine the test section Mach number, the diffuser throat area and diffuser throat Mach number under best operating conditions.
 Question #4
A convergent divergent nozzle has an exit to throat area ratio of 3. The nozzle is driven by air from an initial pressure of 10 atm and initial temperature of $20^\circ$C. If the nozzle exhausts to a pressure of 4 atm, sketch out the Mach number and the pressure distribution along the nozzle.
 Question #5
To obtain hypervelocity flow in a laboratory, a gun tunnel is commonly used, as illustrated below.
Initially, the pressure and temperature are 1 atm and 300 K 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 #6
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.
 10.24.17
 1. 19 kg/s. 2. Stabilize shock downstream of the throat. $M_{\rm throat}=2.1$. 3. $M=2.0$, $A_{\rm d}=277.4$ cm$^2$, $M_{\rm d}=1.75$. 4. 10 atm, 5.3 atm, 0.5 atm, 3.9 atm, 4.0 atm; 1.0, 2.59, 0.51, 0.47. 5. 53.2, 1804 K, 1196 m/s, 13.6 atm. 6. ${M}_1= 0.55$, ${\rm spill} = 17\%$, ${\rm spill} = 20\%$, ${\rm spill} = 14.6\%$, ${\rm spill} = 10.6\%$, ${M}_{1x}= 2.17$.
 $\pi$