Compressible Flow Assignment 3 — Normal Shock Waves I  
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
Starting from the conservation of mass, momentum, and energy in quasi-1D (including the effect of heat transfer and viscous effects): $$ \frac{\rm d}{{\rm d} x}(\rho A v)=0 $$ $$ \rho v \frac{{\rm d} v} {{\rm d} x} + \frac{{\rm d} P} {{\rm d} x} = \frac{{\rm d} } {{\rm d} x}\left(\mu \frac{{\rm d} v} {{\rm d} x} \right) $$ $$ \frac{\rm d}{{\rm d} x} \left( \rho A v \left(h+ \frac{1}{2}v^2\right) \right)-\frac{\rm d}{{\rm d} x} \left( k A \frac{{\rm d} T}{{\rm d} x} \right)=\frac{\rm d}{{\rm d} x} \left( v \mu A \frac{{\rm d} v}{{\rm d} x} \right) $$ Do the following:
(a)  Show that the following equation is correct $$ \frac{1+ \frac{\gamma-1}{2}M_x^2}{1+ \frac{\gamma-1}{2}M_y^2} = \frac{\left(M_x + \frac{1}{\gamma M_x} \right)^2}{\left(M_y + \frac{1}{\gamma M_y} \right)^2} $$ where $M_x$ and $M_y$ are the Mach number upstream and downstream of the shock, respectively.
(b)  Show that the Mach number downstream of a normal shock corresponds to: $$ M_y^2 =\left(M_x^2 + \frac{2}{\gamma-1} \right) \left(\frac{2\gamma}{\gamma-1} M_x^2 -1 \right)^{-1} $$ where $M_x$ and $M_y$ are the Mach number upstream and downstream of the shock, respectively.
Question #2
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 $70$ 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 #3
A certain pitot tube is used for measuring the Mach number $M_\infty$ of a supersonic air stream, and the pressures shown in the sketch are recorded.
pitot.png ./download/file.php?id=1678&sid=b01d19cb2859304ac8ee711ebecaeaa8 ./download/file.php?id=1678&t=1&sid=b01d19cb2859304ac8ee711ebecaeaa8
Estimate the free-stream Mach number.
Question #4
A flow is referred to as “hypersonic” when the Mach number is 4 or above. In order to achieve hypersonic level Mach numbers, a shock tunnel is usually used, as illustrated below. Initially, the pressure and temperature are 0.05 atm and 293 K respectively, and are uniform throughout the whole assembly. A normal shock ($M_1=4.5$) travels towards the nozzle section. If $A_1 \gg A_2$, the nozzle section is effectively like a solid wall, from which the incident shock reflects. The idea of the shock 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.
hypersonictunnel.png ./download/file.php?id=1679&sid=b01d19cb2859304ac8ee711ebecaeaa8 ./download/file.php?id=1679&t=1&sid=b01d19cb2859304ac8ee711ebecaeaa8
(a)  Calculate the pressure behind the reflected shock assuming normal reflection. What are the resulting stagnation pressure and temperature?
(b)  Find the area ratio $A_3/A_2$ required to achieve parallel isentropic flow throughout (i.e. no shocks or underexpanded jets at exit). What is the calculated flow velocity in the test section? A quick estimate from part (a) yields about 8 atm and about 3000 K for the stagnation pressure and temperature, respectively.
(c)  By inserting a razor blade parallel to the flow in the test section, the angle of the Mach wave (with respect to the flow direction) is measured to be 18$^\circ$. What is the efficiency of the hypersonic nozzle? Recall the expression we derived previously for the nozzle efficiency: $$ \eta_{\rm nozzle}=\left( \frac{2}{(\gamma-1)M_{\rm e}^2}+1 \right)^{-1} \left(1-\left(\frac{P_{\rm e}}{P_\circ} \right)^\frac{\gamma-1}{\gamma} \right)^{-1} $$
Question #5
Consider a pipe in which air at 27$^\circ$C and 0.5 bar flows uniformly with a speed of 150 m/s. The end of the pipe is suddenly closed by a valve, and a shock wave propagates back into the pipe. Compute the speed of the wave with respect to the pipe and the pressure and temperature of the air which has been brought to rest.
Question #6
A blast wave (shock wave) has been initiated by an atomic bomb explosion. The wave travels through the air at sea-level conditions with a speed of 60000 m/s. Estimate the change in pressure, temperature, and flow velocity produced by the wave with respect to an observer who is stationary with respect to the undisturbed air.
Question #7
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
10.14.17
Answers
2.  31 N.
4.  (a) $T_\circ=2859$ K, $P=P_\circ=7.43$ atm; (b) $M=4.0$, $A_3/A_2=10.5$; (c) $M_{\rm actual}=3.236$, $\eta=0.89$.
5.  297 m/s, 0.89 bar, 356 K.
7.  $1.24$, $264.6~{\rm kPa}$, $4.504~{\rm kg/m^3}$.
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