2013 Compressible Flow Final Exam  
Compressible Fluid Flow
Final Examination
December 20, 2013
14:00 — 17:00


ANSWER ALL 6 PROBLEMS; TOTAL POINTS: 150; NO NOTES OR BOOKS; TAKE $\gamma=1.4$ IN ALL CASES; USE COMPRESSIBLE FLOW TABLES THAT WERE DISTRIBUTED.
12.10.14
Question #1
Starting from the conservation of mass, momentum, and energy in quasi-1D (including the effect of heat transfer within the flow): $$ \frac{\rm d}{{\rm d} x}(\rho A v)=0 $$ $$ \rho v \frac{{\rm d} v} {{\rm d} x} + \frac{{\rm d} P} {{\rm d} x} =0 $$ $$ \frac{\rm d}{{\rm d} x} \left( \rho A v \left(h+ \frac{1}{2}v^2\right) \right)-\frac{\rm d}{{\rm d} x} \left( k A \frac{{\rm d} T}{{\rm d} x} \right)=0 $$ 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
Consider an essentially frictionless converging-diverging nozzle connected to a pipe in which friction effects take place:
pipe.png  ./download/file.php?id=2293&sid=3d7c1f2836a51cc4cebe50d38a77a30d  ./download/file.php?id=2293&t=1&sid=3d7c1f2836a51cc4cebe50d38a77a30d
Then, what is the Mach number distribution along the pipe if the nozzle exit Mach number is $M_{\rm e}=1.65$ and the discharge tank pressure is 220 kN/m$^2$?
Question #3
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:
supersonicturbojet.png  ./download/file.php?id=2294&sid=3d7c1f2836a51cc4cebe50d38a77a30d  ./download/file.php?id=2294&t=1&sid=3d7c1f2836a51cc4cebe50d38a77a30d
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 #4
Consider the following supersonic airfoil profile:
question04.png  ./download/file.php?id=2295&sid=3d7c1f2836a51cc4cebe50d38a77a30d  ./download/file.php?id=2295&t=1&sid=3d7c1f2836a51cc4cebe50d38a77a30d
where the top surface is expressed by the function $y=-h\left(x/t\right)^{m}$ with $m \ge 1$ and where the bottom surface is a straight line joining the leading edge to the trailing edge. Using linearized theory, find the expression for the lift coefficient in terms of $M_\infty$, $h$, $t$, and $m$. Simplify the expression as much as possible. Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
Question #5
After taking a Schlieren photograph of the flow exiting a nozzle, the following wave pattern is observed:
question05.png
Knowing that the surrounding pressure is of 1 atm, that the pressure in the reservoir driving the nozzle is of 15 atm, and that the nozzle exit area is of $\rm 0.2~m^2$, determine the minimum and maximum nozzle throat area that would yield the wave pattern observed in the Schlieren photo.
Question #6
You are assigned the task of designing the inlet of a ramjet, consisting of a Prandtl-Meyer compression fan followed by an oblique shock, as follows:
question06.png  ./download/file.php?id=2297&sid=3d7c1f2836a51cc4cebe50d38a77a30d  ./download/file.php?id=2297&t=1&sid=3d7c1f2836a51cc4cebe50d38a77a30d
The inlet is to be designed such that the air has a Mach number of 2 at the inlet exit (after the oblique shock), and such that the temperature ratio across the oblique shock is equal to the temperature ratio across the compression fan. For $M_\infty=5$, $P_\infty=0.2$ atm, $T_\infty=250$ K, and $H=1$ m, do the following:
(a)  Find the Mach number, pressure, and temperature after the compression fan ahead of the oblique shock
(b)  Find the pressure and temperature after the oblique shock
(c)  Find the angle $\theta$ (the angle between the cowl and the $x$ axis)
(d)  Find the angle $\delta_1$ (the angle between the ramp and the $x$ axis)
(e)  Find the $x$-$y$ coordinates of point ${\bf C}$
Answers
2.  $M=1.65,~1.23,~0.82,~1.0$; $P_{\rm exit}=252~{\rm kPa}$.
3.  $19.6~{\rm kg/s}$, $0.8127$.
4.  $\frac{4h}{t\sqrt{M_\infty^2-1}}$.
5.  $0.026-0.082~{\rm m^2}$.
6.  $834~{\rm K}$, $457~{\rm K}$, $3.37$, $1.65~{\rm atm}$, $8.71~{\rm atm}$, $20.5^\circ$, $-3.5^\circ$, $4.9~{\rm m}$, $1.0~{\rm m}$.
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