2012 Compressible Flow Final Exam  
Compressible Fluid Flow
Final Examination
June 16, 2012
16:00 — 19:00


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
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 1.5 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 #2
An isentropic nozzle having an area ratio of 2 discharges air into an insulated pipe of length $L$ and diameter $D$. The nozzle is supplied at $700$ kN/m$^2$ and 20$^\circ$C and the duct discharges into a space where the pressure is 275 kN/m$^2$. Calculate $4fL/D$ of the pipe and the mass flow per unit area in the pipe (kg/s$\,$m$^2$) for the cases where a normal shock stands:
(a)  in the nozzle throat
(b)  in the nozzle exit plane
(c)  in the duct exit plane
Question #3
Consider a thin, supersonic airfoil profile expressed by the function $y=-h\left(x/t\right)^{m}$ with $m \ge 1$:
thinairfoil.png ./download/file.php?id=2288&sid=ee65998348384d977051c50e3b94e12e ./download/file.php?id=2288&t=1&sid=ee65998348384d977051c50e3b94e12e
The leading edge of the profile is tangent to the direction of the oncoming air stream. Using linearized theory,
(a)  Find the expressions for the lift and drag coefficients in terms of $M_\infty$ ,  $h/t$,  and $m$.
(b)  Find an expression for the lift over drag ratio in terms of $h/t$ and $m$. Plot the lift over drag ratio versus $m$ for $t/h=5$ and for $t/h=10$. Consider the range $1 \le m \le 4$.
Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
Question #4
A 2-D converging-diverging nozzle is driven from a reservoir at 1000 kPa and 300 K. The exit-to-throat area ratio is 2.4031. The nozzle discharges into a dump tank which is at a pressure of 120 kPa. Calculate and sketch out the Mach number and pressure distributions inside the nozzle and downstream as far as calculable. Identify the main features of the flow field.
Question #5
Supersonic flow at $M=2.0$ and $P=100$ kPa is deflected by a series of ramps but in the end is bent back to its original direction:
question05.png
Is there any loss of stagnation pressure in doing this? If so, quantify (i.e. find the stagnation pressure difference between the final redirected situation and the freestream). Calculate the Mach number and the pressure of the flow at each deflection, including the final redirected situation.
Question #6
After completing a PhD at PNU (during which you have published several papers in the leading Journal in aerospace engineering, the AIAA Journal), you are hired by Samsung Techwin in Changwon. Coincidentally, your first project at Samsung Techwin is related to your PhD thesis topic and concerns the design of an inlet for a supersonic gas turbine engine. Specifically, it is desired to find the inlet geometry that would yield the best performance (minimal stagnation pressure loss) with the following configuration:
question06.png ./download/file.php?id=2290&sid=ee65998348384d977051c50e3b94e12e ./download/file.php?id=2290&t=1&sid=ee65998348384d977051c50e3b94e12e
For $M_1=3$ and $H=1$ m, do the following:
(a)  Find the Mach number in zones 2, 3, and 4
(b)  Find the pressure ratio across each shock
(c)  Find the angles $\theta_1$ and $\theta_2$
(d)  Find the $x$-$y$ coordinates of points ${\bf A}$ and ${\bf B}$
Not being sure how to solve this problem, you contact your PhD thesis advisor for some help. He mentions that the optimal performance of a ramjet inlet is attained when each shock within the inlet (either oblique or normal) has the same pressure ratio. That is: $$ \frac{P_4}{P_3}= \frac{P_3}{P_2}= \frac{P_2}{P_1} $$ He further mentions that when the pressure ratio across each shock is the same, so is the temperature ratio: $$ \frac{T_4}{T_3}= \frac{T_3}{T_2}= \frac{T_2}{T_1} $$ and that the temperature ratio across the last shock is a function of the Mach number in zone 3 according the normal shock relationship: $$ \frac{T_4}{T_3}=\left(\frac{2\gamma}{\gamma+1}M_3^2 - \frac{\gamma-1}{\gamma+1} \right)\left(\frac{\gamma-1}{\gamma+1}+\frac{2}{(\gamma+1)M_3^2} \right) $$ Your PhD advisor goes on saying that given the latter, and knowing that the stagnation temperature remains constant throughout the inlet, it is possible to find $M_2$, $M_3$, and $M_4$. He thinks (but is not sure) that this may require an iterative procedure, and that you should be careful in performing enough iterations so that the temperature ratio is determined with enough accuracy (at least 3 significant digits).
Answers
1.  $\rm Enlarge~throat~by~38.7\%$, $\rm mass~spill~of~35.2\%$
2.  $812.6~{\rm kg/(m^2 s)}$, $5.07$, $0.539$, $812.6~{\rm kg/(m^2 s)}$, $0.225$, $812.6~{\rm kg/(m^2 s)}$
3.  $F_{\rm L}/F_{\rm D}=(2m-1)\times(t/hm^2).$
4.  $M=1.0$, $2.4$, $2.06$, $1.77$, $2.08$; $P=523$, $68.4$, $120$, $194$, $120~{\rm kPa}$
5.  $M=2.07$, $2.22$, $2.47$, $1.98$; $P=89.7$, $70.9$, $48.0$, $100.2~{\rm kPa}$.
6.  $M_3=1.57$, $M_2=2.29$, $M_4=0.68$, $P_4/P_3=2.71$, $\theta_1=14.5^\circ$, $\theta_2=3.5^\circ$, $x_{\rm A}=1.62~{\rm m}$, $y_{\rm A}=1~{\rm m}$, $x_{\rm B}=2.41~{\rm m}$, $y_{\rm B}=0.61~{\rm m}$.
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