2011 Compressible Flow Final Exam  
June 11th 2011
15:00 — 18:20
 
 
ANSWER ALL 6 PROBLEMS; ALL PROBLEMS HAVE EQUAL VALUE; NO NOTES OR BOOKS; TAKE $ \gamma=1.4$ IN ALL CASES; USE GAS TABLES THAT WERE DISTRIBUTED.
05.24.14
Question #1
Consider that you are faced with designing a two-dimensional supersonic diffuser. The main challenge in designing a supersonic diffuser is to minimize the stagnation pressure losses while achieving a high compression ratio. This can be done by compressing the flow through a series of equal strength oblique shocks preceding a normal shock. To minimize the geometric complexity of the inlet, it is here desired to use only three shocks to decelerate the flow to subsonic conditions:
3shockdiffuser.png
The inlet is designed for a flight Mach number of 2.2 and such that the pressure increases twofold across each oblique shock. Find the $x,~y$ coordinates of points A and B in the inlet depicted above for a diffuser height $h$ fixed to 1.0 m.
Question #2
Consider a parallel two-dimensional jet leaving a nozzle at “e” with $M_{\rm e}=2.2$:
nozzle3.png  ./download/file.php?id=1996&sid=bff5bf97331a5e74809b544f1fb81a6b  ./download/file.php?id=1996&t=1&sid=bff5bf97331a5e74809b544f1fb81a6b
(a)  Plot the deflection angle of the jet, $\delta$, as a function of the pressure ratio $P_{\rm a}/P_{\rm e}$ for values of the latter between unity and the value for which a normal shock wave stands in the exit plane.
(b)  Determine the range of pressure ratios, $P_{\rm a}/P_{\rm e}$, for which regular reflection of the oblique shock wave originating at the nozzle exit is not possible.
Question #3
Consider a thin, curved profile of parabolic shape expressed by the curve $y=-h\left(x/t\right)^2$:
thinairfoil.png  ./download/file.php?id=1997&sid=bff5bf97331a5e74809b544f1fb81a6b  ./download/file.php?id=1997&t=1&sid=bff5bf97331a5e74809b544f1fb81a6b
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 $M_\infty$ and the thickness ratio $h/t$.
(b)  Plot the pressure coefficient for the upper and lower surfaces against percent chord, for $M_\infty=2$ and $h/t=0.10$.
Question #4
A converging-diverging axisymmetric (circular cross-section) nozzle has an exit-to-throat area ratio of about 1.44. The nozzle exit diameter is 10 cm and is connected to a length of pipe $L=3.0$ m of the same diameter (i.e. $D=10$ cm) whose friction factor is known to be about $f\approx 0.005$. The pipe discharges into a dump tank whose pressure is 340 kPa. Calculate and sketch out the Mach number and pressure distribution along the nozzle and pipe, if the nozzle supply conditions are 600 kPa and 293 K. Calculate the mass flux as well.
Question #5
Consider an asymmetric diamond-shaped (double-wedged) supersonic airfoil immersed in a stream of $M_\infty=2.2$ and $P_\infty=100$ kPa:
question5.png  ./download/file.php?id=1998&sid=bff5bf97331a5e74809b544f1fb81a6b  ./download/file.php?id=1998&t=1&sid=bff5bf97331a5e74809b544f1fb81a6b
The airfoil is at an angle of attack of $\alpha=4^\circ$. Using exact shock-expansion theory calculate the lift and drag coefficients for the above airfoil.
Question #6
For the airfoil problem outlined in Question 5, calculate the lift and drag coefficient using linearized theory. Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
Answers
1.  $y_{\rm B}=1~{\rm m}$, $x_{\rm B}=1.256~{\rm m}$, $x_{\rm A}=0.889~{\rm m}$, $y_{\rm A}=0.1988~{\rm m}$
2.  $P_{\rm a}/P_{\rm e} \le 1$ or $P_{\rm a}/P_{\rm e} \ge 2.4$
3.  $C_{\rm L}=4 (h/t) (M_\infty^2 -1)^{-1/2}$, $C_{\rm D}=(16/3)\times (h/t)^2 \times (M_\infty^2-1)^{-1/2}$.
4.  $7.74 ~{\rm kg/s}$.
5.  $0.131$, $0.02$.
6.  $0.141$, $0.022$.
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