2017 Compressible Flow Final Exam  
Compressible Flow
Final Examination
Tuesday December 19th, 2017
16:30 — 19:30


ANSWER ALL 6 PROBLEMS; ALL PROBLEMS HAVE EQUAL VALUE; NO NOTES OR BOOKS; TAKE $\gamma=1.4$ IN ALL CASES; USE COMPRESSIBLE FLOW TABLES THAT WERE DISTRIBUTED.
11.30.17
Question #1
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.
Question #2
Consider a continuous supersonic wind tunnel as illustrated below:
question04.png ./download/file.php?id=3975&sid=c39e3b4bb7b40153fd51c6d4e9ad68c2 ./download/file.php?id=3975&t=1&sid=c39e3b4bb7b40153fd51c6d4e9ad68c2
The test section has a cross-sectional area of 5 $\rm m^2$, and the wind tunnel should be designed such that the pressure in the test section is 0.1 atm and the Mach number in the test section is $2.5$. Perform the following tasks:
(a)  For a fixed-geometry nozzle and a fixed-geometry diffuser, find the nozzle throat area ($A_2$) and the diffuser throat area ($A_5$), and sketch the Mach number distribution between stations 1 and 6.
(b)  For a fixed-geometry nozzle and a variable-geometry diffuser, find the nozzle throat area and the minimum and maximum diffuser throat area. As well, sketch the Mach number and pressure distribution between stations 1 and 6 when the wind tunnel operates at maximum efficiency.
(c)  What role does the compressor play for a fixed-geometry diffuser? What role does the compressor play for a variable geometry diffuser?
Question #3
A two-dimensional supersonic diffuser is to be designed as shown below for a Mach number of $2.5$. The ratio $h_2/h_1$ is to be chosen so that the diffuser will barely swallow the initial shock, and the ratio $l/h_1$ is to be selected so as to obtain the wave pattern shown.
diffuser.png ./download/file.php?id=3974&sid=c39e3b4bb7b40153fd51c6d4e9ad68c2 ./download/file.php?id=3974&t=1&sid=c39e3b4bb7b40153fd51c6d4e9ad68c2
(a)  Determine $h_2/h_1$ and $l/h_1$
(b)  Neglecting friction compare the overall stagnation-pressure ratio of this diffuser with the stagnation pressure ratio of a diffuser in which a normal shock occurs at Mach number $2.5$.
Question #4
Consider a supersonic flow with $P_\infty=1$ atm, $T_\infty=300$ K, and $M_\infty=2$ interacting with an airfoil as follows:
Q4.png
Do the following :
(a)  Derive an expression for the chord function of $L$.
(b)  Derive an expression for the coefficient of lift $C_L$ function of $P/P_\infty$ in each region surrounding the airfoil.
(c)  Using linearized theory, find $\theta_{\rm defl}$ and $P/P_\infty$ in each region surrounding the airfoil.
(d)  Calculate the coefficient of lift based on what you obtained in (a), (b), and (c).
Question #5
Consider air at $M_\infty=8$, $P_\infty=10000$ Pa, and $T_\infty=250$ K entering a 3-oblique-shock inlet followed by a long duct in which friction takes place, as follows:
Q5.png ./download/file.php?id=3970&sid=c39e3b4bb7b40153fd51c6d4e9ad68c2 ./download/file.php?id=3970&t=1&sid=c39e3b4bb7b40153fd51c6d4e9ad68c2
For optimal performance, the three oblique shocks have the same strength (Oswatich condition), and are such that $T_3=1000$ K. Knowing that the friction factor in the duct is $f=0.01$, that the length of the duct is of $L=1$ m, and that $H_1=1$ m, do the following:
(a)  Find $P_3$, the pressure after the three oblique shocks.
(b)  Find $M_3$, the Mach number after the three oblique shocks.
(c)  Find $H_2$, the height of the duct following the 3 shocks.
(d)  Find $M_5$, the Mach number at the exit of the duct.
Hint: the effective diameter of the duct can be taken as $D=2 H_2$.

Question #6
Consider the SR-71 Blackbird at takeoff:
sr71.jpg
Based on what you observe in the picture, and knowing that the area ratio between the nozzle exit and throat is 2, estimate as accurately as possible the range of the flow pressure at the nozzle exit. You can assume that $\gamma=1.4$ everywhere.
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