2009 Compressible Flow Final Exam
December 12th 2009
15:00 — 18:00

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.
 05.23.14
 Question #1
Consider the following problem:
 (a) Estimate the maximum flow rate of air through the passage shown assuming that the friction coefficient of the duct is 0.005. (b) For what range of back pressures will this maximum flow rate be achieved?
 Question #2
A two-dimensional supersonic diffuser is to be designed as shown below for a Mach number of 3.0. 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.
 (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 3.0.
 Question #3
Consider the following symmetric double wedge profile:
The airfoil is located in an airstream with a free-stream Mach number $M_\infty=2.8$ and a free stream static pressure $P=1.0133\times 10^5$ N/m$^2$. Using exact shock-expansion theory, calculate the lift and drag per unit width of the wedge for an angle of attack $\alpha=-20^\circ$.
 Question #4
Consider a thin, supersonic airfoil profile expressed by the function $y=-h\left(x/t\right)^{m}$ with $m \ge 1$:
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 #5
A two-dimensional converging-diverging nozzle has an exit-to-throat dimension ratio $H/h=2.5$. The nozzle is supplied by an upstream reservoir at 1000 kPa and 293 K. The nozzle exhausts into a dump tank in such a way that the lower segment of the dump tank provides a continuing straight wall for the nozzle bottom beyond the exit for some distance, as shown in the following sketch:
A pitot tube mounted in the nozzle exit plane records, in a particular test, a reading of about 522 kPa. The dump tank pressure for this particular test is 116 kPa. Calculate and sketch out the Mach number and pressure distributions along the nozzle and downstream of the nozzle exit as far as is calculable. Sketch out the main features of the flow field.
 Question #6
After obtaining a Ph.D. from Pusan National University, you are hired by the Korean Agency for Defense Development (ADD) in Daejeon. Your first design project consists of developing a suitable mixed-compression inlet for a supersonic combustion ramjet (scramjet) engine. In a mixed-compression inlet, some of the compression process takes place externally and some internally. Since the scramjet engine is intended to power a non-reusable hypersonic cruise missile, the geometrical complexity is desired to be kept to a minimum to keep the manufacturing costs low. For this reason, the compression process consists of only two oblique shocks, as depicted in the figure below:
For an inlet exit temperature fixed to 1066 K, design the inlet such that it yields optimal performance (i.e. minimal stagnation pressure loss) for a flight Mach number of 7.5 at an altitude of 15 km. Specifically, perform the following tasks:
 (a) Find the compression ratio and percent stagnation pressure loss through the inlet (b) Find the angles $\delta$ and $\theta$ (c) Find the Mach number at the inlet exit (d) Find the $x,~y$ coordinates of points A and B as a function of the inlet height $h$
Assume frictionless and constant-$\gamma$ flow.
 Answers
 1. $0.0847~{\rm kg/s}$, $0 — 260~{\rm kPa}$ 2. $h_1/h_2=1.39$, $l/h_1=2.27$, $P_4^\circ/P_1^\circ=0.388$, $P^\circ_y/P^\circ_x=0.328$. 3. $-357~{\rm kN/m^2}$, $166~{\rm kN/m^2}$. 4. $F_{\rm L}/F_{\rm D}=(2m-1)\times(t/h)\times(1/m^2)$. 5. $M=2.44$, $2.05$, $1.69$, $2.04$; $P=64.3$, $116$, $199$, $116~{\rm kPa}$. 6. $M_3=2.73$, $P_3/P_1=57.7$, $P_\circ^{\rm loss}=78.1\%$, $\delta=14^\circ$, $\theta=8.3^\circ$, $x_{\rm A}=2.73 h$, $y_{\rm A}=h$, $x_{\rm B}=3.27 h$, $y_{\rm B}=0.82h$.
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