2010 Compressible Flow Final Exam (Including Solutions)  
December 11th 2010
15:00 — 18:30
Question #1
A converging-diverging nozzle of throat to exit area ratio of 0.695 is connected to an insulated duct of 10 cm diameter and 1.5 m long. The duct is known to have an average friction coefficient of the order of $f=0.005$. The nozzle is supplied with air at a pressure of 10 bar and at a temperature of $20^\circ$C. On a relative scale the nozzle can be essentially considered to be frictionless and the flow adiabatic. The back pressure in the dump tank to which the duct exhausts is of the order of 1 bar. Sketch out the Mach number distribution as a function of distance from the nozzle exit to the duct exit (i.e., along the duct).
Question #2
A two-dimensional supersonic diffuser is to be designed as shown below for a Mach number of 2.7. 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.
question2.png  ./download/file.php?id=1883&sid=eb2391488f410fa7305ff74f611cc9f9  ./download/file.php?id=1883&t=1&sid=eb2391488f410fa7305ff74f611cc9f9
(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 2.7.
Question #3
A supersonic stream leaves a nozzle in parallel flow (region “a”) with a Mach number of 2.2 and a pressure of 0.67 bar:
question3.png  ./download/file.php?id=1884&sid=eb2391488f410fa7305ff74f611cc9f9  ./download/file.php?id=1884&t=1&sid=eb2391488f410fa7305ff74f611cc9f9
The pressure of the atmosphere into which the jet discharges is 1 bar.
(a)  Calculate the pressures in regions “b” and “c”.
(b)  Make a sketch to scale showing stream lines and shock lines.
(c)  Assuming the pressure at the nozzle entrance to be maintained constant, what is the maximum atmospheric pressure for which this general type of flow configuration is possible? Describe the nature of the flow pattern when the exhaust-region pressure is raised above the limiting value.
Question #4
After obtaining a Ph.D., 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:
scramjetinlet-scaled.png  ./download/file.php?id=1990&sid=eb2391488f410fa7305ff74f611cc9f9  ./download/file.php?id=1990&t=1&sid=eb2391488f410fa7305ff74f611cc9f9
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 8.0 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.
Question #5
Consider the following problem:
question5.png  ./download/file.php?id=1887&sid=eb2391488f410fa7305ff74f611cc9f9  ./download/file.php?id=1887&t=1&sid=eb2391488f410fa7305ff74f611cc9f9
A symmetric, truncated wedge supersonic aerofoil has included tip angles of $10^\circ$ as shown above. The aerofoil is placed in a flow of $M_\infty=1.8$ and $P_\infty=0.1$ bar at an angle of attack of $\alpha=6^\circ$. Calculate the lift and drag coefficients for the aerofoil using exact shock-expansion theory.
Question #6
For the aerofoil problem outlined in Question 5, calculate the lift and drag coefficient using first order linearized theory. Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
1.  $P_4=3.66~{\rm bar}$, $M=1.8$, $1.44$, $0.72$, $1.0$.
2.  $A_y/A^\star=1.34$, $l/h_1=1.8$, $P_4^\circ/P_1^\circ=0.559$, $P_y^\circ/P_x^\circ=0.4236$.
3.  $P_{\rm b}=1~{\rm bar}$, $P_{\rm c}=1.43~{\rm bar}$, $P_{\rm back}=1.6~{\rm bar}$.
4.  $M_3=3$, $P_3/P_1=57.74$, $P^\circ_{\rm loss}=78.1\%$, $\delta=13.13^\circ$, $\theta=7.4^\circ$, $x_{\rm A}=2.94h$, $y_{\rm A}=h$, $x_{\rm B}=3.518h$, $y_{\rm B}=0.821h$
5.  $C_{\rm L}=0.254$, $C_{\rm D}=0.0478$.
6.  $C_{\rm L}=0.276$, $C_{\rm D}=0.056$.
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