Compressible Flow Assignment 7 — Supersonic Airfoils and Nozzles  
Question #1
Consider the following expansive flow around a curved wall:
expansioncorner.png  ./download/file.php?id=1714&sid=1bf966e80d2ad441e83fa2603cf808fd  ./download/file.php?id=1714&t=1&sid=1bf966e80d2ad441e83fa2603cf808fd
Compare the exact theory with the linear theory by plotting curves of $P_2/P_1$ versus $M_1$ for two values of $\delta$, namely $2^\circ$ and $10^\circ$. Use a range of $M_1$ from 1 to 3.
05.21.14
Question #2
A thin flat plate airfoil is immersed in a supersonic stream of $M_\infty=2.2$ at an angle of attack of $10^\circ$. Determine the lift and drag coefficients per unit length of the flat plate airfoil using the linearized theory and compare with the estimates from exact methods. The width of the airfoil (i.e., distance along the chord) is $l$.
Question #3
Consider the following symmetric double wedge profile:
wedge1.png  ./download/file.php?id=1715&sid=1bf966e80d2ad441e83fa2603cf808fd  ./download/file.php?id=1715&t=1&sid=1bf966e80d2ad441e83fa2603cf808fd
The airfoil is located in an airstream with a free-stream Mach number $M_\infty=3$ and a free stream static pressure $P=1.0133\times 10^5$ $\rm N/m^2$. Calculate the lift and drag per unit width of the wedge for an angle of attack $\alpha$ of $-15^\circ$. Take $\gamma=1.4$.
Question #4
A two-dimensional jet leaves a converging-diverging nozzle in parallel flow and discharges into the atmosphere where the pressure is $1.013 \times 10^5$ $\rm N/m^2$:
nozzle1.png  ./download/file.php?id=1716&sid=1bf966e80d2ad441e83fa2603cf808fd  ./download/file.php?id=1716&t=1&sid=1bf966e80d2ad441e83fa2603cf808fd
The area ratio of the nozzle is 2. Calculate the angle $\delta$ in degrees, if
(a)  $P_\circ=12 \times 10^5$ N/m$^2$,
(b)  $P_\circ$ is infinite.
Question #5
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=1717&sid=1bf966e80d2ad441e83fa2603cf808fd  ./download/file.php?id=1717&t=1&sid=1bf966e80d2ad441e83fa2603cf808fd
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 #6
As shown below, a cambered supersonic aerofoil is simulated by an articulated flat plate where the articulated deflections are $2^\circ$ at each step.
thinairfoil2.png  ./download/file.php?id=1718&sid=1bf966e80d2ad441e83fa2603cf808fd  ./download/file.php?id=1718&t=1&sid=1bf966e80d2ad441e83fa2603cf808fd
If the angle of attack for the aerofoil is $\alpha=4^\circ$, determine the lift and drag forces for the aerofoil per unit span
(a)  using exact shock-expansion theory,
(b)  using first order linearized theory.
Question #7
A supersonic stream leaves a nozzle in parallel flow (region “a”) with a Mach number of $2$ and a pressure of 0.67 bar:
exhaust1.png  ./download/file.php?id=1719&sid=1bf966e80d2ad441e83fa2603cf808fd  ./download/file.php?id=1719&t=1&sid=1bf966e80d2ad441e83fa2603cf808fd
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.
(d)  Compare the results of part (a) with the results of calculations based on linear theory.
Question #8
After taking a Schlieren photograph of the flow exiting a nozzle, the following wave pattern is observed:
question05.png
Knowing that the surrounding pressure is of 1 atm, that the pressure in the reservoir driving the nozzle is of 15 atm, and that the nozzle exit area is of $\rm 0.2~m^2$, determine the minimum and maximum nozzle throat area that would yield the wave pattern observed in the Schlieren photo.
11.30.17
Answers
2.  0.362, 0.064, 0.351, 0.0619.
3.  -225 kN/m, 74.6 kN/m, -279 kN/m, 106.5 kN/m.
4.  $1.7^\circ$, $1.61^\circ$, $98.7^\circ$.
5.  $\frac{4m^2}{(2m-1)\sqrt{M_\infty^2-1}} \left( \frac{h}{t}\right)^2$
6.  8462 Pa/m, 809 Pa/m, 8422 Pa/m, 806 Pa/m.
7.  1 bar, 1.5 bar, 1.3 bar, 1 bar, 1.328 bar.
8.  $0.026 — 0.082$ m$^2$.
Due on December 7th at 16:30. Do Questions #3, #4, #6, and #8 only.
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