Aerospace Propulsion Assignment 8 — Underexpanded and Overexpanded Nozzles  
Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Solve the problems starting from the Aerospace Propulsion Tables. Do not use any other document to solve the problems.
05.21.14
Question #1
A two-dimensional planar jet leaves a converging-diverging nozzle in parallel flow and discharges into the atmosphere at ground level where the pressure is $1.013 \times 10^5$ $\rm N/m^2$:
nozzle1.png ./download/file.php?id=10877 ./download/file.php?id=10877&t=1
The area ratio between the nozzle exit plane and the nozzle throat 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 #2
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=10880 ./download/file.php?id=10880&t=1
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 #3
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 stagnation pressure at the nozzle entrance 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
Question #4
Consider a parallel two-dimensional jet leaving a nozzle at “e” with $M_{\rm e}=2.2$:
nozzle3.png ./download/file.php?id=11231 ./download/file.php?id=11231&t=1
(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.
06.14.23
Question #5
The flow entering a 2-D planar converging-diverging nozzle has a stagnation pressure of 800 kPa and a stagnation temperature of 3000 K. The exit-to-throat area ratio is 2.005. Once the flow exiting the nozzle reaches steady state, a Schlieren photograph is taken:
nozzle.png
Determine quantitatively the range of back pressure for which the shock configuration, illustrated in the Schlieren photograph above, will be observed. The flow within the nozzle can be assumed essentially frictionless.
Answers
1.  $1.7^\circ$, $1.61^\circ$, $98.7^\circ$.
2.  1 bar, 1.5 bar, 1.3 bar, 1 bar, 1.328 bar.
3.  $0.026 — 0.082$ m$^2$.
Due on Tuesday November 10th at 23:59. Do problems 1, 2 and 3 only.
11.05.26
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