2018 Compressible Flow Final Exam  
Compressible Flow
Final Examination
Wednesday December 19th 2018
18:00 — 21:00

Question #1
(a)  Defining the nozzle efficiency as: $$ \eta_{\rm nozzle}\equiv \frac{v_{\rm e}^2}{v_{\rm ei}^2}$$ with $v_{\rm e}$ the measured exit velocity, and $v_{\rm ei}$ the exit velocity that would be obtained if the flow is expanded isentopically to the measured exit pressure. Show that for a perfect gas and for adiabatic wall conditions, the nozzle efficiency becomes: $$ \eta_{\rm nozzle}=\left( \frac{2}{(\gamma-1)M_{\rm e}^2}+1 \right)^{-1} \left(1-\left(\frac{P_{\rm e}}{P_\circ} \right)^\frac{\gamma-1}{\gamma} \right)^{-1} $$ with $P_\circ$ the stagnation pressure (in the reservoir driving the nozzle), $M_{\rm e}$ the measured exit Mach number and $P_{\rm e}$ the measured exit pressure.
(b)  From Schlieren photographs of the flow of air at the exit of a converging-diverging insulated nozzle it is observed that the angle of the Mach wave subtended by a disturbing small obstruction is $\alpha=40^\circ$. The measured static pressure at the exit cross section is 0.198 atm, while the pressure in the reservoir which drives the air flow through the nozzle is 1 atm. Determine the nozzle efficiency and the Mach number at the exit. Take $\gamma=1.4$ for air.
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=4630&sid=bf3d479e933291b59ac208fee781b4c3  ./download/file.php?id=4630&t=1&sid=bf3d479e933291b59ac208fee781b4c3
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 #3
It is proposed to measure the chemical composition of the gases approaching the nozzle of a turbojet engine by withdrawing a sample of gas through a capillary tube 1.25 cm long and 0.25 mm in internal diameter. The capillary will be attached to a much larger tube “A”, as shown in the following sketch, and the gas will be pumped to the chemical apparatus by a vacuum pump.
question10.png  ./download/file.php?id=4631&sid=bf3d479e933291b59ac208fee781b4c3  ./download/file.php?id=4631&t=1&sid=bf3d479e933291b59ac208fee781b4c3
One of the practical questions is whether or not the time required to draw a sample of reasonable size is excessive, and it is therefore desired to estimate the maximum rate at which gas can be drawn through the capillary. The stream in which the probe is placed is at 200 $\rm kN/m^2$ and $650^\circ$C and flows at a speed of 300 m/s. For purposes of estimating, it may be assumed that the gas has properties $\gamma=1.4$ and $R=287$ J/kgK. The viscosity of the gas may be taken as $4 \times 10^{-5}$ kg/ms. Since the flow in the capillary is likely to be laminar, the friction coefficient is given by the Poiseuille relationship $f=16/{\rm Re}_D$.
(a)  Estimate the maximum rate at which gas may be sampled in kg/s.
(b)  What is the pressure at the inlet of the capillary for the condition of maximum flow?
Question #4
Consider a shockwave moving toward a sound wave in a duct as follows:
Q4a.png  ./download/file.php?id=4627&sid=bf3d479e933291b59ac208fee781b4c3  ./download/file.php?id=4627&t=1&sid=bf3d479e933291b59ac208fee781b4c3
where $v_{\rm s}$ and $v_{\rm w}$ are the velocities of the shock and the wave with respect to the ground. In zone 1, the gas consists of air at rest with a pressure of 1 atm and a temperature of 300 K. When the shock and the wave reach each other, they reflect as follows:
Q4b.png  ./download/file.php?id=4629&sid=bf3d479e933291b59ac208fee781b4c3  ./download/file.php?id=4629&t=1&sid=bf3d479e933291b59ac208fee781b4c3
Knowing that $v_{\rm s}=1000$ m/s, do the following:
(a)  Find the flow properties (in the ground frame) after the incident shock. I.e., find $P_2$, $T_2$, and $v_2$ in the ground frame. Clearly indicate the flow direction in zone 2.
(b)  Find the flow properties (in the ground frame) after the sound wave. I.e., find $P_3$, $T_3$, and $v_3$ in the ground frame. Clearly indicate the flow direction in zone 3.
(c)  Using the flow properties found in (a) and (b) find the pressure after the reflected waves $P_4$ and $P_5$. Also, find the velocity of the contact surface in the ground frame, $v_{\rm c}$, as well as the velocities of the reflected waves $v_{\rm R1}$ and $v_{\rm R2}$ in the ground frame. Clearly indicate the flow direction in zones 4 and 5.
Hint: a contact surface separates two flow regions of equal pressure and velocity but of different density.

Question #5
Consider the following airfoil:
It is known that $M_\infty>1$ and that $d<c$ with $c$ the chord of the airfoil. It is also known that the bottom surface of the airfoil for $x>d$ is equal to: $$ y=b-a-\sqrt{b^2-x^2}~~~~{\rm for}~x>d $$ with $b$ and $a$ some constants such that $b>a$ and $a>0$. Determine the lift coefficient $C_{\rm L}$ as a function of $a$, $b$, $c$, and $d$ using linearized theory. Recall that the linearized pressure coefficient collapses to: $$ {C_P}_{f,g}=\mp\frac{ 2 \theta_{\rm defl}}{ \sqrt{M_\infty^2-1}} $$
Question #6
Consider the following rocket-powered supersonic flight vehicle:
Q6.png  ./download/file.php?id=4626&sid=bf3d479e933291b59ac208fee781b4c3  ./download/file.php?id=4626&t=1&sid=bf3d479e933291b59ac208fee781b4c3
The freestream conditions correspond to $P_\infty=0.05$ bar, $T_\infty=250$ K, and $M_\infty=2.8$. The following geometric parameters are also known: $\theta=40^\circ$, $L_1=1$ m, $L_2=0.1$ m, $H_{\rm t}=0.034$ m. Knowing that the combustor stagnation temperature $T_\circ=3000$ K and that the specific heat ratio $\gamma$ and the gas constant $R$ can be taken equal to 1.4 and 287 J/kgK everywhere, do the following:
(a)  Find the combustor stagnation pressure $P_\circ$ that will result in a normal shock standing at the nozzle exit.
(b)  Find the velocity difference across the contact surface separating the flow coming out of the nozzle from the air flowing around the external surface of the flight vehicle.
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