2014 Fluid Mechanics Final Exam  
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The final exam will be such that 3 questions out of 6 are taken from assignments #1 to #9 (including the problems that I did not solve in class). One question will be a derivation, and the other two questions some standard assignment problems. Please make sure you know how to solve all the assignment problems, and can remember all the derivations. Don't skip the hardest problems or derivations. I will ask for sure one hard problem or derivation from the assignments.
 
 
12.11.14
Fluid Mechanics
Final Examination
December 13th 2014
14:00 — 17:00
 
 
NO NOTES OR BOOKS; USE FLUID MECHANICS TABLES THAT WERE DISTRIBUTED; ANSWER ALL 6 QUESTIONS; TOTAL POINTS: 150 PTS.

12.24.14
Question #1
Starting from the $x$-component of the momentum equation: $$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 u}{\partial z^2} +B_x $$ and from the mass conservation equation: $$ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial x}(\rho u) + \frac{\partial }{\partial y}(\rho v) + \frac{\partial }{\partial z}(\rho w) = 0 $$ Show that the skin friction coefficient and the thickness of a laminar boundary layer correspond to: $$ C_f=0.647 \cdot {\rm Re}_x^{-0.5} {\rm ~~~~and~~~~} \delta/x=4.64 \cdot {\rm Re}_x^{-0.5} $$ Outline all assumptions.
Question #2
A model wing is tested fully submerged in a water tunnel (no free surface) at a speed of 8 m/s. The wing is rectangular in planform with a span of 1 m and a chord of 0.15 m. The measured drag is 50 N. The temperature of the water is $15^\circ{\rm C}$. Find the span and chord of the prototype wing which moves at $30$ m/s through air at the same incidence under dynamically similar conditions. The air pressure is 85 kPa and the temperature is $-5^\circ$C. Determine also the drag of the prototype. Use the following viscosities and densities: $\mu_{\rm air} = 1.65\times 10^{-5}$ kg/ms; $\mu_{\rm water} = 1.15\times 10^{-3}$ kg/ms; $\rho_{\rm water}$ = $10^3$ kg/m$^3$.
Question #3
A water tap is fed through a 370 m long pipe from a container. The water surface of the container is $21$ m above the ground, and the opening of the tap is located 3.4 m above the ground. The friction factor of the tube is $f=0.020$. Five bends give a pressure loss of $0.8\cdot \frac{1}{2} \rho u_{\rm b}^2$ each. In addition, there are two valves present in the system that each introduce a pressure loss of $1.5\cdot \frac{1}{2} \rho u_{\rm b}^2$. How much time will it take to fill up a 32 m$^3$ cistern?
Question #4
You perform an experiment in which liquid water at a temperature of 27$^\circ$C flows on a flat plate as follows:
figure5.png
The flat plate has a length $L$ of 2 meters, a height $H$ of 2 mm, and a depth $D$ of 1 meter. Knowing that the boundary layer thickness at the trailing edge of the plate is of $\delta=2.76$ cm, estimate as well as possible the total drag force on the plate caused by the flowing water. Please use the following data for liquid water: density $\rho=1000$ kg/m$^3$, viscosity $\mu=10^{-3}$ kg/ms, heat capacity $c=c_p=c_v=4200~$J/kgK, conductivity $k=0.6~$W/mK.
Question #5
Consider a large pump with the exit pipe being $\Delta H=20$ meters higher than the entrance pipe as follows:
figure4.png  ./download/file.php?id=2312&sid=530ff7689a030ad0fda32fe8293fd713  ./download/file.php?id=2312&t=1&sid=530ff7689a030ad0fda32fe8293fd713
Water enters the pump with a mass flow rate of 100 kg/s, a pressure of 1 atm and a temperature of 300 K. It is observed that the water entering the pump has a uniform velocity profile (constant $q$) while the water exiting the pump has a velocity profile that is fully-developed. The pipe diameter at the pump entrance is of $D_1=8$ cm and the pipe diameter at the pump exit is $D_2=12$ cm. Given a pump power of 140 kW and a pump efficiency of 50%, and knowing that the pump is well insulated and does not lose heat to the environment, do the following:
(a)  Calculate the pressure of the water exiting the pump
(b)  Calculate the temperature of the water exiting the pump
Use the following data for liquid water: density $\rho=1000$ kg/m$^3$, viscosity $\mu=10^{-3}$ kg/ms, heat capacity $c=c_p=c_v=4200~$J/kgK, conductivity $k=0.6~$W/mK.
Question #6
A large amount of snow fell unexpectedly on Busan, and your engineering firm is in charge of clearing the roads. For this purpose, you mount a plow to some truck as depicted below:
figure6.png  ./download/file.php?id=2313&sid=530ff7689a030ad0fda32fe8293fd713  ./download/file.php?id=2313&t=1&sid=530ff7689a030ad0fda32fe8293fd713
Knowing that the truck travels at 40 km/hour, that the width of the snow $W$ is of 1 m, that the height of the snow $H$ is of 30 cm, that the density of the snow can be taken as $100~$kg/m$^3$, and that the height difference between the repelled snow and the incoming snow $\Delta y$ is of 2 m, do the following:
(a)  Find the total power necessary to push the plow
(b)  Find the percentage of the total power that is due to the snow changing height
(c)  Find $\theta$, the angle of the snow plow
Hints: (i) neglect the friction between the snow and the plow; (ii) the angle $\theta$ is not equal to the deflected snow angle of $35^\circ$: it's not so simple..
Answers
2.  0.52 m, 3.47 m, 9.38 N.
3.  4 hours 44 minutes.
4.  0.0246 N or 1.451 N.
5.  764 kPa, 300.167 K.
6.  68.9 kW, 8.5%, 15.8$^\circ$.
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