2013 Fluid Mechanics Final Exam  
Fluid Mechanics
Final Examination
December 15th 2013
18:00 — 21:00

Question #1
Starting from the first law of thermodynamics in integral form: $$ \frac{{\rm d}}{{\rm d}t} \int_{V} \rho E {\rm d}V + \int_{S} (\rho E + P)(\vec{v}\cdot\vec{n}){\rm d}S=\dot{Q}-\dot{W} $$ with: $$ E\equiv e+\frac{1}{2}q^2 + g y $$ Show that conservation of energy within a pipe system can be expressed as: $$ \left( \frac{P_1}{\rho} + g y_1 + \frac{\alpha_1}{2} (u_{\rm b})_1^2\right) -\left( \frac{P_2}{\rho} + g y_2 + \frac{\alpha_2}{2} (u_{\rm b})_2^2\right)=h_{\rm L} $$ Outline all assumptions and provide the definitions for the kinetic energy coefficient $\alpha$ and the head loss $h_{\rm L}$.
Question #2
Water with a density of 1000 kg/m$^3$ flows steadily through the following elbow:
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At the inlet to the elbow (station 1), the pressure is $5\times 10^5$ Pa and the inner diameter is 80 mm. At the outlet to the elbow (station 2), the inner diameter is 50 mm and the pressure is $4.75 \times 10^5$ Pa. The mass flow rate is 15 kg/s. Around the elbow, the atmospheric pressure is 101300 Pa. Knowing that $\theta=60^\circ$, do the following tasks:
(a)  Determine the $x$ and $y$ components of the force vector $\vec{F}$ that is required to hold the elbow in place.
(b)  How much of this force is due to the atmospheric pressure?
Question #3
Consider the following embankment made of concrete, with the shape of a right-angled triangle with the base $L$ and the height $h$, and fixed to the ground at point A:
Determine the minimum length $L$ the concrete embankment must have to avoid tipping over point A. Take $h=1$ m, $\rho_{\rm w}=1000$ kg/m$^3$ and $\rho_{\rm c}=2400$ kg/m$^3$.
Question #4
After obtaining a PhD and graduating from PNU, you are hired by the Korea Aerospace Research Institute (KARI) and assigned the task to design a prototype of a new airship. The prototype has a length $L$ of 100 m and its maximum diameter $D$ is of 20 m:
figure4.png  ./download/file.php?id=1405&sid=f283f250d98752dc286b97e7c2370666  ./download/file.php?id=1405&t=1&sid=f283f250d98752dc286b97e7c2370666
It is wished to estimate the drag force that would act on the prototype when it is at an altitude of 10 km and when the air speed $q_\infty$ is of 5 m/s (at an altitude of 10 km, the air pressure and temperature can be taken equal to 0.26 atm and 223 K respectively). To do so, you decide to build a small scale model and measure its drag in a wind tunnel under dynamically similar conditions, and further use non-dimensional analysis to relate the drag force measured in the model to the one acting on the prototype. For this purpose, do the following:
(a)  Find the non-dimensional numbers associated with this problem, and show how they are related to the criteria of similarity ${\rm Re}_D$ and $C_{\rm D}$
(b)  If the length of the model used in the experiment is fixed to 2 m and the air density in the wind tunnel is of $1.2$ kg/m$^3$, determine the diameter of the model airship and determine the wind tunnel air flow speed
(c)  Knowing that the drag force measured on the model is of 150 N, estimate the drag force acting on the prototype.
Notes: the air viscosity can be taken as $10^{-5}$ kg/ms for both the prototype and the model, and the gas constant for air is $286$ J/kgK.
Question #5
You are waterskiing on a lake being pulled by a boat, and the lake surface is smooth and free of waves. As the boat reaches cruise speed (i.e. constant speed), the force necessary to hang on to the handle pulling you is of 37.3 N. Knowing that the water viscosity is of $10^{-3}$ kg/ms and that the water density is of 1000 kg/m$^3$, and that each of your two waterskis is 2 m long by 0.25 m wide, estimate as well as possible the cruise speed of the boat. Hints: (i) you can assume that the drag acting on your skis is limited to friction drag and that the drag due to air resistance is negligible; (ii) this problem may require iterations.
Question #6
You are working for a power plant, and one assignment given to you is to measure the wall roughness in an old rusted pipe. The pipe has a length of 2 m and a diameter of 2 cm. Because the pipe's length is much greater than its diameter, it is difficult to measure directly the height of the bumps on its interior surface. For this reason, you decide to measure the average height of the bumps indirectly through a fluid dynamics experiment: you attach a small pump to one extremity of the pipe and force water (viscosity of $10^{-3}$ kg/ms and density of 1000 kg/m$^3$) to flow through the pipe. You measure a mass flow rate of 1.57 kg/s and a force acting on the pipe due to fluid friction of 11.78 N. Knowing the latter, do the following:
(a)  Find $e$, the average height of the bumps within the rusted pipe
(b)  Find the percent increase in mass flow rate should the rusted pipe be substituted by a pipe with the same diameter and length but with smooth inner walls ($e \rightarrow 0$)
Hints: (i) Because the length is much greater than the diameter, the flow can be assumed fully-developed throughout; (ii) For fully-developed flow, $dP/dx$ is constant; (iii) Part (a) and part (b) can be answered independently of each other.

2.  $F_x=-1625~{\rm N}$, $F_y=735~{\rm N}$, $F_x^{\rm atm}=410~{\rm N}$, $F_y^{\rm atm}=-172~{\rm N}$
3.  $0.75~{\rm m}$.
4.  $\Pi_1=\mu/\rho_\infty q_\infty D$, $\Pi_2=F_{\rm drag}/\rho_\infty q_\infty^2 D^2$, $\Pi_3=L/D$, $D_{\rm m}=0.4~{\rm m}$, $q_{\rm m}=85.4~{\rm m/s}$, $F_{\rm p}=439~{\rm N}$.
5.  $5.21~{\rm m/s}$.
6.  $8 \times 10^{-3}~{\rm cm}$, $31.2\%$.
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