Fundamentals of Fluid Mechanics Assignment 5 — Stagnation Pressure  
Question #1
A turbine of type Pelton has a radius $R$ of 0.9 m and is turning with a speed of $\dot{\omega}=-250$ rpm as shown below:
figure1.png  ./download/file.php?id=5368&sid=3fc74a41cb351c9d29789d637f232adc  ./download/file.php?id=5368&t=1&sid=3fc74a41cb351c9d29789d637f232adc
In the ground reference frame, the incoming water mass flow rate $\dot{m}_{\rm w}$ is of 120 kg/s, the incoming water speed $q_{\rm w}$ is of 61 m/s, and the angle $\theta$ is of 150$^\circ$. Determine the force on the blade $F$ as well as the power input to the turbine. Outline clearly your assumptions.
Question #2
A flat plate is placed in front of a horizontal jet of water. The plate is inclined with an angle $\theta=135^\circ$ as shown below:
figure3.png  ./download/file.php?id=5369&sid=3fc74a41cb351c9d29789d637f232adc  ./download/file.php?id=5369&t=1&sid=3fc74a41cb351c9d29789d637f232adc
Knowing that $q_1=10$ m/s and that $P_1=P_{\rm atm}=101300$ Pa, determine the ratio of mass fluxes going in the two directions (i.e., find $\dot{m}_3/\dot{m}_1$ and $\dot{m}_2/\dot{m}_1$). The flow is assumed to be inviscid and two-dimensional, and the gravity force can be neglected.
Question #3
Consider water with a density of 1000 kg/m$^3$. The water flows down a pipe, exits the pipe, and then interacts with a wedge, as follows:
figure4.png  ./download/file.php?id=5370&sid=3fc74a41cb351c9d29789d637f232adc  ./download/file.php?id=5370&t=1&sid=3fc74a41cb351c9d29789d637f232adc
Do the following:
(a)  Knowing that the gravitational acceleration $g$ is 9.8 m/s$^2$, that the cross-sectional area at station 1 is 1 m$^2$, that the cross-sectional area at station 2 is 0.1 m$^2$, and that the difference in height between station 1 and 2 is $h=2.6$ m, calculate the pressure at station 1 that will result in a mass flow rate in the pipe of $\dot{m}=800$ kg/s.
(b)  Knowing that the water is deflected by the wedge such that $\theta=65^\circ$, find the force $F$ that must be applied on the wedge to hold it in place. You can assume that viscous effects are negligible and that the change in height between station 2 and the bottom of the wedge is small.

Question #4
A plow mounted on a truck, shown below, clears a 0.6 m wide snow string (i.e. $B=0.6$ m):
The snow is 0.1 m deep (i.e. $d=0.1$ m) and its density is 200 kg/m$^3$. The snow leaves in the direction indicated by the figure. The truck travels at 50 km/hr. Evaluate the force and the power required to push the plow. Neglect the friction between the snow and the plow.
Question #5
Consider a jet of water ($\rho=1000$ kg/m$^3$) interacting with a wedge as follows:
figure3.png  ./download/file.php?id=5372&sid=3fc74a41cb351c9d29789d637f232adc  ./download/file.php?id=5372&t=1&sid=3fc74a41cb351c9d29789d637f232adc
Knowing that the surrounding pressure is equal to 1 atm and that $\dot{m}_1=10$ kg/s, that $\dot{m}_2=2\dot{m}_3$, and that $q_1=30$ m/s, find the components of the force exerted by the water on the wedge.
2.  0.854, 0.146.
3.  107500 Pa, 600 N.
4.  3426.82 N, 47.595 kW.
5.  72.96 N, 135.84 N.
Due on Thursday October 10th at 11:00. Do Questions #1, #3, and #4. The other questions are optional.
Hint for Question #4. Several came to my office with a similar question. Check if your approach to find $u_2$ with the provided angles is valid by calculating $v_2$ and $w_2$ using similar arguments and checking if $u_2^2+v_2^2+w_2^2$ is equal to the expected $q_2^2$. If not, your logic definitely has a flaw that needs to be rectified..
In case the previous hint didn't help, here's another hint. Look at the reflected snow from above first. Then draw a triangle with sides $a$, $u_2$ and $w_2$ such that $a^2=u_2^2+w_2^2$ with $a$ being an unknown length. Then look at the reflected snow from the side. Then draw a triangle with sides $b$, $u_2$ and $v_2$ and such that $b^2=u^2_2+v_2^2$ with $b$ an unknown length. Now you have 5 unknowns: $u_2$, $v_2$, $w_2$, $a$, and $b$. Write down enough equations to solve for these 5 unknowns.
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