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: 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.
 08.01.19
 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: 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: 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: 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.
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.
 $\pi$ 