1) You are developing a sampling protocol whereby you're going to insert a probe into a turbulent flow in a circular conduit of radius R.
a. Using a description of a velocity profile typical of turbulent flow in a circular conduit, solve for that position r/R where the actual flow profile velocity has the same magnitude as that of an average velocity that would produce the same discharge.
b. Sketch the scenario - the average velocity profile, the actual velocity profile, and the point where v(r)=Vaverage.

2) A circular steel gate controls flow from a dam's pool into an irrigation canal system's raceways. When closed, the gate is in the vertical plane. It is hung in the dam's concrete structure and controlled with a shaft that runs along the horizontal center line of the gate.
The gate can be opened (or closed) as a rotational motion about its horizontal center line, by applying a moment on the shaft...
The gate is 1meter in diameter. It's pool depth operating range varies from a free surface at minimum pool that is even with the top of the gate itself to a max pool operating depth of 2 meters above the top of the gate. Determine the necessary moment that must be applied to the gate's centerline shaft to keep it closed at both minimum pool and max pool depths.

3) A modest hydropower development is being considered for a mountainous, rural province in south-central China. The full run-of-river baseflow (70 cms), will be diverted and passed into a circular steel penstock. This penstock will be constructed of older, salvaged riveted steel, 300mm in diameter. It's about as "rough" as it can get and still be serviceable. The penstock will be 2.18 kilometers in length from its diversion to the power house. The diversion inlet is 800 meters vertically above the power house.
a.) What is the "idealized" (inviscid) power in Watts, Kilowatts, Megawatts and Horsepower that this facility is capable of..?
b.) What is the power potential, in these same units, for this facility, including pipe losses over the run of pipe that makes up the penstock? Your pipe friction factor is not given, a priori. You must demonstrate explicitly that the friction factor you ultimately settle upon for use in this determination is accurate and correct for this pipe and it's operating flow velocities.
c.) A modern, high-tech European "small hydro" bulb-turbine will be used for power generation. It's average cross-sectional area available for flow, after taking into account the area of the turbine blades, is 0.04 meters*2. What is the average flow velocity thru the turbine section itself?
d.) Does this seem "fast"? At these velocities, would you expect hydraulic forces in the turbine and on its components to be small or large, qualitatively speaking, and what would be causing them?
e.) How many average "American" homes could be power with this facility?
f.) Extra credit: How many rural, mountain homes, typical of south central China could be powered with this facility?

4) In a food processing facility, water at 68 degrees (Fahrenheit) passes thru a 60 degree reducing bend in what is otherwise a continuous pipe. The system operating discharge is 3.0 cfs. The bend is in the horizontal plane and its weight is supported from below. The reducing bend is attached at it's inlet and outlet with bolted flange fittings. The inlet diameter is 6 inches. The outlet diameter is 4 inches. Gage pressure at the 6 inch section is 30 psi.
a. Find the net and component (make your frame-of-reference obvious with a sketch) forces that must be restrained. Use an analysis that does not include frictional losses as the water passes through the reducing bend.
b. There are two dominant processes that are producing the bolts' forces: pressure-area forces and flux or momentum transport forces due to the fluid's motion and change of direction. Which, if either are dominating?
c. If you had used an analysis that included frictional losses, would the component forces to be restrained have been less than or greater than those which you calculated and why (short of a complete recalculation, be specific - which terms of the analysis would change, how would they change and how would that effect the final answers with respect to the hydraulic forces in need of restraint)?
d. Given your answer to c.), above - was your choice of an analysis that did not include frictional losses, leading to the bolt forces, a "conservative" engineering choice?

5) A 72 inch concrete culvert (n=0.012) storm sewer is installed at a slope of 0.00063. What is the discharge in cfs that this culvert can manage when it is flowing at a depth of 90% of its diameter (i.e. nearly full flowing)?