Homework #9 2015, Earth Surface Processes, Humphrey

These questions are designed to make you think about the meanings of some of the stuff we have done in class. The calculations are not difficult, but some of the questions require some understanding.

1. Reproduce the argument for Stoke’s Law for settling velocity, and therefore reproduce the Stoke’s equation for low Reynolds number particles (minus the value of the coefficient). You might find it useful to follow the following line of logic:

a. Scale of Gravity force on particle

b. Find a scale for the viscous shear stresses (hint, we used our work on debris flows to find the shear stress as a function of velocity)

c. Total drag force on particle due to surface shear stress

d. Balance gravity and viscous drag to get Stoke’s Law (with an incalculable constant)

You need to do more than just write the equations down; you need to label in words the important terms.

2 The most common criteria for initiation of sediment motion (or loosely: sediment transport competence) is the Shields plot. This uses the Shields criterion or parameter: Q= t_b / (( r_sed -r_fluid ) g D_sed). The Shields parameter predicts when the basal shear stress is just enough that sediment of a particular size will be transported. There is a set of plots in the readings that describe the behavior of the parameter. However if we stand back a bit we see that the parameter is approx. 0.05 over a wide range of flows (Reynolds numbers) typical in rivers and for a range of sediments also typical of rivers.

a) Use the Shields criteria to estimate the maximum size of sediment in motion in the Laramie river during a bankfull flood. [Just to introduce some consistency, assume the depth of the Laramie river in minor flood is 1.5m and check the slope using Google Earth or some other tool.]

b)  (a mini geomorph puzzle, asking you to read a little on the web or in the library) Look at the Shield’s plot (Google, or your text), the deviations from a flat line at high Reynolds number are caused by changes in the type of boundary layers around a particle. Why, at different Reynolds numbers, can a boundary layer either increase or decrease drag, if the layer is laminar or turbulent. If you don’t like that vague a question, you can answer this instead: Why do golf balls have dimples?

3 In class we discussed that the Rouse number [the ratio of settling velocity (w) to shear velocity (v*)] as a discriminator of types of sediment transport. Sediment with a Rouse number of about 1 is predicted to move in suspension.

a) Calculate the shear velocity (v*) for the Laramie river.

b) Calculate the approx. size of particles that you would expect to find in suspension in the Laramie River. Fortunately you can get away with using the Stoke’s fall velocity for the settling velocity since the particles will be small enough to have a fairly low particle Reynolds number.

c) We also talked about the ratio of t_b to t_c as a discriminator for types of sediment transport. Are these 2 ratios basically saying the same thing? To be precise, the questions is: what is the difference between the 2 ratios for high Reynolds number particles? (hint, try writing out the full equations for the 2 ratios, you have the settling velocity equation for high Re in your notes)

4 What is the order of magnitude of the lift force on a 1cm diameter piece of gravel on a river bed, with a 0.5m/s water flow over it and stagnant water under it. Use the Bernoulli equation we developed in class, (it is also in the equation list on the web page). Is the force enough to lift the particle? (to get a scale for the lift force from the Bernoulli effect, you can assume the water velocity difference is the full 0.5m/s) [hint: the Bernoulli equation gives you a way to calculate the water pressure; that water pressure can be turned into a force by multiplying by the cross-sectional area of the particle]

5. (this weeks geomorphic puzzle) Many lakes in cold regions of the world turn-over.  The phrase describes a phenomena that occurs once (or twice) a year; the water in the lake moves so that in a short time the water at the top of the lake sinks and flows under the water at the lake bottom, while the water that was at the bottom circulates onto the top of the lake.  This process is hugely important for nutrient cycling.  Why does it occur, where does the head gradient come from?  If you can, explain why the event is often sudden, not gradual, and why it doesn’t occur in warmer regions.  And as a bonus, why doesn’t the ocean do the same thing.  (hint: think of temperature and density gradients)