Homework
#7 2013, Earth Surface Processes,
Humphrey
Don’t forget
to start your term paper/PPT!
1. Reproduce the argument for Stoke’s Law for settling velocity, and therefore reproduce
the Stoke’s equation (minus the value of the
coefficient). You might find it useful
to follow the following line of logic:
a. Gravity force on particle
b. Total drag force on particle due to
surface shear stress
c. A scale for the viscous shear stresses
d. Balance gravity and viscous drag to get
Stoke’s Law (with an incalculable constant
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= tb / (( rsed -rfluid ) g Dsed). 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 1.0m and check the slope using
Google Earth or some other tool.] If you
can, comment on why the sediment size so calculated is probably somewhat too
large?
b) Look back
in your homework to where you calculated the velocity of the Laramie
River. Now use that velocity and Manning's equation to back-calculate the
value of n for the
3 In
class we discussed that the ratio of settling velocity (w) to shear velocity (v*) as a discriminator
of types of sediment transport. Sediment
with a ratio of less than about 0.2 is predicted to move in suspension.
a)
Calculate the shear velocity (v*) for the Laramie river.
b)
Calculate the maximum 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.
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. (a mini geomorph puzzle) Why do
golf balls have dimples? (just a sentence or two
on why making something rougher, makes it go further)
6. (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)