Homework #8 2013, Earth Surface Processes, Humphrey
1 a) Use Manning’s
equation to discuss how basal shear stress would change in a river that enters
a reach that is twice as steep? At first glance it might appear that the
slope goes up, but the depth goes down, so the shear stress might not
change. Not true. This is a hard question… a couple of hints,
use depth instead of hydraulic radius in Manning’s equation, and assume the
width does not change, so that the discharge (Q=v*d*w) stays constant. Typically the width actually decreases, which
increases the shear stress.
b) Show that the Shields parameter and equation implies that otherwise
similar alluvial rivers (alluvial rivers are rivers that are reworking sediment
that the river itself is moving) must be steeper for coarser sediment. (Hint,
write out the terms for tb and appeal to the previous question)
c) If discharge in a river increases downriver, as is usually the case,
but the sediment size in transport stays the same: what will probably happen to
the river slope?
d) Again looking at the previous parts to this question, if the river
gets steeper, and the bed sediment gets coarser (as we would expect), what
would be the effect of the coarse sediment on the basal shear stress (as
opposed to the sediment staying the same size)?
2. In
class we used the energy in river flow, and showed that the Froude number can
be interpreted as the ratio of kinetic energy to potential energy in the
flow. Many questions in river flow can be addressed by examining the
total energy of the flow. Energy, per unit width and unit length,
(in other words a square meter column of water the depth of the river) of the
flow can be expressed as a height of water (similar to the concept of hydraulic
head in Darcy’s groundwater flow). To be precise the energy in a
column is equal to the potential energy above the bed (rgh)
plus the kinetic energy (rv2/2), ‘h’ is the depth, and ‘v’ is the
velocity of the water. Dividing this by density*gravity turns this into
the energy head of the flow: E = h + v2/(2g)
In most river flows, by far the
largest part of this energy head is just the depth of flow, with the kinetic
energy only a few extra centimeters of head. Because of this we
often ignore the kinetic energy.
To get a sense for the amount of
energy in river flow, we calculate several energies (expressed as water
head). Use the Laramie river in
flood, with a depth of 1m, flow velocity of 1m/s, slope of 2x10-4. Calculate:
a. the potential
energy per width, per meter length of the Laramie river (as a head).
b. the kinetic
energy of the flow (as a head)
c. the potential
energy lost by a column of water per meter of flow down river (as a head)
d. the Froude
number for the Laramie river
e. the super
elevation expected on a bend in the Laramie river (assume the bend has a radius
of 20m and the width is about 4 meters). Does this elevation of the
water come from the potential energy of the flow, or the kinetic energy?
f. Extensive
high Froude number flow (shooting flow) is relatively rare and usually only
found in steep bedrock rivers. Calculate the velocity that would be
needed in the Laramie river to achieve super
critical flow.
g. (hard, mini
puzzle) Assuming the discharge, width and the roughness stay the same, how
steep would the Laramie river have to be to reach a Fr of 1?
3. While we are
talking about energy: energy is expended by a column of river water to move the
load of sediment. Analyzing this is very difficult to do correctly,
but we can approximate the energy by saying that the water flow has to
counteract the settling velocity. We will make several assumptions:
the only sediment in transport is sand (0.2mm), and the amount in transport
(mass qsed) is 0.1kilograms/(m*s). Remember, energy is Force *
distance. Force is easy; it is the weight of the
sediment, distance is less obvious, but think of the settling
velocity of the sand.
Compare the energy to move the
sediment with the energy of the flow from ‘c’ above.
4.
If a minor flood occurs in the Upper Yellowstone and Missouri drainage,
how long will it take for that kinematic flood wave take to reach New Orleans?
You should use several locations down the river system to get the relevant
data. Dams, of course will totally change
this, however, try to get an estimate as if the dams weren’t there. This would give an idea of how quickly floods
used to drain from the continent, while current floods take much longer.
5 We talked about the logarithmic
velocity profile that develops above a rough bed. An equation that works for rivers such as the
6 Rivers are mainly transporting mechanisms for moving sediment
from the hillslope sources to the lake and ocean
sinks. To illustrate this, try to get an
estimate of the percentage of the land in the US that is actually overlain by
flowing water (creeks rivers etc.). I have never tried this question before, see what you can find out.