Homework #8 GEOL 4880 Humphrey Fall 2012
Fairly short, but question 2 is
difficult, and question 3 will require some web surfing
1.
I talked in class about 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) 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.
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 and the roughness stay the same, how steep
would the Laramie river have to be to reach a Fr of 1?
2.
While
we are talking about energy: energy is expended by a column of river water to
move the load of sediment. 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.
3.
(Geomorphic
puzzle) Magnus (and Bernoulli) effect on particles and eddies. When
particles are lifted from the bed by the Bernoulli effect, they are often given a spin, typically in the
same direction that most eddies spin. So let us investigate the effect of
spinning on particle motion.
a) Spinning round particles follow curved paths,
you already know this from Baseball, Tennis and many other ball sports. The
effect is usually referred to as the Magnus effect, although similar effects
are observed in Flettner
Rotors. If a large rough object travels through water or air, and is
spinning, what direction does it tend to curve? And of course why? And make sure you include a diagram of the curve
direction, since words are difficult in this case.
b) So, the real question, do particles lifting from the bed of a river
experience positive or negative lift from the Magnus effect?
c) (hard) What is the difference between the Bernoulli effect and the Magnus effect?