Homework
#9 GEOL 4880 Humphrey Fall 2014
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 change)
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).
f.
Does
this elevation of the water come from the potential energy of the flow, or the
kinetic energy?
g.
Raising
the water surface obviously takes energy, where does it go?
h.
High
Froude number flow (super critical or 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.
i.
(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 river water to move its load
of sediment. This sort of calculation 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 try this for the Laramie River (flow
parameters in previous question). We
will make several assumptions: the only sediment in transport is sand (0.2mm),
and the amount in transport (mass qsed) is
0.1kilograms/(m3*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. (You will need to calculate the
energies in the same units!)
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)
Having answered (b), can you predict the vertical motion of an eddy, based on
its spin direction?
c) (hard) What is the difference between the Bernoulli effect and the Magnus effect?