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.  For a full answer, you need to include a comment on boundary layers, which is the concept that a disturbance in a flow caused by some boundary (or solid surface) will grow downstream.  The region of disturbed flow is called a boundary layer.  Precisely defining boundary layers is impossible in real flows, but the concept is very useful.  (you can just paraphrase some web sites to answer this section)

            b) So,( the real question), do particles lifting from the bed of a river experience positive or negative lift from the Magnus effect?  This is asking you to consider the effect of the shearing of the fluid on the Magnus effect.  This is a difficult question.

            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?