Homework #9            2022, Earth Surface Processes, Humphrey

In this homework set several questions don’t have plug and chug answers. Question 1c and question 3 especially require some very careful thinking. Although question 3 is difficult, the results are important to understanding some of the interactions between the river slope and the sediment in transport in alluvial rivers (alluvial rivers are rivers that are reworking sediment that the river itself is moving).

1.   In small alluvial rivers, we often see “riffle-pool” sequences (look it up on google).  This means the stream alternately flows through deeper parts (the “pools”) and shallower parts (the “riffles”).  This is well known to fishermen since it determines where to find fish.  Since the discharge of the river stays constant as you go downstream (and for this question, assume the width is constant), answer the following if the riffle is ½ as deep as the pool:

            a. Which has the fast water flow, pools or riffles?

           b. Which has a steeper water surface, pools or riffles?

           c. (Hard) which has the higher basal shear stress (show your work)?  [hint, we discussed in class that in turbulent flow, the velocity goes as the square root of the slope depth product]

 

2.    Reproduce the argument for the settling velocity of large (high Reynolds number) particles.  You only need to come up with the approximate form, not the value of the coefficient.  You might find it useful to follow the following line of logic:

1.    Gravity force on particle

2.    A scale for the pressure stresses on the particle surface (remember this is scaled by the K.E. of the water that is slowed down and pushed out of the way)

3.    Total drag force on particle due to pressure stresses acting on the area of the particle

4.    Balance gravity and pressure drag to get a turbulent settling ‘Law’ (with a constant not calculable by this “scaling” argument)

 

3.     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 (this is expressing energy as pressure, similar to the concept of hydraulic head in Darcy’s groundwater flow).  To be precise the total energy in a water column in a stream is equal to the potential energy above the bed (rgh*1/2, the ½ comes from the average height, not the total height) plus the kinetic energy (rv2/2), ‘h’ is the depth, and ‘v’ is the mean velocity of the water.  Dividing this by density*gravity turns this into the energy head of the flow: E = h/2 + v2/(2g) with units of meters of water height.

In most river flows, by far the largest part of this energy head is just the potential energy from the depth of water, 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 low flood, with a depth of 1m, flow velocity of 0.5m/s, slope of 5x10-4.  Calculate:

a.    the potential energy per square meter of bed of the Laramie river (as a head above the bed of the river).

b.    the kinetic energy of the flow, per square meter (as a head)

c.    the potential energy lost by a square meter column of water per meter of flow down river, again express this as head.

d.   Most of the energy lost by the flowing water goes to create turbulence.  What is the ultimate sink for this energy?

e.   Now if part c is expressed as Joules/m2 s (do the conversion!  Remember to divide by time), it is what is referred to as unit STREAM POWER, the maximum energy of the flow per square meter of bed per second that is available to do geomorphic work.  What is the unit stream power of the Laramie river?

f.    the Froude number for the Laramie river?

4 a) Show that the Shields parameter and equation implies that otherwise similar alluvial rivers must be steeper for coarser sediment, if the sediment is being transported. (Hint, write out the terms for the basal shear stress)

   b) 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?

   c) When you produced the logarithmic velocity profile of the Laramie river in the previous homework, you should have found that the average velocity was about 0.65m/s. Manning’s equation in case you forgot to write it carefully in your notes is: average velocity = ( slope^1/2 * Depth^2/3 ) / n.  Where the “n” is called Manning’s n and is a measure of roughness of the channel.  Back calculate what “n” needs to be for the Laramie river.  Use a depth of 1m and a slope of 5x10^-4.  Look up Manning’s n on Google or look in your text an see if this is a reasonable number.

  d) if the Laramie river at the flow level (near flood stage) in the part c) is about 10m wide, what would the Discharge of the river be, in m^3 /s.

5 You have told me of the general direction of your term paper (and I may have given you feedback).  Now you need to write an outline of your approach.  This can be in form of several sentences, or even a bulleted outline.