Homework #9 2019, Earth Surface Processes, Humphrey
This is a hard homework set since the questions don’t have plug and chug answers. Question 3 especially requires some careful thinking.
1. 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 pushed out of the way)
3. Total drag force on particle due to surface shear stress times the surface area
4. Balance gravity and pressure drag to get a turbulent settling ‘Law’ (with an incalculable constant)
2. I talked in class about energy in river flow, and showed that the Froude number (introduced on Thursday) 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*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 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 low flood, with a depth of 1m, flow velocity of 1m/s, slope of 2x10-4. Calculate:
a. the potential energy per meter width, per meter length of the Laramie river (as a head above the bed of the river).
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, express this as head.
d. Now if part c is expressed as Joules/m2 s (do the conversion!), it is what is referred to as STREAM POWER, the energy of the flow per square meter of bed that is available to do geomorphic work.
e. the Froude number for the Laramie river
f. (Hard) High Froude number flow (super critical or shooting flow) is relatively rare and usually only found in steep bedrock rivers. Assume the width, discharge and the roughness stay the same, how steep would the Laramie river have to be to reach a Fr of 1? You can use Manning’s equation, but you will have to adjust the depth to keep the discharge constant. You can use 0.025 for manning’s n.
3 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 of the reach 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)?
4 (Hard)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.
5 Write a couple of logical sentences, or a paragraph, on the main processes that you are going to be talking about in your project. Or outline what approach you are going to take to talk about your project.