hw1

Isotope Hydrology HW-1, due: Tu 2/5

1) (12 points) The dimensionless Reynolds number R is defined as R= ULr/m, where U = a characteristic velocity [L T^-1]; L = a characteristic length [L]; r = fluid density [M L^-3]; m = viscosity [M L^-1 T^-1]. Fig shows the relationship between friction factor f ( a measure of the friction in pipes)and Reynolds number R for smooth pipes which is measured in laboratory experiments over a range of Reynolds numbers. The break in the measurements at Reynolds numbers between 2000 and 4000 marks the transition from laminar to turbulent flow.

a) Calculate the Reynolds number for typical groundwater flow in a clean sand. Use as characteristic length L a typical pore size for sand (Fig) and determine the velocity U from Darcy's law. Assume a hydraulic gradient of 1/100, a typical hydraulic conductivity (Fig), and density/viscosity (Fig). Do you expect to find laminar or turbulent flow?

b) Imagine flow of water through a limestone cave. Could the flow be turbulent there? Make a quantitative argument.

2) (12 points) Flood estimate
A construction project is planned for a small non-monitored stream near Oracle Arizona. This stream is typically dry during most of the year, but floods during extreme presipitation events. The planning agency is concerned that their construction site might be flooded during extreme years and wants to know how high the water might be able to rise in the stream. One way is to estimate the maximum amount of precipitation to be expected in a day, assume that all the water makes it into the stream (no evaporation) and then convert the discharge rate into a depth using Manning's equation.

Info you'll need:

size of the watershed: 42.3 miles²
streambed is rocky, the roughness coefficient was estimated as 0.05
the stream bed drops in elevation at a rate of 2m per 100m
daily precipitation data, Oracle, AZ, 1984-91
Manning's equation:

a) Use the maximum daily precipitation rate in the record as best estimate for the worst case scenario.

b) Calculate the maximum discharge in the stream assuming that all that precipitation comes down and enters the stream in 2 hours. Express the result in metric SI units.

c) Assume that the channel has a V-shape cross section, with a 90^o angle at the bottom (draw a little sketch). Let us call the length of the sides of the "V": L and the depth of the channel: h. Then R_H = area/wetted perimeter = L²/(2*2*L) = L/4. Remember that Q=A*U, with A being L²/2. Rearange Mannings equation to get L. Then convert L into h, using L² = 2h², valid for this particular triangle. So, how deep can the stream get and how far from the center do you need to be in order to avoid flooding?