Homework #1 1) Consider a plant whose drag coefficient, Cd, is given by the equation Cd = -.087*ln(Re)+1.43 where Re is the Reyonolds number. The vegetation can be approximated as a cylinder with a diameter, D, of 4 cm and height of 20cm. The vegetation is uprooted when the force on the vegetation exceeds 4.5 Newtons. Determine at what velocity the drag force exceeds the uprooting force. 2) Assume the drag force for a given area is defined as FD = ρ*(a/(2*(1-φ))*Cd*u2 where φ is the canopy density and a is the frontal area as defined in Nepf 2012. Using the plant properties and assuming the Cd values from problem 1 is valid for multiple stems, plot the drag force for a given area (m2) given an average spacing, ∆S of 100cm, 50 cm, and 25 cm for a velocity range, u, of 1 to 300 cm/s. 3) Take the trapezoidal channel constructed on a slope S = 0.005, with a bottom width, w, of 4m and a side slope of 1:4 (1 meter vertical rise for 4 meters of horizontal distance). Calculate the discharge for a water depth, h, of 1 m if constructed of concrete (Mannings coefficient n=.013) and if the channel was vegetated with a Mannings coefficient n = 0.045. What depth would the vegetated channel need to be to have the same discharge as a concrete channel flowing at 1 m depth. h w 4) Calculate the Froude number for the vegetated and concrete channel for a water depth of 1 m and list whether the flow is supercritical, critical, or subcritical. 5) The vegetated channel described above has a discharge, Q that varies monthly as defined in the table below. The vegetation grows to a height of 1.25 m. For the vegetation to live successfully it needs to be emergent (stems/leaves above the water surface) 8 months of the year. Is this vegetation suitable for the given monthly discharges? If not, propose a possible change to the channel that would allow the selected vegetation to work. Month Jan Feb Mar Apr May Jun Q (m^3/s) 12.37 15.3 21.4 36 25 20 Month Jul Aug Sep Oct Nov Dec Q (m^3/s) 14 9 5 3 2 1.5