CE 576 Environmental Flows
Spring 2012
Homework 6
30. A river with depth 100 cm has the mean velocity profile shown below. Some values
are given in the table.
a. Estimate the depth-averaged velocity.
b. Estimate the shear velocity by fitting a logarithmic profile to the data.
c. Estimate the roughness length y0 and the equivalent sand grain roughness ks .
z (cm)
0.5
10.0
20.0
40.0
60.0
80.0
100.0
U (cm/s)
22.7
41.4
45.7
50.0
52.6
54.4
55.8
100
90
Distance above bottom (cm)
80
70
60
50
40
30
20
10
0
20
25
30
35
40
45
Velocity (cm/s)
50
55
60
31. Show that the Reynolds-averaged equation (in two dimensions) for a contaminant of
concentration C and molecular diffusivity D is
∂C
∂C
∂C
+u
+w
=D
∂t
∂x
∂z
∂ 2C
∂2C
+
∂x2
∂z 2
−
∂ ∂ uC −
wC
∂x
∂z
*32. You collected several time series of velocity in the flume in the basement of Town
Engineering Building. Use that data to determine the shear velocity in three ways.
(If you would like an example Matlab code to help you with the processing, please let
me know.)
a. Estimate the depth-averaged velocity U and estimate the shear velocity as u∗ ≈
U/20.
b. Fit a logarithmic profile to the time-averaged velocity measurements and compute
u∗ and y0 .
c. Compute the Reynolds stress −u v , fit a line to the profile, and extrapolate to
the bottom, to get u2∗ .
d. Compare your estimates and explain which method you trust most.