Lab 3
Fluvial Geomorphology
September 16, 2013
Due September 24, 2013
Fluids and Velocity Profiles
Field site: Little Bear River downstream from Porcupine Reservoir near Paradise, UT
Overview:
You will be divided into teams of 3 or 4. One team will measure a channel cross-section,
and the other team will measure the water-surface slope in a similar manner to Lab 2 and
as described by Harrelson et al (1994, p. 26-32). Bed material grain size will also be
determined.
After the cross section and slope are measured, each team will measure 3 vertical velocity
profiles along the cross-section. One profile will be in the deepest, most uniform, portion
of the cross-section, and the other 2 profiles will at locations that are hydraulically
“different” (i.e. not uniform, more turbulence, etc).
Based on these profiles, determine (1) the velocity at 0.6 times the depth, and (2) the
average of the velocity at 0.8 and 0.2 times the depth.
Each team will calculate the shear stress along their cross-section using the mesured
reach-averaged value of slope. Compare this value with the estimated shear stress at each
measured velocity profile, based on the procedure described below. Each team will
calculate the roughness height of the bed (point at which the velocity goes to 0), using the
measured velocity profile and using measured grain size data.
Background
Flow that does not vary in time is described as steady. Flow that does not
vary along stream is termed uniform. For steady, uniform flow, the stress acting
on the bed is:
τ0 = ρgRS
(1)
where R is the hydraulic radius, in length, S is the water-surface slope, ρ is the density
of water (1g/cm3 or 1000kg/m3), and g is acceleration due to gravity (9.81 m/s2).
The problem often encountered in calculations using (1) is that near bed velocity
measurements are not sufficiently precise to determine the velocity gradient near the
channel bed. By empirically quantifying the vertical velocity structure near the bed, and
assuming a logarithmic profile, you can make a sufficient estimation.
For cases where we have velocity profile data, and have it going all the way to the
channel bed (or at least fairly close to it), we will have an empirical equation of the form:
u(z)  alnz b
(2)
where u(z) is the velocity, u, at a depth, z above the bed. The equation for the law of the
wall in fluids is
u
u  * ln y  C1

(3)
Now, examine what happens when u goes to zero, as this tells us empirically what is the
length of the roughness height, z0. This value is defined as that height at which velocity
goes to zero. If we substitute zero in for u, then we have
u ( z )  a ln z  b
0  a ln z 0  b
b
a
b
z 0  exp

 a 
Note that when we look specifically at the case of velocity going to zero, we use z0
instead of z to denote that height.
ln z 0 
AN EXAMPLE:
Suppose that we have velocity profile data for a stream that is 0.84 m deep, with 6
velocity measurements through the velocity profile:
Depth(m)
0.02
0.12
0.24
0.37
0.55
0.73
0.84
Vel (m/s)
0.36
0.93
1.45
1.42
1.37
1.45
If we take these data and graph them in excel, with velocity as a function of depth, then
we get the following:
1.6
1.4
Vel (m/s)
1.2
1
0.8
0.6
0.4
0.2
0
0.00
0.20
0.40
0.60
0.80
Dist from bed (m)
We notice two important things about these data. First, the data do not go all the way to a
depth of zero. Second, we do not have a velocity measurement at the water surface.
So first, we want to use excel to put a logarithmic regression line through the data.
1.8
1.6
Vel (m/s)
1.4
1.2
1
0.8
y = 0.3359Ln(x) + 1.6755
R2 = 0.8972
0.6
0.4
0.2
0
0.00
0.20
0.40
0.60
0.80
Dist from bed (m)
Note that this gives us the equation of velocity as a function of depth (where in excel
velocity is the y variable, and depth is the x variable)
y  0.3359ln x 1.6755
We remember that z0 is defined as the depth for which velocity goes to zero:
y  0.3359ln x  1.6755 vel  0.3359ln(depth)  1.6755
vel  0  0  0.3359ln(depth)  1.6755
 1.6755 0.3359ln(depth)
 1.6755
 ln(depth)
0.3359
 4.98809 ln(depth)
depth  exp(4.98809)
depth( when _ vel  0)  z 0  0.006819
These data give a z0 of 0.0068. This is very small for a roughness height, but it is based
on empirical data, so it must be right for the data that we have. Now we need to get an
estimate of the actual shear stress. We use the law of the wall equation
u
u*
ln y  C1

u* = √( τ/ρ)
And we realize that with our velocity profile (regression equation in the graph) we have
the equation for this line empirically as well. So, we note that our slope from these data is
y  0.3359ln x 1.6755
So we can now say for this particular data set that (remembering that  = 0.4 and  =
1000 kg/m3)
u
u  * ln y  C1

u  0.3359 ln y
u
 *  0.3359

u*    0.3359

   0.3359

     2  0.3359 2
   18.1
This gives us a total shear stress of 18.1 N/m2.
Assignment:
In the field:
One team will measure a channel cross-section, 1 team will measure the channel slope in
a similar manner to Lab 2 and as described by Harrelson et al (1994, p. 26-32).
Each team will measure 3 velocity profiles along the cross-section line. For the first
profile, select the deepest portion along the cross-section and measure the depth using
your top-setting rod. With your flow meter, collect 30 second time-averaged velocity
measurements at 8 equally spaced intervals upward through the flow at that location.
Record the depths and velocity of each of those measurements. Without moving your
wading rod, also collect a velocity measurement at the theoretical point of the average
velocity for a logarithmic velocity profile (i.e, 0.6 of the depth for depths less than 2.5
feet, or an average of measurements taken at 0.2 and 0.8 of the depth for depths greater
than 2.5 feet).
Repeat this for two other profiles that display differing flow properties such as
turbulence, eddies, etc.
Next, each team should conduct a pebble count along the cross section line in order to
determine the grain size distribution of the substrate. The activity of actually measuring
individual particles that comprise the bed is a trivial activity involving measuring the baxis of each particle or passing the particle through a field sieve, known as a
gravelometer. This activity is described in Harrelson et al. (1994, p. 49-52).
Make a sketch of the site that includes the location of your cross section line and the
approximate location of the area of which you conducted your pebble count.
In the Office:
Each team shall share their data with the other.
Plot the longitudinal profile of the water surface. Show each data point that you survey
and connect straight lines between surveyed points. In the event that you survey any
points that are clearly in error, develop a second plot in which you censor these points
from your final data set.
Determine the slope of the water surface by fitting a linear least squares relation to your
data.
Plot your cross-sections, looking downstream. Determine the water surface width, crosssection area, wetted perimeter, and hydraulic radius of each cross-section that you survey.
Determine the reach-averaged shear stress using the depth-slope product determined by
τ0 = ρgRS
Next, plot your measurements of the velocity profiles using your 8 equally spaced
measurements of velocity. Plot depth on the x axis and velocity on the y axis. Also, plot
your average measurement taken at 0.6 times the depth, or the average of your
measurement taken at 0.2 and 0.8 times the depth. Fit a logarithmic regression line
through your data, and display the equation.
According to the example above, calculate the roughness heights (the height at which
velocity is zero) at each profile, and derive the shear stress for that vertical profile.
Next, plot the grain size distribution according to the example in Harrelson p. 51. Make
the x-axis logarithmic. Estimate grain size distribution parameters (D16, D50, D65, D84,
D90)
Some scientists have suggested that zo can be approximated as 0.1*D84 or 0.1*D90.
Calculate a roughness height using these relations.
In your write-up:
Include a location map, a site map, your plots of the longitudinal water surface profile
with the linear regression equation, a plot of your cross section, a plot of vertical velocity
distribution with logarithmic regression equations, and a plot of your grain size
distribution.
How does your average velocity as determined from your velocity profile compare with
the average velocity taken from 1 measurement (at 0.6 times the depth, or 0.2 and 0.8
times the depth).
How do your velocity profiles differ from each other? How variable are your estimates
of the roughness heights from the different profiles?
Compare your shear stress calculation from the velocity profile taken at the deepest, most
uniform location. How does your reach-averaged calculation of shear stress compare to
the shear stress as determined from this vertical velocity profile? Explain these
differences and what may cause them. Why might you use 1 method for calculating
shear stress over another?
How do your calculated roughness heights differ using your velocity data and your grains
size data? What may cause these differences? Why might you use one method over
another?