Lab 4
Fluvial Geomorphology
September 17, 2012
Due September 24, 2012.
Roughness and Shear Stress
Field site: Blacksmith Fork above U P and L Company’s Dam near Hyrum (Station
number 10113500)
Overview:
Today we will make field measurement sufficient to back-calculate channel roughness.
Unfortunately, there is no direct way to measure channel roughness, and the best way is
to back-calculate this attribute at a known flow.
We will also determine the average boundary shear stress at two locations in the channel,
and conduct velocity measurements of a vertical profile to calculate the roughness height
of the channel bed and an alternative measure of shear stress using measured velocity
data.
Assignment
Field Determination of Channel Roughness
Remembering that the discharge form of the Manning equation is:
Q=(AR0.67S0.5)/n
We can rearrange this equation and solve for n, which is what you should do in this
exercise. You should use the reported discharge at the time of the exercise, and focus on
determining a robust measure of reach average hydraulic radius and reach-average slope.
Thus, enough channel cross-sections need to be measured to determine a reach-average
hydraulic radius. You will be divided into teams of 2. Each team will measure one
channel cross section which will be shared with the other teams. You might look at the
reach first and determine an adequate location to measure a cross-section that seems
representative of the reach. When doing this, note the channel shape and flow velocity
through the cross section. You will survey the cross section using a tape, rod, and
engineers level. Since we are only concerned with the present flow conditions, there is
no need to survey any additional topography beyond the wetted channel. Using the
measured cross sections, determine an average hydraulic radius (R) for the existing flow
condition.
Next, you will need to survey S, the reach-averaged water slope, which should be roughly
20 channel widths in length. The primary challenge here is that you will need to move
the instrument several times in order to survey reach slope over the required distance.
In the calculation of n, convert your units to the SI system (Q,R) where necessary; S is
unitless.
Field Determination of Roughness Height and Boundary Shear Stress
Flow that does not vary along stream is termed uniform. For steady, uniform flow,
the stress acting on the bed is:
τ0 = ρgRS
where R is the hydraulic radius, given by ratio of flow area A to wetted perimeter
P, and S is the bed slope; ρ is the density of water (1g/cm3 or 1000kg/m3) which remains
constant, and g is acceleration due to gravity (9.81 m/s2) which also remains constant
unless we are taking measurements on Mars or the moon.
The problem often encountered is that the above method is not precise enough to
determine the stress that is acting on a specific region of the channel bed. By empirically
quantifying the vertical velocity structure at those locations, and assuming a logarithmic
profile, you can get around that problem. A description of the theory and an example are
provided at the end of this handout.
Collect Velocity Data
At your measured cross section, each team will measure 1 velocity profile along deepest,
fastest portion along the cross section. 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 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).
Calculate an average velocity for your profile based on these data.
Calculate the average boundary shear stress for the reach using
τ0 = ρgRS
Next, by following the theory and example below, calculate the shear stress along your
cross-section using a reach-averaged value of slope, and by empirically calculating it at
the location of your vertical velocity profile.
Calculate the roughness height of the bed (point at which the velocity goes to 0), using
the measured velocity profile by following the example below.
Make a sketch of the site that includes the location of your cross section line, the location
of the other team’s cross section line, the location of the gage, and the approximate
location of the three measured velocity profiles along your cross-section.
Theory
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
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

Now, what we do is take the measured velocity data, and look at what happens when u
goes to zero, as this tells us empirically what the roughness height, z0, is (roughness
height is formally defined as that height at which velocity goes to zero). So, if we
substitute zero in for U in the equation from the graph, 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  0.3359
ln 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  0.3359
lnx 1.6755
So we can now say for this particular data set that (remembering that kappa = 0.4 and rho
= 1000 kg/m3)
u
u  * ln y  C1

u  0.3359 ln y
u
 *  0.3359

u*    0.3359

   0.3359

     2  0.33592
   18.1
This gives us a totally empirical Shear Stress of 18.1 N/m2.
To Hand In:
Each team shall share their cross section 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 profile 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.
In your write-up, be sure to 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, and a plot of vertical velocity distribution with logarithmic regression equations.
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).
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? Is the shear stress
calculated at your profile larger or smaller than the reach-averaged shear stress? Explain
these differences and what may cause them. Why might you use 1 method for calculating
shear stress over another?
Describe the importance of the calculated roughness height. Will the roughness height
be greater or smaller if the rocks on the bed of the stream are larger?