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The Open-Channel Toolbox TM Peter Wilcock
• Conservation Relations– Conservation of Mass
(Continuity)– Conservation of
Energy– Conservation of
Momentum
• Constitutive Relations– Flow Resistance– Sediment Transport
Conservation of Mass (Continuity)
• Mass is neither created nor destroyed
• Inputs = outputs• Inputs and outputs for
fluid flow are discharge– Vel x Flow Area
U1A1 = U2A2
A
NCED and “beige” slides by Peter Wilcock/
Johns Hopkins Univ
Conservation of Momentum (Force-balance)
Unsteady, nonuniform flow
• Flow accelerates in space and time
1-d St. Venant eqn.
Rearranged 1-d St. Venant eqn.
Potential Energy and Kinetic Energy
• Bernoulli energy equation– H = d + Z + V2/2g + losses– d = depth– Z = elevation above datum,
e.g. sea level– V = velocity of flow– g = gravity
H1
H1
• Energy is neither created nor destroyed• Two components
– kinetic ( )– potential (z+h)
• Energy is also converted to heat, hf
• H1 =H2 + hf
Conservation of Energy
g
U
2
2
http://ga.water.usgs.gov/edu/hyhowworks.html
Flow Resistance
• Relation between velocity, flow depth, basal shear stress, and hydraulic roughness
• A variety of relations exist including– Manning’s– Chezy
• Empirical• The big unknown: n
n
RSU
32
Using continuity,
ARn
SUAQ 3
2
(Metric)Multiply by 1.49 for English units
LWD covering less than 2% of the streambed can provide half
the total roughness or flow resistance. This results in a finer
streambed substrate.
Buffington and Montgomery 1999, WRR 36, 3507-3521Manga and Kirchner, 2000, WRR 36, 2373-2379.
Sediment Load• Sources:
– Chemical weathering (dissolved)
– Human activity– Mass wasting– Slopewash– Rill and gully
formation– Channel scour
• Bed• Cutbanks
What does transport depend on?
• The strength of the flow, the fluid, and the sediment
• Strength of the flow = shear stress• The sediment = grain size and density• The fluid = water density and water viscosity (its
resistance to deformation)
Sediment transport
• Directly expressed in terms of sediment supply and water supply
• Shear stress is a descriptor of transport rate
bcaq )( ***
2/3*** )(8 cq General Form:
Meyer-Peter and Muller:
Emmett and Wolman (2001)
How to measure sediment transport?
Sediment Transport and Incipient Motion
• They are not the same– sed trans = mass flow rate per unit time– incipient motion = moves or not moves (binary: 0 or 1)
• What they share– f(shear stress)– transport depends on the fluid force applied to the bed
Tractive Force or Incipient Motion
• Grain motion is driven by shear stress, – Units of force/unit area: psf,
psi, Pa
• Critical shear stress, c
– Shear stress needed to get a grain of a given size moving
• Shield’s number ordimensionless shear stress
*
gDs )1(*
gDscc )1(*
gDso
)1(
*
Shield’s equation
Shield’s diagram
motion
no motion
http://www.uwsp.edu/gEo/faculty/lemke/geomorphology/lecture_outlines/03_stream_sediment.html
=
Problem Site 9 on the White River
Try it
5 m2m
For a rectangular channel 5 m. wide by2 m. deep (running full), with a slope of 2%
What is the basal shear stress?
gRS
Remember, hydraulic radius R = A/WP
Pasm
kg
m
m
m
m
s
m
m
kg218218
50
1
9
1
1
1081.910002
2
23
2222
1
sm
kg
ms
mkg
m
NPa
Try It
• A basketball has a diameter of ~ 10 inches.
• What is the critical shear stress required to just move a basketball-sized rock in a river?
• For the channel in the last problem is the rock mobile?