The Open-Channel Toolbox TM Peter Wilcock Conservation Relations –Conservation of Mass (Continuity) –Conservation of Energy –Conservation of Momentum.

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?