15. Physics of Sediment Transport

William Wilcock (based in part on lectures by Jeff Parsons)

OCEAN/ESS 410

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Lecture/Lab Learning Goals• Know how sediments are characterized (size and

shape)• Know the definitions of kinematic and dynamic

viscosity, eddy viscosity, and specific gravity• Understand Stokes settling and its limitation in real

sedimentary systems.• Understand the structure of bottom boundary layers

and the equations that describe them• Be able to interpret observations of current velocity in

the bottom boundary layer in terms of whether sediments move and if they move as bottom or suspended loads – LAB

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Sediment Characterization

φ Diameter, D

Type of material

-6 64 mm Cobbles

-5 32 mm Coarse Gravel

-4 16 mm Gravel

-3 8 mm Gravel

-2 4 mm Pea Gravel

-1 2 mm Coarse Sand

0 1 mm Coarse Sand

1 0.5 mm Medium Sand

2 0.25 mm Fine Sand

3 125 μm Fine Sand

4 63 μm Coarse Silt

5 32 μm Coarse Silt

6 16 μm Medium Silt

7 8 μm Fine Silt

8 4 μm Fine Silt

9 2 μm Clay

• There are number of ways to describe the size of sediment. One of the most popular is the Φ scale.

φ= -log2(D)D = diameter in millimeters.

• To get D from φD = 2-φ

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Sediment CharacterizationSediment grain smoothness

Sediment grain shape - spherical, elongated, or flattened

Sediment sorting

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Grain size

% F

iner

Sediment TransportTwo important concepts•Gravitational forces - sediment settling out of suspension•Current-generated bottom shear stresses - sediment transport in suspension (suspended load) or along the bottom (bedload)

Shields stress - brings these concepts together empirically to tell us when and how sediment transport occurs

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Definitions

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1. Dynamic and Kinematic Viscosity

The Dynamic Viscosity μ is a measure of how much a fluid resists shear. It has units of kg m-1 s-1

The Kinematic viscosity ν is defined

where ρf is the density of the fluidν has units of m2 s-1, the units of a diffusion coefficient. It measures how quickly velocity perturbations diffuse through the fluid.

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2. Molecular and Eddy Viscosities

Molecular kinematic viscosity: property of FLUID

Eddy kinematic viscosity: property of FLOW

In flows in nature (ocean), eddy viscosity is MUCH MORE IMPORTANT!

About 104 times more important

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3. Submerged Specific Gravity, R

Typical values:

Quartz = Kaolinite = 1.6Magnetite = 4.1Coal, Flocs < 1

f

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Sediment Settling

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Settling Velocity: Stokes settling

Settling velocity (ws) from the balance of two forces - gravitational (Fg) and drag forces (Fd)

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Settling SpeedBalance of Forces

Write balance using relationships on last slide

k is a constant

Use definitions of specific gravity, R and kinematic viscosity ν

k turns out to be 1/1812

Limits of Stokes Settling Equation

1. Assumes smooth, small, spherical particles - rough particles settle more slowly

2. Grain-grain interference - dense concentrations settle more slowly

3. Flocculation - joining of small particles (especially clays) as a result of chemical and/or biological processes - bigger diameter increases settling rate

4. Assumes laminar settling (ignores turbulence)

5. Settling velocity for larger particles determined empirically

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Boundary Layers

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Bottom Boundary Layers

• Inner region is dominated by wall roughness and viscosity• Intermediate layer is both far from outer edge and wall (log layer)• Outer region is affected by the outer flow (or free surface)

The layer (of thickness δ) in which velocities change from zero at the boundary to a velocity that is unaffected by the boundary

δ is likely the water depth for river flow.

δ is a few tens of meters for currents at the seafloor

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Shear stress in a fluid

τ= shear stress = = forcearea

rate of change of momentumarea

Shear stresses at the seabed lead to sediment transport

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The inner region (viscous sublayer)

• Only ~ 1-5 mm thick• In this layer the flow is laminar so the molecular

kinematic viscosity must be used

Unfortunately the inner layer it is too thin for practical field measurements to determine τ directly

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The log (turbulent intermediate) layer

• Generally from about 1-5 mm to 0.1δ (a few meters) above bed

• Dominated by turbulent eddies• Can be represented by:

where νe is “turbulent eddy viscosity”

This layer is thick enough to make measurements and fortunately the balance of forces requires that the shear stresses are the same in this layer as in the inner region

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Shear velocity u*

Sediment dynamicists define a quantity known as the characteristic shear velocity, u*

The simplest model for the eddy viscosity is Prandtl’s model which states that

Turbulent motions (and therefore νe) are constrained to be proportional to the distance to the bed z, with the constant, κ, the von Karman constant which has a value of 0.4

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Velocity distribution of natural (rough) boundary layers

z0 is a constant of integration. It is sometimes called the roughness length because it is often proportional to the particles that generate roughness of the bed (a value of z0 ≈ 30D is sometimes assumed but it is quite variable and it is best determined from flow measurements)

From the equations on the previous slide we get

Integrating this yields

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What the log-layer actually looks like

Plot ln(z) against the mean velocity u to estimate u* and then estimate the shear stress from Z0

lnz0 lnz0

Slope = κ/u*

= /u*

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Shields Stress

When will transport occur and by what mechanism?

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Hjulström Diagram

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Shields stress and the critical shear stress

• The Shields stress, or Shields parameter, is:

• Shields (1936) first proposed an empirical relationship to find θc, the critical Shields shear stress to induce motion, as a function of the particle Reynolds number,

Rep = u*D/ν26

Shields curve (after Miller et al., 1977) - Based on empirical observations

Sediment Transport

No Transport

Transitional

Transitional

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Initiation of Suspension

Suspension Bedload

No Transport

If u* > ws, (i.e., shear velocity > settling velocity) then material will be suspended.

Transitional transport mechanism. Compare u* and ws

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