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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 φ D = 2-φ

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Sediment Characterization Sediment grain smoothness

Sediment grain shape - spherical, elongated, or flattened

Sediment sorting

4 Grain size

% F

iner

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Sediment Transport Two 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 fluid. ν has units of m2 s-1, the units of a diffusion coefficient. It measures how quickly velocity perturbations diffuse through the fluid.

ν =µρ f

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2. Molecular and Eddy Viscosities

The molecular kinematic viscosity (usually referred to just as the ‘kinematic viscosity’), ν is an intrinsic property of the fluid and is the appropriate property when the flow is laminar. It quantifies the diffusion of velocity through the collision of molecules. (It is what makes molasses viscous).

The Eddy Kinematic Viscosity, νe is a property of the flow and is the appropriate viscosity when the flow is turbulent flow. It quantities the diffusion of velocity by the mixing of “packets” of fluid that occurs perpendicular to the mean flow when the flow is turbulent

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

R =ρp − ρ f

ρ f

ρa

ρpTypical values: Quartz = Kaolinite = 1.6 Magnetite = 4.1 Coal, Flocs < 1

f

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Sediment Settling

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Settling Velocity: Stokes settling

Fg ∝ Excess Density ( ) × Volume( )× Acceleration of Gravity( )

∝ ρp − ρ f( )Vg ∝ ρp − ρ f( )D3g

Fd ∝ Diameter( ) × Settling Speed( )× Molecular Dynamic Viscosity( )

∝ Dwsµ

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

�

∝means "proportional to"11

Settling Speed Fd = Fg

Dwsµ = k ρp − ρ f( )D3g

ws = kρp − ρ f( )D2g

µ

ws = kρp − ρ f( )ρ f

ρ f

µD2g

ws =118

RgD2

ν

Balance 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 1/18 12

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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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Outer region

Intermediate layer

Inner region

δ

z ~ O(δ)

u

xy

z

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

xy

z

τ = shear stress = = force area

rate of change of momentum

τ = µ ∂u∂z

= ρ fν∂u∂z

area

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

τ = µ ∂u∂z

= ρ fν∂u∂z

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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

zu

e ∂∂= ρντ

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Shear velocity u*

u*2 = νe

∂u∂z

τ = ρνe∂u∂z

= ρu*2 = Constant

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

zue *κν =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)

u z( )u*

=1κln zz0

⇒ ln z = ln z0 +κu*u z( )

2** u

dzduzu ρρκ =

From the equations on the previous slide we get

Integrating this yields

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What the log-layer actually looks like lnz

U

~30D

slope = u* /κnot applicable becauseof free-surface/outer-flow effects

0.1δ

~ 30Dviscous sublayer

z

U

log layer

not applicable becauseof free-surface/outer-flow effects

0.1δ

~ 30Dviscous sublayer

z

U

log layer

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

τ = ρ f u*2

Z0

lnz0 Slope = κ/u* = 0.4/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/ν

θ f =τ

ρp − ρ f( )gD

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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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