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Basic Sediment Transport and Boundary Layer Theory

Environmental Hydraulics

Sediment Transport

Interaction between fluid flow (water or air) and its loose boundaries.

Strong interaction between the flow and its boundaries.

Professor H.A. Einstein: ”my father had an early interest in sediment transport and river mechanics, but after careful thought opted for the simpler aspects of physics”

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Overview

• introduction

• properties of sediment

• current boundary layers

• threshold of motion

• bed features

• suspended load transport

• bed load transport

• total load transport

Importance of Sediment Transport

• erosion (land, rivers, coast)

Examples: soil erosion, local scour around structures, beach erosion

• accumulation (rivers, lakes, reservoirs coast)

Examples: reservoir sedimentation, infilling of harbors, navigation channels, and water intakes

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Classification of Sediment

• cohesive (clay)

electro-chemical forces; form a coherent mass

• non-cohesive (sand)

frictional forces; collection of individual particles

Classification of Sediment Transport

• bed load

along the bottom; particles in contact; bottom shear stress important

• suspended load

in the water column; particles sustained by turbulence; concentration profiles develop

• wash load

more or less evenly distributed through the water column; little relationship with (river) flow properties

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Agents for Sediment Transport

• wind

• water

uni-directional flow (rivers)

oscillatory flow (waves)

combined flow (waves + current)

Properties of Sediment

• grain size (diameter)

• density

• porosity

• concentration (by volume or mass)

• angle of repose

• permeability (fluidization)

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Grain-Size Classification

φ log d=− 2

Phi scale:

Grain-Size Fractions

dn = the grain diameter for which n% of the grains by mass is finer

Geometric standard deviation: σ /g d d= 84 16

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Sediment Sorting

/ well sorted

/ well mixed

d d

d d

<>

84 16

84 16

2

16

Grain-size distribution often follows a log-normal distribution

Density, Concentration and Porosity

Density of sand grains (quartz): 2650 kg/m3

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Bulk Density

mass of mixtureρ

volume of mixtureB =

Angle of Respose

The angle to the horizontal at which grains start to roll on a flat bed of sediment that is gradually tilted from the horizontal.

A representative value on the angle of repose is 32 deg.

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Permeability and Fluidization

ρυP

B

K dpV

dz=

Bulk velocity:

Current Boundary Layers

Velocity profile over a loose boundary (bed).

Vertical gradients are much larger than horizontal ones.

Logarithmic velocity profiles:

*

*

( ) lnκ

τρ

o

o

u zU z

z

u

=

=

Mean velocity: ( )h

U U z dzh= ∫

0

1

κ .=0 4

Von Karman’s constant

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Bed Roughness Length

*

*

*

*

νν

ν

ν

so

so

s so

u kz

u

kz

u

k u kz

= <

= +

= >

59

30 9

7030

Smooth flow

Transitional flow

Rough flow

Nikuradse roughness:

.sk d= 502 5

Current Skin Friction Shear Stress (Flat Bed)

Shear stress: τ ρo DC U= 2

Drag coefficient:( )κ

ln /Do

Cz h

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜ +⎝ ⎠

2

1

Relationship with other friction coefficients:

/D

f g gnC

C h= = =

2

2 1 38

( Darcy-Weisbach / Chezy / Manning )

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Current Total Shear Stress (Non-Flat Bed)

τ τ τo os of= +

Shear stress from skin friction and form drag:

ripples

High flow speed => sheet flow => additional roughness

o os of otz z z z= + +

Wilson (1989): τ(ρ ρ)

osot

s

zg

=−

530

Threshold of Motion

Conditions for initiation of motion.

Shields diagram

*τρ ( ) ν

cr u df

g s d

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠−50

501(Shields 1936)

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Threshold current speed (Soulsby 1997):

( )( )

( )( )

//

*

* **

/

*

( )

.( ) . exp .

.

( )ν

cr

hU g s d f D

d

f D DD

g sD d

⎛ ⎞⎟⎜= −⎟⎜ ⎟⎟⎜⎝ ⎠

= + − −+

⎛ ⎞− ⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

1 71 2

5050

1 3

502

7 1

0 300 055 1 0 020

1 1 2

1

Threshold bed shear stress (Soulsby and Whitehouse 1997):

( )( )**

.θ . exp .

.cr DD

= + − −+0 30

0 055 1 0 0201 1 2

τθ

(ρ ρ)cr

crsg d

=− 50

Threshold Shields parameter:

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Bed Features

A variety of features appears on a loose bed exposed to flowing water:

• ripples

• dunes

• antidunes

Example of Bed Features

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Characterization of Bed Forms

Current Ripples

Geometric properties (Soulsby 1997):

rλ d= 501000

Δ rr

λ=7

Ripple wavelength

Ripple height

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Dunes

Geometric properties (Soulsby 1997):

dλ h=7

Δ .d dλ=0 07

Dune wavelength

Dune height

(more complex formulas exist)

Friction due to Bed Forms

Skin friction and form drag contributes.

Δrof r

r

z aλ

=2

Form drag given by (Soulsby 1997):

ar typically about 1.0