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Bed Load and Suspended Load.Sediment Transport Formulas

Environmental Hydraulics

Sediment Transport Modes

• 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

bed load suspended load sheet flow

Increasing Shields number

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Suspended Load

Settling velocity less than upward turbulent component of velocity (for grains to remain in suspension).

Important parameter: ws/u*

( ) ( )a

h

ssz

q c z u z dz= ∫

Sediment Concentration Profile

Balance between sediment settling and upward sediment diffusion from turbulence:

s s

dCw C K

dz=−

( ) expa

zs

az

s

wC z C dz

K

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠∫

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Sediment Diffusivity

*

*

κ

κ

s o

s

s

K K

K u z

zK u z

h

=

=

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠1

Constant

Linear

Parabolic

Different expression for the diffusivity:

Suspended Sediment Concentration Profiles

Exponential (constant diffusivity):

( ) exp sR

o

wC z C z

K

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

if ws/Ko> 4: weak suspension

if ws/Ko < 0.5: strong suspension

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Power-law (linear diffusivity):

( )κs *w / u

aa

zC z C

z

−⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

Rouse number (suspension parameter):

κs

*

wb

u=

b > 5: near bed suspension (h/10)

5 > b > 2: suspension through bottom half of boundary layer

2 > b >1: suspension throughout boundary layer

1 > b: uniform suspension throughout boundary layer

Power-law (parabolic diffusivity):

( )κs *w / u

aa

a

h zzC z C

h z z

−⎛ ⎞− ⎟⎜= ⎟⎜ ⎟⎟⎜ −⎝ ⎠(Rouse profile)

For power-law profiles za is an additional parameter to estimate besides Ca.

More complicated diffusivity relationships exist (e.g., Van Rijn).

=> More complicated concentration profiles.

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Comparison between concentration profiles

Rouse profile

Different profiles

Comparison with Data(Camenen and Larson 2007)

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Comparison with Data

Exponential Power-law (linear)

Rouse profile

Similar fit for all concentration profiles (Camenen and Larson 2007)

Settling Velocity

Depends on:

• particle diameter

• particle density

• particle concentration

• particle shape

• viscosity of water (temperature)

• turbulence

/

*

( )ν

g sD d

⎛ ⎞− ⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

1 3

502

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Dimensionless grain size for characterization of settling velocity:

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Settling Velocity

( )νs *w . . D .

d= + −2 310 36 1 049 10 36

D*

Dim

ensi

onle

ss fa

ll sp

eedSoulsby (1997):

Reference Concentration and Height

..

. τρ ( )

sa

s

cr sa

TC

T

T dz

g s

=+

= +−

50

0 01561 0 0024

26 31 12

Smith and McLean (1977) (power-law/linear):

τ ττ

os crs

cr

T−=

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Suspended Load Transport

expo sss c R

s o

K w hq U c

w K

⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥= − − ⎟⎜ ⎟⎢ ⎥⎟⎜⎝ ⎠⎣ ⎦1

Integrate product between concentration and velocity over the vertical.

For the exponential concentration profile and constant velocity:

θθexp .

θcr

R cRc A⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠4 5

( )*. exp .cRA D−= ⋅ −33 5 10 0 3

Reference concentration (Camenen and Larson 2007):

Bed Load

Threshold of motion exceeded (to-tcr > 0) => sediment movement along bottom as bed load.

Rolling, sliding, and hopping (saltation) of grains along the bed.

Weight of the grains is borne by contact with other grains.

Bed load occurs:

• over flat beds at low flows

• in conjunction with ripples for stronger flows

• over a flat bed for very strong flows (sheet flow)

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Bed load dominates for low flows and/or large grains.

( )Φ sbq

s g d=

− 3501

Parameters to characterize bed load:

Shields number( )τ

θρ ρ

o

s gd=

− 50

Dimensionless transport number

Bed Load Transport Formulas

( )Φ θ θ/

cr= − 3 28

Meyer-Peter and Müller (1948):

( )Φ θ θ θ/cr= −1 212

Nielsen (1992):

/ θΦ θ exp .

θcr

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠3 212 4 5

Camenen and Larson (2006):

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Nielsen (1992)

Camenen and Larson (2006)

Total Load Transport

Or: Predict bed load and suspended load at the same time (one formula for both transport modes).

Resolves the physics to a lesser degree, but practical.

Distinction between bed load and suspended load often hard to make.

Example of such total load formulas:

• Engelund-Hansen (1972)

• Ackers-White (1973)

(based on flow velocity)

Add bed load and suspended load => total load

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Example: Engelund-Hansen total load formula

( )( )

/D

t

. C Uq

g s d=

−

3 2 5

2

50

0 05

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Comparison between EH, VR, and AW formulas