bed load and suspended load. sediment transport … nielsen (1992) camenen and larson (2006) total...
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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