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
1 Suspended Load
Settling velocity less than upward turbulent component of velocity (for grains to remain in suspension).
Important parameter: ws/u*
h q = czuzdz ss ∫ ()() za
Sediment Concentration Profile
Balance between sediment settling and upward sediment diffusion from turbulence:
dC wC= − K ssdz
⎛⎞z ⎜ ws ⎟ Cz()= Ca exp⎜− dz⎟ ⎜ ∫z ⎟ ⎝⎠a Ks
2 Sediment Diffusivity
Different expression for the diffusivity:
K so= K Constant
Kuzs = κ * Linear
⎛⎞ ⎜ z⎟ Parabolic Kuzs = κ * 1− ⎟ ⎝⎠⎜ h⎟
Suspended Sediment Concentration Profiles
Exponential (constant diffusivity):
⎛⎞w Cz()= Cexp⎜− s z⎟ R ⎜ ⎟ ⎝⎠Ko
if ws/Ko> 4: weak suspension
if ws/Ko < 0.5: strong suspension
3 Power-law (linear diffusivity):
−w/κ u ⎛⎞z s* Cz()= C⎜ ⎟ a ⎜ ⎟ ⎝⎠za
Rouse number (suspension parameter): w b = s κu*
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):
−w/uκ ⎛⎞z hz− s* Cz()= C⎜ a ⎟ (Rouse profile) a ⎜ ⎟ ⎝⎠hzz− a
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.
4 Comparison between concentration profiles
Different profiles
Rouse profile
Comparison with Data (Camenen and Larson 2007)
5 Comparison with Data
Exponential Power-law (linear)
Similar fit for all concentration profiles (Camenen and Larson 2007)
Rouse profile
Settling Velocity
Depends on: • particle diameter • particle density • particle concentration • particle shape • viscosity of water (temperature) • turbulence
Dimensionless grain size for characterization of settling velocity:
⎛⎞− 13/ ⎜gs()1 ⎟ Dd* = ⎟ 50 ⎝⎠⎜ ν2 ⎟
6 Settling Velocity
Soulsby (1997):
ν w..D.=+10 3623 1 049− 10 36 s*d() Dimensionless fall speed Dimensionless
D*
Reference Concentration and Height
Smith and McLean (1977) (power-law/linear):
0. 0156Ts Ca = ττos− cr 1+ 0. 0024Ts T = s τ 26. 3τ Td cr z =+cr s 50 a ρgs()−112
7 Suspended Load Transport
Integrate product between concentration and velocity over the vertical. For the exponential concentration profile and constant velocity:
K ⎡⎤⎛⎞wh qUc= os⎢⎥1−−exp⎜ ⎟ ss c R ⎢⎥⎜ ⎟ wKso⎣⎦⎝⎠
Reference concentration (Camenen and Larson 2007): ⎛⎞θ cA= θexp⎜−45 . cr ⎟ RcR ⎝⎠⎜ θ ⎟ −3 ADcR = 35.exp.⋅− 10() 03 *
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)
8 Bed load dominates for low flows and/or large grains.
Parameters to characterize bed load:
q Φ = sb Dimensionless transport 3 number (s −1)g d50
τo θ = Shields number ()ρρs − gd50
Bed Load Transport Formulas
Meyer-Peter and Müller (1948):
32/ Φθθ= 8()− cr
Nielsen (1992):
12/ Φθθθ=12 ()− cr
Camenen and Larson (2006):
⎛⎞θ Φθ=1232/ exp⎜− 4 . 5 cr ⎟ ⎝⎠⎜ θ ⎟
9 Camenen and Larson (2006)
Nielsen (1992)
Total Load Transport
Add bed load and suspended load => total load
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)
10 Example: Engelund-Hansen total load formula
32/ 5 005.CD U qt = 2 ()g()sd−1 50
Comparison between EH, VR, and AW formulas
11