Suspended Load

Above certain critical shear stress conditions, sediment particles are maintained in suspension by the exchange of momentum from the fluid to the particle.

To predict the flux of suspended sediment, we need:

velocity profile

sediment concentration profile

0

CdzuQs

Threshold for suspension:

Suspension occurs under high shear stress flows, when turbulent fluctuations in the vertical velocity of the flow (w’) are as strong, or stronger than the settling velocity of sediment (Ws).

The vertical velocity fluctuations scale with the mean velocity and thus the friction velocity of the flow [w’ (0.8 to 1.0) x u*], so the criteria for incipient suspension is Ws u*.

Rouse Parameter, P, defines the ratio of settling velocity to vertical fluctuations:

*uWsP

8.0

0.28.0

0.2

*

*

*

uWsP

uWsP

uWsP

no suspension, “stream weak” - bedload

incipient suspension

full suspension, “stream strong” - suspended load

When compared to the threshold of motion criteria, it is seen that, for fine-grained sediment (D < 63 - 88 m) the suspension threshold is less restrictive than that for the transport threshold.

Silts & Clays will be suspended as soon as they become mobile, and will not tend to be transported as bedload.

Some flows have transport capacity, but can not resuspend.

Zone 1: Shear stress sufficient to erode the bed.

Shear stress sufficient for significant suspended load transport.

Zone 2: Shear stress not sufficient for erosion.

Shear stress sufficient for suspended load maintenance.

Zone 3: Shear stress not sufficient for erosion.

Transport along bottom maintained as bedload.

Material placed in suspension at higher shear stress levels could move into the bedload layer.

Zone 4: Shear stress not sufficient to resuspend or maintain in suspension any suspended load.

Conservation of Mass for Suspended Sediment

Full equation:

Where us, vs, ws are the components of sediment velocity

Assume: steady state

uniform flow in horizontal

Next remember that the flow is turbulent, and we can treat the mean and fluctuating components separately.

0

zwc

yvc

xuc

tc sssssss

'''

''

wwwvvvuuuwwwccc

sss

sss

0''

ssss wwccz

0''

ssss wcwcz

Conservation of suspended sediment for steady, uniform flow becomes:

Time-averaging:

Vertical sediment velocity = Ws (time-averaged) and can be pulled out of the differential:

Eddy Diffusivity for Mass:

We can use the same gradient diffusion argument as we did for the mixing of momentum:

In analogy, for sediment:

0''

zwc

zcWs ss

zuAwu z

''

zcKwc s

ss

''

0

zcKcWs

zs

ss

Integrating ..

Need a form for the eddy diffusivity for mass (Ks)

zuAK zs *

z

z sa

s

a

dzK

Wscc 1ln

Solve for the concentration profile:

where P is the Rouse parameter (P = Ws/u* )

and za is the reference height where concentration, ca, can be specified.

*uWs

a

a

s

zz

cc

P

aa

s

zz

cc

Notes:

For a grain-size distribution of sediment, calculate profile for each size, then add together.

To get profiles that are valid throughout a thicker region of the bottom boundary layer, different values of the eddy diffusivity must be used.

At high concentrations, conservation of mass should be rewritten to take both water mass and sediment mass into account.

Rouse Parameter:

if P is high:

if P is low:

Where do we position “a”

• top of bedload layer ~ 2D

• practically - 1-2 cm above bed

C

z

C

z

Holding the grain size constant:

very fine sand, D = 0.1 mm

ca = 1 at a = 2 cm

at higher u*, more sediment in water column

at lower u*, turbulent diffusion can’t support sediment grains

Holding the shear velocity constant:

u* = 2 cm/s

ca = 1 at a = 2 cm

finer grain sizes distributed throughout bottom boundary layer

coarser sizes suspended only very near bed.

We still need to know the concentration at a reference elevation (ca) to compute the concentration profile.

Reference height can be a location at which you have some idea of concentration from data or from theory.

Yalin - proposed that, near the bed at steady state, concentrations should increase with excess shear stress,

cs is proportional to S =

Smith & McLean - proposed that za be taken at the top of the bedload layer (which they related to zo) and formed the relationship:

SScc

o

oba

1

cr

crb

SScc

o

oba

1

Where cb is the concentration of the bed

ca is taken at zo

o is the “resuspension parameter”, in sands 2.4 x 10-3 (Smith & McLean)

experimental values have been shown to range : 1.6 x 10-5

5.4 x 10-3