1. 1B Clastic Sediments Lecture 27 SEDIMENT TRANSPORT Onset of motion
Mode of transport
Estimation of bedload transport
Estimation of suspended load transport NH

2. SEDIMENT CONTINUITY For constant sediment concentration C,
(qs)1 < (qs)2: erosion of channel bed.
(qs)1 > (qs)2: deposition on channel bed.
Bed volume change in time dt:
dydx = -1/(1-l) dqsdt ,
where dy is thickness eroded/aggraded
is porosity of bed material.
Bed volume change may also result
from change in sediment concentration:
dydx = -1(1-l) (hdC)dx ,
where h is flow depth.
dydt = -1/(1-l) (dqs/dx + hdC/dt)
Net change in bed elevation
over time is related to down-
stream change of sediment
transport rate and the change
in suspended sediment
concentration in time.

3. TRESHOLD OF SEDIMENT MOVEMENT
Stream power w = t0u
is rate of work done on channel bed.
Below threshold shear stress tc:
no sediment motion and transport.
Effective shear stress: t – tc

4. FORCES ON PARTICLE ON BED
Cylinder on horizontal bed
below inviscid fluid.
No viscosity: no drag force.
Stream lines converge and then
diverge over cylinder in symmetrical
fashion: pressure distribution results
in net upward force: lift force.

5. FORCES ON PARTICLE ON BED
Cylinder on horizontal bed
below inviscid fluid.
No viscosity: no drag force.
Stream lines converge and then
diverge over cylinder in symmetrical
fashion: pressure distribution results
in net upward force: lift force.
Cylinder under viscous flow:
Flow separation behind cylinder
causes drag force in addition to lift.
Resultant fluid pressure force FF has
upward and downstream component.

6. FORCES ON PARTICLE ON BED
Resisting grain motion:
Gravitational force FG; neighbouring grains.
Entrainment of grain by rotation about pivot.
Angle of easiest movement: a.
Threshold of movement: balance of moments
about pivot point:
FG(sin a)a1 = FF(cos a)a2 ,
where a1 and a2 are moment arms.
FG = c1D3g’ ,
where g’ = (rs – rf)g, and constant
c1 accounts for flow variability
grain characteristics.
FF = c2D2t0 ,
where constant c2 accounts for
grain shape and packing.
Ignore lift force: vanishes after
entrainment.

7. FORCES ON PARTICLE ON BED
Threshold of movement: balance of moments
about pivot point:
FG(sin a)a1 = FF(cos a)a2 ,
FG = c1D3g’ ,
FD = c2D2t0 .
Combine and regroup:
[a1c1/a2c2 tan a] = t0/Dg’ .
b ~ grain characteristics
local flow:
boundary Reynolds no.
Definition of threshold condition
relies on experimentation.

8. DIMENSIONAL ANALYSIS OF MOTION THRESHOLD
Variables:
Critical bed shear stress, tc [ML-1T-2] repeat
Fluid density, rf [ML-3] repeat
Fluid viscosity, m [ML-1T-1]
Grain diameter, D [L] repeat
Submerged specific weight of grain, g’ [ML-2T-2]
Sought: balance of inertial and viscous forces (Reynolds number),
balance of gravitational and fluid forces.
Combine g’ with repeating variables:
t0/g’D = b (Shields’ stress).
Combine m with repeating variables:
(D√rf√t0)/m = (rfu*D)/m = u*D/n = Re* ,
where n = m/r is the kinematic viscosity.
Remember: shear velocity u*2 = t0/r .
Re* is boundary Reynolds number.
and Re* fully characterise
onset of sediment motion.
Their relation was constrained
experimentally by Shields.

9. SHIELDS’ DIAGRAM
Re* < 10: fine grain sizes: well-packed, cohesive sediment, enclosed
within viscous sublayer. Entrainment more difficult than fine sand.
Shields’ stress b increases with decreasing Re*
Re* > 10: Non-cohesive silt and sand.
Entrainment more difficult with increasing grainsize.
Expected: Shields’ stress b increases with increasing Re*
Experimental results: flat trend.
Shear stress and grain size on both axes.

10. YALIN’S DIAGRAM b
In Shields’ diagram: shear stress and grain size on both axes.
Solve by combining boundary Reynolds number with Shields’ b to
eliminate shear stress:
F = Re*2/b .
Yalin’s plot of b against √F has same general form as Shields’ curve.

11. MODES OF SEDIMENT TRANSPORT
BED LOAD
Sliding, rolling, saltation
SUSPENDED LOAD
Mode of transport depends on
grain density
grain size
flow hydraulics
Conditions vary in space and time:
Modes of transport change frequently.
Distinction between bed load and
suspended load is not easy.
Transport stage

12. TRANSPORT STAGE u*/w , With increasing shear velocity,
where u* is the shear velocity, t0/r
w is the settling velocity (cf. Stoke’s law).
With increasing shear velocity,
proportion of load moving in suspension
increases.
Therefore dimensionless grain velocity
ug/U increases with transport stage.
Here, ug is the grain velocity, and U is the
flow velocity.
u* = w approximates saltation –
suspension threshold.
When u* > w, then grains move with
approximately the velocity of the flow.
Results shown for quartz sand in flow 48 mm deep.

13. BEDLOAD TRANSPORT Bedload transport rate ~ stream power, w = t0u .
conversion factor to be constrained empirically
Prediction of bedload transport complicated by:
bed armouring and consolidation of gravels.
resistance of bedforms in sand and gravel rivers.
lack of constraints on threshold of sediment motion.
unsteadiness in high stage flows.
has dimensions [ML2T-3] over unit area of stream bed [L2]:
Proportional to the cube of (excess) flow velocity.
Distribution of flow velocity in open channels.

14. FLOW RESISTANCE IN BEDLOAD RIVERS
Force driving flow down inclined plane: downslope component of
gravity acting on the mass of water.
Flow resistance: frictional energy loss during flow on bed and banks.
Parameters of the problem:
Flow depth h Combine two length scales in problem in
flow velocity u dimensionless variable: h/ks, the relative roughness.
fluid density rf
fluid viscosity m Express fluid flow in a Reynolds number: Re = rfuh/m .
basal shear stress t0
roughness height ks Make shear stress dimensionless by dividing by ru2:
8t0/rfu2 = f f is the friction factor.
Using t0 = rgSh, rfgh sin a = 1/8 rfu2f .
Solving for u, u = √(8 gh sin a)/f .
Usually written as: u = C(h sin a)0.5 , C = (8g/f)0.5
Chezy coefficient.
Flow depth h is inadequate length scale.

15. FLOW RESISTANCE IN BEDLOAD RIVERS
Only bed and banks exert friction.
Together they are termed:
Wetted perimeter.
Hydraulic radius RH =
channel cross section area
wetted perimeter
RH is better length scale for calculation of flow resistance.
u = (RH2/3 sin a1/2)/n ,
where n is Manning’s roughness coefficient.
n = 1/C RH4/3 ; C = (8g/f)0.5 ; f = 8t0/rfu2
Use to estimate formative discharges in (paleo) channels:
Measure channel slope from exposed geometry or terrace form.
Largest clasts are assumed to represent maximum discharge conditions.
Critical bed shear stress is estimated from particle size using tc = bg’D .

16. SUSPENDED LOAD
The distribution of suspended sediment in a flow can be treated as
diffusion problem, with high concentration at bed, and low concentration
near surface. Mass flux Q is linearly proportional to concentration gradient:
Q = -k∂C/∂y , where k is a diffusivity constant.
Assuming conservation of mass,
∂C/∂t = ∂Q/∂y .
Combined: ∂C/∂t = ∂/∂y (k∂C/∂y).
If diffusivity constant in y, then ∂C/∂t = k(∂2C/∂y2).
If concentration is constant in time, then ∂C/∂t = 0, and ∂2C/∂y2 = 0.
Concentration profile obtained by twice integrating for boundary conditions:
C = K1y + K2
It can be shown that K1 = -C0/h, where C0 is the concentration at the bed.
K2 = C0 .
C = C0 (1 – y/h) This profile is linear in depth.

17. Mississippi River at St. Louis
SUSPENDED LOAD
C = C0 (1 – y/h)
This profile is linear in depth.
Observed suspended sediment concentration
profiles are not linear in depth.
We have ignored the settling of grains.
Concentration profile reflects balance of
upward diffusion and gravitational settling
of grains.
When C is constant in time, then any loss of sediment due to settling is
balanced by upward diffusion of sediment.
Settling flux is wC. Upward diffusion flux Q = -k dC/dy.
wC = -k dC/dy ,
or dC/C = -wdy/kt , where kt = be, and e is the kinematic eddy viscosity.
b ≈ 1, and e = u*[(h-y)/h]ky . Von Karman’s k = 0.4 .
dC/C = whdy/[bku*(h-y)y] .
Mississippi River at St. Louis

18. SUSPENDED LOAD dC/C = whdy/[bku*(h-y)y] At reference height a,
C = Ca . Integrate:
C/Ca = [h-y/y × a/h-a]w/bku*.
This gives suspended sediment
concentration at any depth in
flow in relation to concentration
at reference depth. Only need to
know a and Ca.
The grouping w/bku* is the Rouse number.
Since b ≈ 1, and k = 0.4, w = u* for a Rouse number of 2.5.
This is the criterion for suspension.
Rouse number > 2.5: w > u* with bedload transport dominant.
High bed roughness → high shear velocity → high suspended sediment conc.
High viscosity → low settling velocity → high suspended sediment conc.

19. SUSPENDED LOAD dC/C = whdy/[bku*(h-y)y] At height a close to bed,
C = Ca
Then, integration gives:
C/Ca = [h-y/y × a/h-a]w/bku*
This gives suspended sediment
concentration at any depth in
flow in relation to concentration
at reference depth. Only need to
know Ca.
Suspended sediment transport rate is product of the mass of suspended
sediment ms in a column of water over a unit area of bed and the depth-
averaged flow velocity U at the station:
Qs = msU
Assumes that all sediment is
mobilized from bed.