Physics of sediment transport

1. 15. Physics of Sediment
Transport
William Wilcock
(based in part on lectures by Jeff Parsons)
OCEAN/ESS 410
1

2. Lecture/Lab Learning Goals
• Know how sediments are characterized (size and
shape)
• Know the definitions of kinematic and dynamic
viscosity, eddy viscosity, and specific gravity
• Understand Stokes settling and its limitation in real
sedimentary systems.
• Understand the structure of bottom boundary layers
and the equations that describe them
• Be able to interpret observations of current velocity in
the bottom boundary layer in terms of whether
sediments move and if they move as bottom or
suspended loads – LAB
2

3. Sediment
Characterization
 Diameter,
D
Type of
material
-6 64 mm Cobbles
-5 32 mm Coarse Gravel
-4 16 mm Gravel
-3 8 mm Gravel
-2 4 mm Pea Gravel
-1 2 mm Coarse Sand
0 1 mm Coarse Sand
1 0.5 mm Medium Sand
2 0.25 mm Fine Sand
3 125 m Fine Sand
4 63 μm Coarse Silt
5 32 m Coarse Silt
6 16 m Medium Silt
7 8 m Fine Silt
8 4 m Fine Silt
9 2 m Clay
• There are number of
ways to describe the
size of sediment.
One of the most
popular is the Φ
scale.
 = -log2(D)
D = diameter in
millimeters.
• To get D from 
D = 2-
3

4. Sediment Characterization
Sediment grain smoothness
Sediment grain shape - spherical, elongated, or flattened
Sediment sorting
4
Grain size
%
Finer

5. Sediment Transport
Two important concepts
•Gravitational forces - sediment settling out of
suspension
•Current-generated bottom shear stresses -
sediment transport in suspension (suspended
load) or along the bottom (bedload)
Shields stress - brings these concepts together
empirically to tell us when and how sediment
transport occurs
5

6. Definitions
6

7. 1. Dynamic and Kinematic Viscosity
The Dynamic Viscosity  is a measure of how much a
fluid resists shear. It has units of kg m-1 s-1
The Kinematic viscosity  is defined
where  f is the density of the fluid has units of m2 s-1,
the units of a diffusion coefficient. It measures how
quickly velocity perturbations diffuse through the fluid.
n =
m
rf
7

8. 2. Molecular and Eddy Viscosities
Molecular kinematic viscosity:
property of FLUID
Eddy kinematic viscosity:
property of FLOW
In flows in nature (ocean), eddy
viscosity is MUCH MORE
IMPORTANT!
About 104 times more important
8

9. 3. Submerged Specific Gravity, R
R =
rp - rf
rf
ra
rp
Typical values:
Quartz = Kaolinite = 1.6
Magnetite = 4.1
Coal, Flocs < 1
f
9

10. Sediment Settling
10

11. Settling Velocity: Stokes settling
Fg µ Excess Density
( )´ Volume
( )
´ Acceleration of Gravity
( )
µ rp - rf
( )Vg µ rp - rf
( )D3
g
Fd µ Diameter
( )´ Settling Speed
( )
´ Molecular Dynamic Viscosity
( )
µ Dwsm
Settling velocity (ws) from the balance of two forces -
gravitational (Fg) and drag forces (Fd)
µmeans "proportional to"
11

12. Settling Speed
Fd = Fg
Dwsm = k rp - rf
( )D3
g
ws = k
rp - rf
( )D2
g
m
ws = k
rp - rf
( )
rf
rf
m
D2
g
ws =
1
18
RgD2
n
Balance of Forces
Write balance using
relationships on last slide
k is a constant
Use definitions of specific
gravity, R and kinematic
viscosity 
k turns out to be 1/18
12

13. Limits of Stokes Settling
Equation
1. Assumes smooth, small, spherical particles - rough
particles settle more slowly
2. Grain-grain interference - dense concentrations
settle more slowly
3. Flocculation - joining of small particles (especially
clays) as a result of chemical and/or biological
processes - bigger diameter increases settling rate
4. Assumes laminar settling (ignores turbulence)
5. Settling velocity for larger particles determined
empirically
13

14. Boundary Layers
14

15. 15

16. Outer region
Intermediate layer
Inner region
d
z ~ O(d)
u
x
y
z
Bottom Boundary Layers
• Inner region is dominated by wall roughness and viscosity
• Intermediate layer is both far from outer edge and wall (log layer)
• Outer region is affected by the outer flow (or free surface)
The layer (of thickness ) in which velocities change from zero at the boundary
to a velocity that is unaffected by the boundary
is likely
the water
depth for
river flow.
is a few
tens of
meters for
currents
at the
seafloor
16

17. Shear stress in a fluid
x
y
z
= shear stress = =
force
area
rate of change of momentum
t = m
¶u
¶z
= rfn
¶u
¶z
area
Shear stresses at the seabed lead to sediment transport
17

18. The inner region (viscous sublayer)
• Only ~ 1-5 mm thick
• In this layer the flow is laminar so the molecular
kinematic viscosity must be used
Unfortunately the inner layer it is too thin for practical field
measurements to determine  directly
t = m
¶u
¶z
= rfn
¶u
¶z
18

19. The log (turbulent intermediate) layer
• Generally from about 1-5 mm to 0.1(a few meters)
above bed
• Dominated by turbulent eddies
• Can be represented by:
where e is “turbulent eddy viscosity”
This layer is thick enough to make measurements and
fortunately the balance of forces requires that the
shear stresses are the same in this layer as in the
inner region
z
u
e
¶
¶
= rn
t
19

20. Shear velocity u*
u*
2
= ne
¶u
¶z
t = rne
¶u
¶z
= ru*
2
= Constant
Sediment dynamicists define a quantity known as the
characteristic shear velocity, u*
The simplest model for the eddy viscosity is Prandtl’s
model which states that
z
u
e *
k
n =
Turbulent motions (and therefore e) are constrained to be
proportional to the distance to the bed z, with the constant,
 , the von Karman constant which has a value of 0.4
20

21. Velocity distribution of natural (rough)
boundary layers
z0 is a constant of integration. It is sometimes called the
roughness length because it is often proportional to the
particles that generate roughness of the bed (a value of
z0 ≈ 30D is sometimes assumed but it is quite variable
and it is best determined from flow measurements)
u z
( )
u*
=
1
k
ln
z
z0
Þ lnz = lnz0 +
k
u*
u z
( )
2
*
* u
dz
du
z
u r
rk =
From the equations on the previous slide we get
Integrating this yields
21

22. What the log-layer actually looks like
lnz
U
~30D
slope = u*/k
not applicable because
of free-surface/
outer-flow effects
0.1d
~ 30D
viscous sublayer
z
U
log layer
not applicable because
of free-surface/
outer-flow effects
0.1d
~ 30D
viscous sublayer
z
U
log layer
Plot ln(z) against the mean velocity
u to estimate u* and then estimate
the shear stress from
t = rf u*
2
Z0
lnz0
Slope =  /u*
= 04/u*
22

23. 23

24. Shields Stress
When will transport occur and by
what mechanism?
24

25. Hjulström Diagram
25

26. Shields stress and the critical shear stress
• The Shields stress, or Shields parameter, is:
• Shields (1936) first proposed an empirical
relationship to find  c, the critical Shields shear stress
to induce motion, as a function of the particle
Reynolds number,
Rep = u*D/
qf =
t
rp - rf
( )gD
26

27. Shields curve (after Miller et al., 1977)
- Based on empirical observations
Sediment Transport
No Transport
27

28. Initiation of Suspension
Suspension Bedload
No Transport
If u* > ws, (i.e., shear velocity > settling velocity) then
material will be suspended.
Transitional transport
mechanism. Compare
u* and ws
28