This document summarizes key concepts related to sediment transport in rivers and oceans. It defines terms like viscosity, boundary layers, shear stress, and Shields stress. It explains how sediment size is characterized and settles according to Stokes' law. The structure of bottom boundary layers is described, including the log layer where shear stress can be measured. The Shields curve is introduced to predict whether sediment will move by bedload, suspension or not at all based on shear stress and particle size.
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This years’ keynote speaker of the Delft3D Users Meeting is Prof. Rudy L Slingerland, Pennsylvania State University, USA.
For more than 37 years, Prof. Rudy L. Slingerland of the Pennsylvania State University, USA, has been active as a scientist, educator and academic leader. He has held numerous positions within the University including head of the Department of Geosciences. His research group currently studies the evolution of morphodynamic systems, including tectonically-driven landscapes, deltas, rivers and shallow marine shelves by coupling theory with observations in the field and subsurface. The group’s ultimate goal is to develop predictive theories for the behavior of these systems and the record of their deposits. He was also recognized with the 2012 G. K. Gilbert Award for Geomorphology from the American Geophysical Union (AGU), which honors a scientist who has made a significant contribution to the field of earth and planetary surface processes. In 2013, he was elected a Fellow of the AGU.
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YouTube talk https://youtu.be/cUkXYlXpW4E
https://meetings.aps.org/Meeting/MAR21/Session/J25.4
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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
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
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
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
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*
= 04/u*
22
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