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

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

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

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

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

6. Definitions

7. 1. Dynamic and Kinematic Viscosity
The Dynamic Viscosity μ is a measure of how much a fluid resists shear. It has units of kg m-1 s-1
The Kinematic viscosity ν is defined
where ρf is the density of the fluid. ν has units of m2 s-1, the units of a diffusion coefficient. It measures how quickly velocity perturbations diffuse through the fluid.

8. 2. Molecular and Eddy Viscosities
Molecular kinematic viscosity: property of FLUID
The molecular kinematic viscosity (usually referred to just as the ‘kinematic viscosity’), ν is an intrinsic property of the fluid and is the appropriate property when the flow is laminar. It quantifies the diffusion of velocity through the collision of molecules. (It is what makes molasses viscous).
The Eddy Kinematic Viscosity, νe is a property of the flow and is the appropriate viscosity when the flow is turbulent flow. It quantities the diffusion of velocity by the mixing of “packets” of fluid that occurs perpendicular to the mean flow when the flow is turbulent
Eddy kinematic viscosity:
property of FLOW
In flows in nature (ocean), eddy viscosity is MUCH MORE IMPORTANT!
Like, 104 times more important

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

10. Sediment Settling

11. Settling Velocity: Stokes settling
Settling velocity (ws) from the balance of two forces - gravitational (Fg) and drag forces (Fd)

12. Settling Speed Balance of Forces
Write balance using relationships on last slide
k is a constant
Use definitions of specific gravity, R and kinematic viscosity ν
k turns out to be 1/18

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

14. Boundary Layers

16. Bottom Boundary Layers
The layer (of thickness δ) in which velocities change from zero at the boundary to a velocity that is unaffected by the boundary
δ is likely the water depth for river flow.
δ is a few tens of meters for currents at the seafloor
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)

17. Shear stress in a fluid
Shear stresses at the seabed lead to sediment transport
force
rate of change of momentum
τ = shear stress = =
area
area

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 τ directly

19. The log (turbulent intermediate) layer
Generally from about 1-5 mm to 0.1δ (a few meters) above bed
Dominated by turbulent eddies
Can be represented by:
where νe 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

20. Shear velocity u*
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
Turbulent motions (and therefore νe) are constrained to be proportional to the distance to the bed z, with the constant, κ, the von Karman constant which has a value of 0.4

21. Velocity distribution of natural (rough) boundary layers
From the equations on the previous slide we get
Integrating this yields
z0 is a constant of integration. It is sometimes called the roughness length because it is generally proportional to the particles that generate roughness of the bed (usually z0 = 30D)

22. What the log-layer actually looks like
Slope = κ/u*
= 0.4/u*
lnz0
Plot ln(z) against the mean velocity u to estimate u* and then estimate the shear stress from Z0

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

25. Hjulström Diagram

26. Shields stress and the critical shear stress
The Shields stress, or Shields parameter, is:
Shields (1936) first proposed an empirical relationship to find θc, the critical Shields shear stress to induce motion, as a function of the particle Reynolds number,
Rep = u*D/ν

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

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