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Physics of sediment transport

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