15. Physics of Sediment Transport William Wilcock
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1 15. Physics of Sediment Transport William Wilcock (based in part on lectures by Jeff Parsons) OCEAN/ESS 410 Lecture/Lab Learning Goals Know how sediments are characteried (sie 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 Sediment Characteriation There are number of ways to describe the sie of sediment. One of the most popular is the Φ scale. φ = -log 2 (D) D = diameter in millimeters. To get D from φ D = 2 -φ φ Diameter, D Type of material mm Cobbles mm Coarse Gravel mm Gravel -3 8 mm Gravel -2 4 mm Pea Gravel -1 2 mm Coarse Sand 0 1 mm Coarse Sand mm Medium Sand mm Fine Sand µm Fine Sand 4 63 µm Coarse Silt 5 32 µm Coarse Silt 6 16 µm Medium Silt 7 8 µm Fine Silt 8 4 µm Fine Silt 9 2 µm Clay Sediment Characteriation Sediment grain smoothness Sediment grain shape - spherical, elongated or flattened Sediment sorting 1
2 Sediment Transport Two important concepts Gravitational forces - sediment settling out of suspension Current-generated bottom shear stresses - sediment transport in suspension or along the bottom (bedload) Definitions Shield stress - brings these concepts together empirically to tell us when and how sediment transport occurs 1. Dynamic and Kinematic Viscosity 2. Molecular and Eddy Kinematic Viscosities 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! = µ " f 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). where ρ f is the density of the fluid. ν has units of m 2 s -1, the units of a diffusion coefficient. It measures how quickly velocity perturbations diffuse through the fluid The Eddy Kinematic Viscosity, ν e 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 2
3 3. Submerged Specific Gravity, R R =! p "! f! f! p Typical values: Sediment Settling! a f Quart = Kaolinite = 1.6 Magnetite = 4.1 Coal, Flocs < 1 Stokes settling Settling velocity (w s ) from the balance of two forces - gravitational (F g ) and drag forces (F d ) ( ) " ( Settling Speed) "( Molecular Dynamic Viscosity) F d! Diameter! Dw s µ ( ) " ( Volume) "( Acceleration of Gravity) F g! Excess Density! (# p $ # f )Vg! # p $ # f ( ) D 3 g! means "proportional to" F d = F g Dw s µ = k (! p "! f ) D 3 g ( w s = k! p "! f ) D 2 g µ ( ) w s = k! p "! f! f! f µ D2 g w s = 1 RgD 2 18 # 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 3
4 Limits of Stokes Settling Equation 1. Assumes smooth 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 and has a bigger effect than decrease in specific gravity as a result of voids in floc. 4. Assumes laminar flow (ignores turbulence) Shear Stresses y Bottom Boundary Layers The layer (of thickness δ) in which velocities change from ero at the boundary to a velocity that is unaffected by the boundary x u Outer region ~ O(!) Intermediate layer Inner region 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)! δ is likely the water depth for river flow. δ is a few tens of meters for currents at the seafloor Shear stress in a fluid Shear stresses at the seabed lead to sediment transport y x force τ = shear stress = = area! = µ "u " = # f$ "u " rate of change of momentum area 4
5 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! = µ "u " = # f$ "u " Unfortunately the inner layer it is too thin for practical field measurements to determine τ directly 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:! u $ = "# e! where ν 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 Shear velocity u * Sediment dynamicists define a quantity known as the characteristic shear velocity, u * "u u 2 * =! e " "u # = $! e " = $u 2 * = Constant The simplest model for the eddy viscosity is Prandtl s model which states that " =! u e * Turbulent motions (and therefore ν e ) are constrained to be proportional to the distance to the bed, with the constant, κ, the von Karman constant which has a value of 0.4 Velocity distribution of natural (rough) boundary layers From the equations on the previous slide we get du 2!" u * =! u * d Integrating this yields ( ) = 1 u *! ln " ln = ln 0 +! u ( ) 0 u * u 0 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 0 = 30D) 5
6 What the log-layer actually looks like Z 0 0.1! ~ 30D not applicable because of free-surface/ outer-flow effects 0.1! ~ 30D not applicable because of free-surface/ outer-flow effects log layer viscous sublayer viscous sublayer U U ln ~30D ln 0 slope = u * /! Plot ln() against the mean velocity u to estimate u * and then estimate the shear stress from! = " f u * 2 Slope = κ/u * = 0.4/u * U Shields Stress When will transport occur and by what mechanism Shields stress and the critical shear stress The Shields stress, or Shields parameter, is: "! f = (# p $ # f )gd Shields curve (after Miller et al., 1977) - Based on empirical observations Transitional Sediment Transport Shields (1936) first proposed an empirical relationship to find θ c, the critical Shields shear stress to induce motion, as a function of the particle Reynolds number, Re p = u * D/ν No Transport Transitional 6
7 Initiation of Suspension If u* > w s, (i.e., shear velocity > Stokes settling velocity) then material will be suspended. Suspension Transitional transport mechanism. Compare u* and w s Bedload No Transport 7
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