12.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 IV. Essentials of Sediment Transport A. Non-Dimensional Variables Motivational Example: Synthetic Sediment Transport Data (PowerPoint Slides) 1. Reynold’s Numbers All Reynold’s numbers are of the form: viscositylengthvelocityR*=a. Channel Reynold’s number (turbulence). A channel Reynold’s number marks the onset of turbulence !µ"huhuRe==Re > 500: turbulent open channel flow; Re > 2000: turbulent pipe flow Since Re is dimensionless, it applies equally to all flows; it is exactly equivalent to double velocity u, double depth h, double density ρ, or halve viscosity µ. Non-dimensional variables useful precisely because of this generality. b. Particle Reynold’s number (particle suspension, initiation of motion). The Particle Reynold’s number, Rp, uses settling velocity, ws, and particle diameter, D, as the velocity and length scales: Rp=wsD!c. Shear Reynold’s number (initiation of motion). The Shear Reynold’s number, R*, uses shear velocity, u*, and particle diameter, D, as the velocity and length scales: !DuR**=d. Explicit Particle Reynold’s number (initiation of motion, settling velocity). The Explicit Particle Reynold’s number, ( )( )gDs!!!"Rep, uses the expression , which has units of velocity, and particle diameter, D, as the velocity and length scales: ( )( )!"""DgDRsep#=112.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 2. Froude Number The Froude Number is the ratio of inertial to gravitational forces: ghuFr=Note: ghis the celerity of waves Fr < 1: ghu <Fr = 1: ghu =Fr > 1: ghu >; “sub-critical”, waves (and other information) can travel upstream (normal alluvial conditions, Fr < 0.5). ; “critical”, standing waves ; “super-critical”, waves (and other information) can not travel upstream. (Steep channels, bedrock channels) Sub-critical flow transitions to critical when “shooting” over a wier: The flow suddenly transitions back to sub-critical and thus must suddenly increase in depth – this is called a “hydraulic jump”. Discharge over a weir is easily determined by measuring flow depth and width of the weir, because velocity is known ( ) because Fr = 1 at the weir. ghu =3. Rouse Number (mode of sediment transport) The Rouse number dictates the mode of sediment transport. It is the ratio of particle settling velocity to the shear velocity (rate of fall versus strength of turbulence acting to suspend particles): *#kuwRouses=; k = 0.4 (Von Karman’s constant) Bedload: 5.2*>kuws50% Suspended: 5.22.1*<<kuws100% Suspended: 2.18.0*<<kuwsWash Load: 8.0*<kuws212.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 4. Non-Dimensional Settling Velocity Several different non-dimensional groupings are used in describing the controls on settling velocity. The standard non-dimensional settling velocity uses the group ( )( )gDs!!!"to accomplish the non-dimensionalization: ( )( )gDwwsss!!!"=*Dietrich et al (1983) is a key paper tabulating particle settling velocity dependencies on grain size and shape and uses a related variable W* as their non-dimensional settling velocity: W*= ws*2Rp=ws3!s"!( )!( )g#However, there is a complication since both W* and the drag coefficient CD depend on particle Reynold’s number, Rp. Therefore, some workers use the Explicit Particle Reynold’s number, which is related to the non-dimensional settling velocity, ws*, by the relation: Rp=wsD!= ws*RepExcel spreadsheet for calculating settling velocity using equations in Dietrich et al (1983) is available on the class website. 5. Shield’s Stress (sediment transport, initiation of motion). Initiation of motion and sediment transport must depend on, at least: boundary shear stress, sediment and fluid density (buoyancy), and grain-size. Early 1900’s Shields (German) did many experiments on sediment transport and determined a non-dimensional grouping that combines these factors and served to collapse a great range of experimental data to a single curve: ( )gDsb!!""#=*Where boundary shear stress can be approximated by the relation for steady-uniform flow, Shields Stress, τ*, can be written as: ( )( )DhSs!!!"#=*312.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 At the critical condition for initiation of motion, shear stress = τcr, the critical Shields stress is of course: !*cr=!cr"s#"( )gDShields plotted against the Shear Reynold’s number, R*, in his original work. This nicely collapses the data, but is difficult to work with in practice, because both τ* and R* depend on u*, meaning iteration is required to find τcr from the plot (recall !"bu =*). Therefore, Shield’s diagram is usually recast in terms of the Explicit Particle Reynold’s number by plotting against: ( )( )232**!"""#gDRDsep$===6. Non-Dimensional Sediment Transport Rate Qs = total volumetric sediment transport rate through a given river cross section. Sediment flux per unit channel width is by definition: wQqss=Einstein (the son) worked on the sediment transport problem and first defined the non-dimensional volumetic sediment flux as: qs*=qs!s"!( )!( )gD D=qsRep#We will write all sediment transport relationships in terms of this (or very similar) non-dimensional group. 7. Transport Stage Transport stage describes the intensity of sediment transport and is defined simply as the ratio of boundary shear stress to the critical boundary shear stress: crcrbsT**!!!!==412.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 B. Sediment Transport Relations 1. Bedload Transport: rolling, sliding, saltating Generally: ( )( )!!!"/,,**#=sepsRfqTheoretical relations have been developed, and volumetric flux solved by integrating individual grain motions – much is known about bedload sediment transport. In this class we will restrict ourselves to empirical relations determined in the lab and in the field. They must be applied only to conditions similar to those under which they were determined. a. Meyer-Peter Mueller (1948) (generalized) ( )23***8crsq!!"=Where for gravels, τcr* is a constant: Shields (gravel) ~ 0.06; Parker ~ 0.03 (mixed size gravel); Meyer-Peter Mueller = 0.047 (well sorted fine gravel, at moderate transport stage, Ts ~ 8). b. Fernandez-Luque and van Beck (1976) ( )23***7.5crsq!!"=conditions similar to M-P-M, only at low transport stage (Ts ~ 2). c. Wilson (1966) (