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Bed particle stability The Shields criteiron Forces on a submerged particle PARTICLE (SUBMERGED) WEIGHT: π = π2 πΎπ β πΎπ€ π 3 HYDRAULIC FORCE: πΉπ· = π1 π0 π 2 DRIVING FORCES: gravitational plus hydraulic force πΉπ· + π sin π½ RESISTING FORCES: frictional resistance π cos π½ tan π π½ Incipient motion of a submerged particle Motion occurs when driving forces = resisting forces πΉπ· + π sin π½ =1 π cos π½ tan π Substitute and simplify, taking ππ = π0 at incipient motion π2 ππ = πΎ β πΎπ€ π cos π½(tan π β tan π½) π1 π For flat bed, π½ = 0. Combining π1 and π2 , ππ = π tan π πΎπ β πΎπ€ π Or ππ = π tan π = constant πΎπ β πΎπ€ π π½ Shields criterion DEFINE: Shields stress (or Shields number) π 0 β π = πΎπ β πΎπ€ π And the βcriticalβ Shields stress: ππ β ππ = πΎπ β πΎπ€ π MEANING: The critical shear stress for incipient motion of cohesionless bed material normalized to the particlesβ submerged unit weight. ππβ ranges from 0.03-0.06 for coarse sand-gravel rivers Shieldsβ diagram Relates dimensionless critical shear stress to the boundary Reynoldβs number for flow over submerged grains. Dimensional analysis Buckingham Ξ theorem: if there is a physically meaningful equation involving a certain number π of physical variables represented by π physical dimensions, then the original equation can be rewritten in terms of a set of π = π β π dimensionless parameters Ξ 1 , Ξ 2 , β¦ Ξ π constructed from the original variables. For incipient motion problem on level bed: PHYSICAL VARIABLES: ππ , πΎπ β πΎπ€ , ππ€ , π, π; π = 5 DIMENSIONS: mass, length, time [π, πΏ, π]; π = 3 By BPT, π = 2, i.e., the system can be represented with two dimensionless parameters constructed from the 5 physical variables Constructing the Shields diagram From dimensional analysis, the equation relating incipient particle motion to fluid flow properties is: ππ 1/2 π ππ ππ€ πΉ , =0 πΎπ β πΎπ€ π π Since π’β = π/π, this can be written ππ π’πβ π =πΉ πΎπ β πΎπ€ π π Or more compactly ππβ = πΉ(π π β ) This final function is what the Shields diagram shows Shieldsβ diagram ππβ ranges from 0.03-0.06 for coarse sand-gravel rivers. For large π π β , ππβ is constant! Vertical axis: Critical Shields stress for particle motion Horizontal axis: Boundary Reynolds number. Note that πΏ = 11.6π/π’β , so the horizontal axis may also be interpreted more intuitively as (1/11.6) times the ratio of particle diameter to the thickness of the viscous sublayer. πΉπβ π = π. πππ πΉ Using Shields criterion 1. Find ππβ : Consider coarse silica sand, π = 2 mm in a flow with π’β = 0.1 m/s. 0.1 × 0.002 π π = = 199.2 1.004 × 10β6 β From diagram, ππβ β 0.05 2. Compute π β and compare β2 π π π’ 0 π€ πβ = = πΎπ β πΎπ€ π 16187 × 0.002 10 = = 0.31 32.37 3. Since π β > ππβ , this sand will be transported. How does the Shields criterion compare with Hjulstrom? The upper βerosion velocityβ curve from Hjulstrom is similar to a particular case of a dimensionalized Shields curve.