Bed particle stability The Shields criteiron

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Bed particle stability
The Shields criteiron
Forces on a submerged particle
PARTICLE (SUBMERGED) WEIGHT:
π‘Š = 𝑐2 𝛾𝑠 − 𝛾𝑀 𝑑 3
HYDRAULIC FORCE:
𝐹𝐷 = 𝑐1 𝜏0 𝑑 2
DRIVING FORCES: gravitational plus
hydraulic force
𝐹𝐷 + π‘Š sin 𝛽
RESISTING FORCES: frictional
resistance
π‘Š cos 𝛽 tan πœ™
𝛽
Incipient motion of a submerged particle
Motion occurs when driving forces = resisting
forces
𝐹𝐷 + π‘Š sin 𝛽
=1
π‘Š cos 𝛽 tan πœ™
Substitute and simplify, taking πœπ‘ = 𝜏0 at
incipient motion
𝑐2
πœπ‘ =
𝛾 − 𝛾𝑀 𝑑 cos 𝛽(tan πœ™ − tan 𝛽)
𝑐1 𝑠
For flat bed, 𝛽 = 0. Combining 𝑐1 and 𝑐2 ,
πœπ‘ = 𝑐 tan πœ™ 𝛾𝑠 − 𝛾𝑀 𝑑
Or
πœπ‘
= 𝑐 tan πœ™ = constant
𝛾𝑠 − 𝛾𝑀 𝑑
𝛽
Shields criterion
DEFINE: Shields stress (or Shields number)
𝜏
0
∗
𝜏 =
𝛾𝑠 − 𝛾𝑀 𝑑
And the “critical” Shields stress:
πœπ‘
∗
πœπ‘ =
𝛾𝑠 − 𝛾𝑀 𝑑
MEANING: The critical shear stress for incipient
motion of cohesionless bed material normalized
to the particles’ submerged unit weight.
πœπ‘∗ ranges from 0.03-0.06 for coarse sand-gravel rivers
Shields’ diagram
Relates dimensionless critical shear stress to the boundary Reynold’s
number for flow over submerged grains.
Dimensional analysis
Buckingham Π theorem:
if there is a physically meaningful equation involving a certain
number 𝑛 of physical variables represented by π‘˜ physical
dimensions, then the original equation can be rewritten in
terms of a set of 𝑝 = 𝑛 − π‘˜ dimensionless parameters
Π1 , Π2 , … Π𝑝 constructed from the original variables.
For incipient motion problem on level bed:
PHYSICAL VARIABLES: πœπ‘ , 𝛾𝑠 − 𝛾𝑀 , πœŒπ‘€ , 𝑑, 𝜈; 𝑛 = 5
DIMENSIONS: mass, length, time [𝑀, 𝐿, 𝑇]; π‘˜ = 3
By BPT, 𝑝 = 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:
πœπ‘ 1/2
𝑑
πœπ‘
πœŒπ‘€
𝐹
,
=0
𝛾𝑠 − 𝛾𝑀 𝑑
𝜈
Since 𝑒∗ =
𝜏/𝜌, this can be written
πœπ‘
𝑒𝑐∗ 𝑑
=𝐹
𝛾𝑠 − 𝛾𝑀 𝑑
𝜈
Or more compactly
πœπ‘∗ = 𝐹(𝑅𝑒 ∗ )
This final function is what the Shields diagram shows
Shields’ diagram
πœπ‘∗ ranges from 0.03-0.06 for coarse sand-gravel rivers.
For large 𝑅𝑒 ∗ , πœπ‘∗ is constant!
Vertical axis: Critical Shields
stress for particle motion
Horizontal axis: Boundary
Reynolds number. Note that
𝛿 = 11.6𝜈/𝑒∗ , 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.
𝑹𝒆∗
𝒅
= 𝟎. πŸŽπŸ–πŸ”
𝜹
Using Shields criterion
1. Find πœπ‘∗ :
Consider coarse silica sand, 𝑑 = 2 mm in a
flow with 𝑒∗ = 0.1 m/s.
0.1 × 0.002
𝑅𝑒 =
= 199.2
1.004 × 10−6
∗
From diagram, πœπ‘∗ ≃ 0.05
2. Compute 𝜏 ∗ and compare
∗2
𝜏
𝜌
𝑒
0
𝑀
𝜏∗ =
=
𝛾𝑠 − 𝛾𝑀 𝑑 16187 × 0.002
10
=
= 0.31
32.37
3. Since 𝜏 ∗ > πœπ‘∗ , this sand will be
transported.
How does the Shields criterion compare with Hjulstrom?
The upper “erosion velocity” curve from Hjulstrom is similar to a particular
case of a dimensionalized Shields curve.
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