Bed particle stability
The Shields criteiron
Forces on a submerged particle
PARTICLE (SUBMERGED) WEIGHT:
𝑊 = 𝑐2 𝛾𝑠 − 𝛾𝑤 𝑑 3
HYDRAULIC FORCE:
𝐹𝐷 = 𝑐1 𝜏0 𝑑 2
DRIVING FORCES: gravitational plus
hydraulic force
𝐹𝐷 + 𝑊 sin 𝛽
RESISTING FORCES: frictional
resistance
𝑊 cos 𝛽 tan 𝜙
𝛽
Incipient motion of a submerged particle
Motion occurs when driving forces = resisting
forces
𝐹𝐷 + 𝑊 sin 𝛽
=1
𝑊 cos 𝛽 tan 𝜙
Substitute and simplify, taking 𝜏𝑐 = 𝜏0 at
incipient motion
𝑐2
𝜏𝑐 =
𝛾 − 𝛾𝑤 𝑑 cos 𝛽(tan 𝜙 − tan 𝛽)
𝑐1 𝑠
For flat bed, 𝛽 = 0. Combining 𝑐1 and 𝑐2 ,
𝜏𝑐 = 𝑐 tan 𝜙 𝛾𝑠 − 𝛾𝑤 𝑑
Or
𝜏𝑐
= 𝑐 tan 𝜙 = constant
𝛾𝑠 − 𝛾𝑤 𝑑
𝛽
Shields criterion
DEFINE: Shields stress (or Shields number)
𝜏
0
∗
𝜏 =
𝛾𝑠 − 𝛾𝑤 𝑑
And the “critical” Shields stress:
𝜏𝑐
∗
𝜏𝑐 =
𝛾𝑠 − 𝛾𝑤 𝑑
MEANING: The critical shear stress for incipient
motion of cohesionless bed material normalized
to the particles’ submerged unit weight.
𝜏𝑐∗ 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 𝑘 physical
dimensions, then the original equation can be rewritten in
terms of a set of 𝑝 = 𝑛 − 𝑘 dimensionless parameters
Π1 , Π2 , … Π𝑝 constructed from the original variables.
For incipient motion problem on level bed:
PHYSICAL VARIABLES: 𝜏𝑐 , 𝛾𝑠 − 𝛾𝑤 , 𝜌𝑤 , 𝑑, 𝜈; 𝑛 = 5
DIMENSIONS: mass, length, time [𝑀, 𝐿, 𝑇]; 𝑘 = 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:
𝜏𝑐 1/2
𝑑
𝜏𝑐
𝜌𝑤
𝐹
,
=0
𝛾𝑠 − 𝛾𝑤 𝑑
𝜈
Since 𝑢∗ =
𝜏/𝜌, this can be written
𝜏𝑐
𝑢𝑐∗ 𝑑
=𝐹
𝛾𝑠 − 𝛾𝑤 𝑑
𝜈
Or more compactly
𝜏𝑐∗ = 𝐹(𝑅𝑒 ∗ )
This final function is what the Shields diagram shows
Shields’ diagram
𝜏𝑐∗ ranges from 0.03-0.06 for coarse sand-gravel rivers.
For large 𝑅𝑒 ∗ , 𝜏𝑐∗ is constant!
Vertical axis: Critical Shields
stress for particle motion
Horizontal axis: Boundary
Reynolds number. Note that
𝛿 = 11.6𝜈/𝑢∗ , 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. 𝟎𝟖𝟔
𝜹
Using Shields criterion
1. Find 𝜏𝑐∗ :
Consider coarse silica sand, 𝑑 = 2 mm in a
flow with 𝑢∗ = 0.1 m/s.
0.1 × 0.002
𝑅𝑒 =
= 199.2
1.004 × 10−6
∗
From diagram, 𝜏𝑐∗ ≃ 0.05
2. Compute 𝜏 ∗ and compare
∗2
𝜏
𝜌
𝑢
0
𝑤
𝜏∗ =
=
𝛾𝑠 − 𝛾𝑤 𝑑 16187 × 0.002
10
=
= 0.31
32.37
3. Since 𝜏 ∗ > 𝜏𝑐∗ , 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.