457.562 Special Issue on River Mechanics (Sediment Transport)

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Seoul National University

457.562 Special Issue on River Mechanics (Sediment Transport)

.05 Macroscopic view of sediment transport

Prepared by Jin Hwan Hwang

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1. Submerged Angle of Repose

§ If granular particles are allowed to pile up while submerg ed in a fluid, there is a specific slope angle beyond which spontaneous failure of the slope occurs.

§ This angle is the angle of repose (or friction angle).

§ The coefficient of Coulomb friction

µ = tangential resistive force downward normal force

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§ The forces acting on the particle along the slope are:

– The submerged force of gravity, which has a downslope co mponent F_gt and normal component F_gn

– A tangential resistive force F_r due to Coulomb friction.

§ The condition for incipient motion is

– Down slope impelling force of gravity should just barely bal ance with the Coulomb resistive force.

F_gt =

(

ρ_s − ρ

)

^gV^p ^sin^φ

F_gn =

(

ρ_s − ρ

)

^gV^p ^cos^φ

F_r = µF_gn F_gt = F_r

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§ From the previous four relations, it is found

§ The angle of repose is an empirical quantity.

§ Test with well-sorted material indicate that this angle is n ear 30 degree for sand, gradually increasing to 40 for gra vel.

§ Poorly-sorted angular material tends to interlock, giving g reater resistance to failure, and as a result, a higher fricti on angle of repose.

µ = tanφ

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2. Critical Stress for Flow over a Granular Bed

§ When a granular bed is subjected to a turbulent flow, it is found that virtually no motion of the grains is observed at some flows, but that the bed is noticeably mobilized at ot her flows.

§ Factor that affect the mobility of grains subjected to a flo w are

– Randomness: including grain placement, and turbulent – Forces on grains: Gravity and fluid

• Fluid : lift and drag (mean & turbulent)

§ In the presence of turbulent flow,

– random fluctuations typically prevent the clear definition of a critical or threshold condition for motion

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2. Critical Stress for Flow over a Granular Bed

§ But, still we need some conditions for describing it.

§ Ikeda-Coleman-Iwagaki model for threshold conditions.

§ Assumptions:

– The forces acting on a particle taken to “dangerously place d.

– The flow is taken to follow the turbulent rough logarithmic l aw near the boundary.

– Turbulent forces on the particle are neglected.

– Drag and lift forces act through the particle center – On drag force, boundary effects can be neglected – Very low streamwise slopes

– Roughness height is equated to diameter of sediment

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2. Critical Stress for Flow over a Granular Bed

§ Particle is centered at z=D/2

§ To solve this problem, it is ne cessary to use some informati on about turbulent boundary l ayers to to define effective flui d velocity u

_f

acting on the part icle.

§ In the viscous sublayer, a linear velocity profile

u

u_* = u_*z ν

u_f

u_* = 1 2

u_*D

ν when D

δ_ν = u_*D

ν < 0.5

⎛

⎝⎜

⎞

⎠⎟

D /δ_ν > 2 then no viscous sublayer exist.

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2. Critical Stress for Flow over a Granular Bed

§ If no viscous sublayer exists, apply the rough pipe log v elocity profile

§ Then the general form for the both cases,

§ Where

F = 1 2

u_*D

ν for u_*D

ν <13.5 F = 6.77 for u_*D

ν >13.5 u_f

u_* = 2.5 ln 30 z D

⎛⎝⎜ ⎞

⎠⎟ _z=¹

2D

= 6.77

u_f

u_* = F u_*D ν

⎛⎝⎜ ⎞

⎠⎟

Since rough flow

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§ However, it would be more convenient to have a contin uous function

§ Forces acting on the particle

§ Coulomb resistive force

F = 2ν u_*D

⎛

⎝⎜

⎞

⎠⎟

10/3

+ κ ⁻¹ln 1+

9 2

u_*D ν 1+ 0.3u_*D

ν

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎡

⎣

⎢⎢

⎢

⎤

⎦

⎥⎥

⎥

−10/3

⎧

⎨⎪⎪

⎩⎪

⎪

⎫

⎬⎪⎪

⎭⎪

⎪

−0.3

D_f = ρ 1

2π D 2

⎛⎝⎜ ⎞

⎠⎟

2

c_Du²_f , L_f = ρ 1

2π D 2

⎛⎝⎜ ⎞

⎠⎟

2

c_Lu²_f, F_g = ρRg 4

3π D 2

⎛⎝⎜ ⎞

⎠⎟

3

F = µ

(

F − L

)

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§ For a channel with a downstream slope angle, the dow nslope effect of gravity has to be included in the force b alance presented earlier, resulting in

!

_"^∗

= 4 3

'

(

₎

+ '(

₊

1

-

^.

(0

_∗"

1/3)

§ This is dimensionless critical shear stress and called as

“the Shields parameter”

τ_c^* = τ_bc ρgRD

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§ The critical condition for incipient motion of the particle i s that the impelling drag force is just balanced by the re sisting Coulomb frictional force:

§ With all forces above,

§ Since

§ Ikeda-Coleman-Iwagaki

u²_f

RgD = 4 3

µ c_D + µc_L D_f = F_r

u_f

u_* = F u_*D

⎛ ν

⎝⎜ ⎞

⎠⎟ and τ_b = ρu_*² τ_c^* = 4

3

µ c + µc

( )

^F²

(

^u ¹^D ^/^ν

)

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§ Comparison of Ikeda-Coleman-Iwagaki model for initiatio n of motion with Shields data

φ = 40° and 60°

k_s = 2D C_L = 0.85C_D

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3. Shields Diagram

§ Shields elucidate the condition for which sediment grins would be at the verge of moving.

§ He used the fundamental concepts of similarity and dime nsional analysis.

§ He said shield number should be function of shear Reyn olds number

§ Key parameters for the system are

§ So simply

τ_c^* ~ function u_*cD

⎛ ν

⎝⎜ ⎞

⎠⎟

τ_bc, γ and γ _s, D, u_*c = τ_bc / ρ,ν τ_c^* = τ_bc

ρgRD = F_* u_*cD

⎛ ν

⎝⎜ ⎞

⎠⎟

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3. Shields Diagram

§ But this form is not practical,

§ Therefore, Vanoni (1964) proposed

§ Which was used in falling velocity relations

τ_c^* = τ_bc

ρgRD = F_* u_*cD

⎛ ν

⎝⎜ ⎞

⎠⎟

u_*D

ν = u_* gRD

gRDD

ν =

( )

τ^* ^1/2 ^R^ep

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3. Shields Diagram

§ .

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4. Granular Sediment on a Sloping Bed

§ The work of Shields on initiation of motion applied only to the case of nearly horizontal slopes.

§ Most streams, particularly in mountain areas, have steep gradients, creasing a need to account for the effect of th e downslope component of gravity on the initiation motio n.

§ Wiberg and Smith (1987)

τ_c^* = 4 3

tanφ₀ cosα − sinα

( )

C_D + tanφ₀^C_L

( )

^F²^(z¹^/ ^z⁰⁾

α =Bed slope angle

φ₀ = Angle of repose of the grains

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§ The effect of the streamwise bed slope on incipient sedi ment motion can be illustrated by considering the forces active on a particle lying in a bed consisting of similar par ticles over which water flows.

§ Chiew and Parker (1994)

τ_c^*_α

τ_co^* ⁼ ^cosα ¹− tanα tanφ

⎛

⎝⎜

⎞

⎠⎟

φ ^: Repose angle

τ_c^*_α : critical shear stress for sediment on a bed with a longitudinal slope angle α

τ_co^* : critical shear stress for a bed with very small slope

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§ In terms of shear velocities,

§ Right figure was done by Chiew and Parker in duct laboratory experiment

u_*c

u_*o = cosα 1− tanα tanφ

⎛

⎝⎜

⎞

⎠⎟

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5. Threshold Condition on Side Slopes

§ Until now, we just talk the horizontal in the transverse dir ection.

§ In engineering application, side wall slope must be impor tant.

§ Simplification of the analysis:

– Logarithmic upward normal from the bed.

D_f = ρ 1

2π D 2

⎛⎝⎜ ⎞

⎠⎟

2

c_Du²_f e

1 (Drag force) F_g = F_g₂e

2 + F_g₃e

3 (Gravitational force) F_g2,F_g₃

( )

⁼ ⁻^ρ^Rg ⁴₃^π ^⎛_⎝⎜ ^D₂ ^⎞_⎠⎟³

(

^sinθ^,cosθ

)

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5. Threshold Condition on Side Slopes

§ The lift force is given as

§ Coulomb resistive force must balance the sum of the imp elling forces du to flow and transverse downslope pull of gravity at the critical conditions

§ With the previous terms

L_f = ρ ¹

2π ^D 2

⎛⎝⎜ ⎞

⎠⎟

2

c_Lu²_f e

3

µ² F_g₃e

3 + L_f ² = D_f ² + F_g₂e

3 2

u²_f RgD

⎛

⎝⎜

⎞

⎠⎟

2

+ 4

3c_D sinθ

⎛

⎝⎜

⎞

⎠⎟

⎡ 2

⎣

⎢⎢

⎤

⎦

⎥⎥

1/2

= µ ⁴

3c_D cosθ − c_L c_D

u²_f RgD

⎛

⎝⎜

⎞

⎠⎟

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5. Threshold Condition on Side Slopes

§ With some manipulations

§ The case of a transversely horizontal bed is recovering b y setting θ=0. The critical Shields stress if found as in th e previous

§ And denotes with the subscript of o.

τ_c^*

( )

² ⁺ _3c⁴

DF² sinθ

⎛

⎝⎜

⎞

⎠⎟

⎡ 2

⎣⎢

⎢

⎤

⎦⎥

⎥

1/2

= 4µ

3c_DF² cosθ − µ ^c^L c_D τ_c^*

τ_c^* = 4 3

µ^cosα −sinα

( )

c_D + µ^c_L

( )

^F²

(

^u^*c¹^D ^/^ν

)

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5. Threshold Condition on Side Slopes

§ Then

τ_c^* τ_co^*

⎛

⎝⎜

⎞

⎠⎟

2

+

(

1+ µc_L /c_D

)

µ ^sinθ

⎛

⎝⎜

⎞

⎠⎟

⎡ 2

⎣

⎢⎢

⎤

⎦

⎥⎥

1/2

= 1+ µ ^c^L c_D

⎛

⎝⎜

⎞

⎠⎟ cosθ − µ ^c^L c_D

τ_c^* τ_co^*

⎛

⎝⎜

⎞

⎠⎟

When µ = 0.84 (φ = 40°), c_L = 0.85c_D. (c_D can be found in diagram)