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National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
Summary of Lectures on Transport of Non-Cohesive Sediment
•What is Morphodynamics?•Sediment Properties•Modes of Transport of Sediment•Equations for Conservation of Bed Sediment•Overview of Fluid Dynamics•Threshold of Motion•Skin Friction and Form Drag•Relations for Bed Load Transport•Relations for Entrainment of Bed Sediment into Suspension•Formulation for Suspended Sediment•Sediment Transport in Wave Boundary Layers•Formulation for Wave-Current Boundary Layers
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
WHAT IS MORPHODYNAMICS?THE ORIGINS OF
MORPHODYNAMICS:DUNE ASYMMETRY
Felix Maria Exner, an Austrian physicist, asked the following question circa 1920.
Why do river dunes have gentle stoss (upstream) faces and steep lee (downstream) faces?
Looking upstream: Lab (SAFL)
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
MORE DUNES: NOTE THE ASYMMETRY
Looking downstream: Field (Amazon basin)Looking upstream: Lab (H. Ikeda)
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
THE PARAMETERSx = streamwise distance [L]t = time [T]η= bed elevation [L]qt = volume total sediment transport rate per unit stream width [L2/T]λp = bed porosity [1]g = acceleration of gravity [L/T2]H = flow depthU = depth-averaged flow velocity [L/T]Cf = bed friction coefficient [1]
The flow changes the bedThe bed changes the flow
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
THE STAGEEnxer equation of bed sediment
continuity
xq
t)1( t
p ∂∂
−=∂η∂
λ−FELIX EXNER WAS
THE FIRST MORPHODYNAMICIST
Sediment transport relation
)U(qq tt =
St. Venant shallow water equations
2f
2
UCx
gHxHgH
21
xHU
tUH
0x
UHtH
−∂η∂
−∂∂
−=∂
∂+
∂∂
=∂∂
+∂∂
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
EXNER REDUCED THE PROBLEM TO A PROBLEM OF NONLINEAR WAVE DYNAMICS
He found1. Dunes like Froude-subcritical flow.2. Dunes migrate downstream as mass waves.3. Dunes are nonlinear waves: migration speed changes with elev, c = c(η)4. In particular, wave speed increases with elevation5. Voilà, the asymmetry evolves on its own!
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
MORPHODYNAMICS: EXPLAIN HOW WATER AND SEDIMENT INTERACT TO MAKE THESE BEAUTIFUL PATTERNS
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
SEDIMENT PROPERTIES
Rio Cordon, Italy
ρs = density of sediment [ML-3]commonly 2.5 ~ 2.8 g/cm3
quartz: 2.65 g/cm3
ρ = density of water [ML-3], ~ 1g/cm3
R = ρs/ρ - 1, [1], ~ 1.65(submerged specific gravity)
D = characteristic grain size [L], mmvs = fall velocity of sediment [LT-1]
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
SEDIMENT SIZE: LOGARITHMIC PHI AND PSI SCALES
)2(n)D(n)D(n2
l
ll ==φ−=ψφ−ψ == 22D
D (mm) ψ φ
4 2 -2
2 1 -1
1 0 0
0.5 -1 1
0.25 -2 2
0.125 -3 3
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
SEDIMENT SIZE RANGES
Type D (mm) ψ φ
< -9 > 9
4 ~ 9
-1 ~ 4
-6 ~ -1
-8 ~ -6
< -8
-9 ~ -4
-4 ~ 1
1 ~ 6
6 ~ 8
> 8
Notes
Clay < 0.002 Usually cohesive
Silt 0.002 ~ 0.0625 Cohesive ~ non-cohesive
Sand 0.0625 ~ 2 Non-cohesive
Gravel 2 ~ 64 “
Cobbles 64 ~ 256 “
Boulders > 256 “
Non-cohesive coastal morphodynamics is mostly about sand
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
SEDIMENT GRAIN SIZE DISTRIBUTIONS
Sample Grain Size Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.01 0.1 1 10
Grain Size mm
Frac
tion
Fine
r
Characterize grain size distribution in terms of N+1 sizes Db,I such that ff,i denotes the fraction in the sample that is finer than size Db,i
i Db,I mm ff,i12345678
0.03125 0.0200.0625 0.0320.0125 0.100
1 0.9702 0.990
0.25 0.420.5 0.834
4 1.000Use logarithmic scale!
Db,4
ff,4
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
CHARACTERISTIC SIZES BASED ON PERCENT FINER
Sample Grain Size Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.01 0.1 1 10
Grain Size mm
Frac
tion
Fine
r
Dx is size such that x percent of the sample is finer than DxExamples:D50 = median sizeD90 ~ roughness height
D50
D90 To find Dx (e.g. D50) find i such that
1i,fi,f f100
xf +≤≤
Then
x2D
f100
xff
x
i,fi,f1i,f
i,b1i,bi,bx
ψ
+
+
=
⎟⎠⎞
⎜⎝⎛ −
−ψ−ψ
+ψ=ψ
D50 = 0.286 mm; D90 = 0.700 mm
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION
Sample Grain Size Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.01 0.1 1 10
Grain Size mm
Frac
tion
Fine
r
N+1 bounds defines N grain size ranges. The ith grain size range is defined by (Db,i, Db,i+1)and (ff,i, ff,i+1)
f4
f4 = 0.444; D4 = 0.354 mm
( )
( ) 2/11i,bi,bi
1i,bi,bi
DDD21
+
+
=
ψ−ψ=ψ
i,f1i,fi fff −= +
ψI (Di) = characteristic size of ith grain size range
fi = fraction of sample in ithgrain size range
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION
Sample Grain Size Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.01 0.1 1 10
Grain Size mm
Frac
tion
Fine
r
= mean grain size on psi scaleσ = standard deviation on psi scaleψ
( )
σ
ψ
=
=
=σ
=
ψ−ψ=σ
ψ=ψ
∑
∑
2
2D
f
f
g
g
N
1ii
2i
2
N
1iii
Dg = geometric mean sizeσg = geometric standard deviation( ≥ 1)Sediment is well sorted if σg < 1.6Dg = 0.296 mm, σg = 1.71
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
SEDIMENT FALL VELOCITY IN STILL WATER
Assume a spherical particle with diameter DThe downstream impelling force of gravity Fg is:
( )ν
==⎟⎠⎞
⎜⎝⎛πρ=
Dv,cc,v2Dc
21F s
vpvpDD2s
2
DD ReRe
3
g 2DRg
34F ⎟
⎠⎞
⎜⎝⎛πρ=
The resistive drag force is
where ν is the kinematic viscosity of the water and cDis specified by the empirical drag curve for spheres
DF
gF
Condition for equilibrium:
gDvs
f R=R2/1
vpDf ]
)(c34[Re
R =Dg FF = where
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
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pDf ]
)(Rec34[R =
SEDIMENT FALL VELOCITY IN STILL WATERUntangle the relation
ν=
DvsvpRe
gDvs
f R=R2/1
vpDf ]
)(c34[Re
R = where andDF
gF ν=
DRgDpRe
pfds
vpDRgD
RgDvDv ReRRe =
ν=
ν= where
Reduce to Rf = Rf(Rep)
Relation of Dietrich (1982):b1 2.891394b2 0.95296b3 0.056835b4 0.002892b5 0.000245
})](n[b)](n[b
)](n[b)(nbb{exp4
p53
p4
2p3p21f
ReRe
ReReR
ll
ll
+−
−+−=
Original relation also includes correction for shape
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
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pDf ]
)(Rec34[R =
SOME SAMPLE CALCULATIONS OF SEDIMENT FALL VELOCITY (Dietrich Relation)
g = 9.81 ms-2
R = 1.65 (quartz)ν = 1.00x10-6 m2s-1 (water at 20 deg Celsius)ρ = 1000 kgm-3 (water)
D, mm vs, cm/s
0.0625 0.330
0.125 1.08
0.25 3.04
0.5 7.40
1 15.5
2 28.3
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
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pDf ]
)(Rec34[R =
MODES OF TRANSPORT OF SEDIMENT
Bed material load is that part of the sediment load that exchanges with the bed (and thus contributes to morphodynamics).Wash load is transported through without exchange with the bed.In rivers, material finer than 0.0625 mm (silt and clay) is often approximated as wash load.
Bed material load is further subdivided into bedload and suspended load.
Bedload:sliding, rolling or saltating just above bedrole of turbulence is indirect
Suspended load:feels direct dispersive effect of eddiesmay be wafted high into the water column
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
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pDf ]
)(Rec34[R =
TRANSPORT DOMINATED BY BEDLOAD(Delta progradation at SAFL: M. Kleinhans)
videoclip
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
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pDf ]
)(Rec34[R =
TRANSPORT DOMINATED BY BEDLOAD(courtesy Vicenzo D’Agostino)
videoclip
National Center for Earth-surface Dynamics:Renesse 2003: Non-cohesive Sediment Transport
TRANSPORT DOMINATED BY SUSPENDED LOAD(Sand-mud turbidity current at SAFL: J. Marr)
videoclip