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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004
1
CHAPTER 2:
CHARACTERIZATION OF SEDIMENT AND GRAIN SIZE DISTRIBUTIONS
Sediment diameter is denoted as D; the parameter has dimension [L].
Since sediment particles are rarely precisely spherical, the notion of “diameter”
requires elaboration.
For coarse particles, the “diameter” D is often defined to be the dimension of
the smallest square mesh opening through which the particle will pass.
For fine particles, “diameter” D often denotes the diameter of the equivalent
sphere with the same fall velocity vs [L/T] as the actual particle.
Grain size is often specified in terms of a base-2 logarithmic scale (phi scale or
psi scale). These are defined as follows: where D is given in mm,
22D)2(n
)D(n)D(og2
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SEDIMENT SIZE RANGES
Type D (mm) Notes
Clay < 0.002 < -9 > 9 Usually cohesive
Silt 0.002 ~ 0.0625 -9 ~ -4 4 ~ 9 Cohesive ~ non-
cohesive
Sand 0.0625 ~ 2 -4 ~ 1 -1 ~ 4 Non-cohesive
Gravel 2 ~ 64 1 ~ 6 -6 ~ -1 “
Cobbles 64 ~ 256 6 ~ 8 -8 ~ -6 “
Boulders > 256 > 8 < -8 “
Mineral clays such as smectite, montmorillonite and bentonite are cohesive, i.e.
characterized by electrochemical forces that cause particles to stick together.
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SEDIMENT GRAIN SIZE DISTRIBUTIONS
The grain size distribution is
characterized in terms of N+1
sizes Db,i such that ff,i denotes the
mass fraction in the sample that is
finer than size Db,i. In the
example below N = 7.
i Db,i mm ff,i
1 0.03125 0.020
2 0.0625 0.032
3 0.125 0.100
4 0.25 0.420
5 0.5 0.834
6 1 0.970
7 2 0.990
8 4 1.000
Note the use of a logarithmic
scale for grain size.
Sample Grain Size Distribution
0
10
20
30
40
50
60
70
80
90
100
0.01 0.1 1 10
Grain Size mm
Perc
en
t F
iner
Db,4 = 0.25 mm
100 x ff,4 = 42
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WHY CHARACTERIZE GRAIN SIZE DISTRIBUTIONS IN TERMS OF A
LOGARITHMIC GRAIN SIZE?
Consider a sediment sample that is half sand, half gravel (here loosely interpreted as
material coarser than 2 mm).
Plotted with a logarithmic grain size scale, the sample is correctly seen to be half
sand, half gravel. Plotted using a linear grain size scale, all the information about the
sand half of the sample is squeezed into a tiny zone
Grain Size Distribution: Half Sand, Half Gravel
0.0625 mm ~ 64 mm, Logarithmic Scale
0
10
20
30
40
50
60
70
80
90
100
0.01 0.1 1 10 100
D mm
Pe
rce
nt
Fin
er sand gravel
Grain Size Distribution: Half Sand, Half Gravel
0.0625 ~ 64 mm, linear scale
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70
D mm
Pe
rce
nt
Fin
er
sand
gravel
Logarithmic scale for grain size Linear scale for grain size
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UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS
The fractions fi(i) represent a discretized version of the continuous function f(), f
denoting the mass fraction of a sample that is finer than size . The probability
density pf of size is thus given as p = df/d.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-4 -3 -2 -1 0 1 2
f()
p()
The example corresponds to a
Gaussian (normal) distribution
with mean Dg = 0.5 mm
STDEV = 0.8mm
2
2
1exp
2
1p
The grain size distribution is
called unimodl because the
function p() has a single mode,
or peak.
The following approximations are valid for a
Gaussian distribution:
16
84g1684g
D
D,DDD
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UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS contd.
A sand-bed river has a characteristic
size of bed surface sediment (D50 or Dg)
that is in the sand range.
A gravel-bed river has a characteristic
bed size that is in the range of gravel or
coarser material.
The grain size distributions of most
sand-bed streams are unimodal, and
can often be approximated with a
Gaussian function.
Many gravel-bed river, however, show
bimodal grain size distributions, as
shown to the upper right. Such streams
show a sand mode and a gravel mode,
often with a paucity of sediment in the
gravel size (2 ~ 8 mm).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-4 -2 0 2 4 6 8 10
f()
p()
Sand mode Gravel mode
A bimodal (multimodal) distribution can
be recognized in a plot of f versus in
terms of a plateau (multiple plateaus)
where f does not increase strongly
with .
Plateau
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GRAVEL-SAND TRANSITIONS
As rivers flow from mountain reaches to plains
reaches, sediment tends to deposit out, creating
a pattern of downstream fining of bed sediment.
It is common (but by no means universal) for
fluvial sediments to be bimodal, with sand and
gravel modes
In such cases a relatively sharp transition from a
gravel-bed stream to a sand-bed stream is often
found, often with a concomitant break in slope
(Sambrook Smith and Ferguson, 1995, Parker
and Cui, 1998).
Long profiles of bed elevation, bed slope and median grain size
for the Kinu River, Japan. Adapted from Yatsu (1955)
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VERTICAL SORTING OF SEDIMENT
Gravel-bed rivers such as the River Wharfe
often display a coarse surface armor or
pavement. Sand-bed streams with dunes
such as the one modeled experimentally
below often place their coarsest sediment in a
layer corresponding to the base of the dunes.
Sediment sorting in a laboratory flume. Image courtesy A. Blom.
River Wharfe, U.K. Image courtesy D. Powell.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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CHAPTER 6:
THRESHOLD OF MOTION AND SUSPENSION
Rock scree face in Iceland.
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ANGLE OF REPOSE
A pile of sediment under water at resting at
the angle of repose r represents a threshold
condition; any slight disturbance causes a
failure.
Consider the indicated grain. The net
downslope gravitational force acting on the
grain (gravitational force – buoyancy force) is
The net normal force is
The net Coulomb resistive force to motion is
Force balance requires that
1R,sin2
DRg
3
4
sin2
Dg
3
4sin
2
Dg
3
4F
sr
3
r
3
r
3
sgt
r
3
gn cos2
DRg
3
4F
r
3
cc cos2
DRg
3
4F
r
FgtFgn
Fc
0FF cgt
or thus:
which is how c is measured (note
that it is dimensionless). For
natural sediments, r ~ 30 ~ 40
and c ~ 0.58 ~ 0.84.
crtan
D=grain
diameter
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THRESHOLD OF MOTION
DgD,1006.022.0 p
)7.7(6.0
pc
6.0p
RReRe
Re
The Shields number * is defined as
RgD
u
RgD
2
*b
Shields (1936) determined experimentally that a minimum, or critical Shields
number is required to initiate motion of the grains of a bed composed of non-
cohesive particles.
Brownlie (1981) fitted a curve to the experimental line of Shields and obtained the
following fit:
c
Based on information contained in Neill (1968), Parker et al. (2003) amended the
above relation to
]1006.022.0[5.0)7.7(6.0
pc
6.0p
ReRe
In the limit of sufficiently large Rep (fully rough flow), then, becomes equal to
0.03.
c
1
sR
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MODIFIED SHIELDS DIAGRAM
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
1 10 100 1000 10000 100000 1000000
Rep
c*
sandsilt gravel
The silt-sand and sand-gravel
borders correspond to the values
of Rep computed with R = 1.65, =
0.01 cm2/s and D = 0.0625 mm
and 2 mm, respectively.
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)z(u
z
Turbulent flow near a wall (such as the bed of a river) can often be approximated in
terms of a logarithmic “law of the wall” of the following form:
s
s
kuB
k
zn
u
u
1
where denotes streamwise flow velocity
averaged over turbulence, z is a coordinate
upward normal from the bed, u* = (b/)1/2
denotes the shear velocity, = 0.4 denotes
the Karman constant and B is a function of
the roughness Reynolds number (u*ks)/
taking the form of the plot on the next page
(e.g. Schlichting, 1968).
u
LAW OF THE WALL FOR TURBULENT FLOWS
ks is the sand equivalent roughness
function of
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B AS A FUNCTION OF ROUGHNESS REYNOLDS NUMBER
4
6
8
10
12
1 10 100 1000
u*ks/
B
s
s
kuB
k
zn
u
u
1
smooth wall
fully rough wall
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ROUGH, SMOOTH AND TRANSITIONAL REGIMES
Logarithmic form of law of the wall:
s
s
kuB
k
zn
u
u
1
100ku
or62.8k
for5.8k
zn
1
u
u,5.8B s
v
s
s
3ku
or26.0k
for5.5zu
n1
u
u,
kun
15.5B s
v
ss
Viscosity damps turbulence near a wall. A scale for the thickness of this “viscous
sublayer” in which turbulence is damped is v = 11.6 /u* (Schlichting, 1968). If ks/v
>> 1 the viscous sublayer is interrupted by the bed roughness, roughness elements
interact directly with the turbulence and the flow is in the hydraulically rough regime:
If ks/v << 1 the viscous sublayer lubricates the roughness elements so they do not
interact with turbulence, and the flow is in the “hydraulically smooth” regime:
For 0.26 < ks/v < 8.62 the near-wall flow is transitional between the
hydraulically smooth and hydraulically rough regimes.
fully rough wall:
smooth wall
function of
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4
6
8
10
12
1 10 100 1000
u*ks/
B
5.8B
5.5ku
n1
B s
smooth roughtransitional
B AS A FUNCTION OF ROUGHNESS REYNOLDS NUMBER: REGIMES
s
s
kuB
k
zn
u
u
1
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DRAG ON A SPHERE
Consider a sphere with diameter D immersed in a Newtonian fluid with density
and kinematic viscosity (e.g. water) and subject to a steady flow with velocity uf
relative to the sphere. The drag force on the sphere is given as
2
f
2
DD u2
Dc
2
1F
where the drag coefficient
cD is a function of the
Reynolds number (ufD)/,
as given in the diagram to
the right.
Note the existence of an
“inertial range” (1000 <
ufD/ < 100000) where cD
is between 0.4 and 0.5.
Drag Curve for Sphere
0.1
1
10
100
1000
10000
0.1 1 10 100 1000 10000 100000 1000000
ufD/
cD
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THRESHOLD OF MOTION: TURBULENT ROUGH FLOW, NEARLY FLAT BED
This is a brief and partial sketch:
The flow is over a granular bed with sediment size D. The mean bed slope S is small,
i.e. S << 1. Assume that ks = nkD, where nk is a dimensionless, O(1) number (e.g. 2).
Consider an “exposed” particle the centroid of which protrudes up from the mean bed
by an amount neD, where ne is again dimensionless and o(1). The flow over the bed
is assumed to be turbulent rough, and the drag on the grain is assumed to be in the
inertial range. Fluid drag tends to move the particle; Coulomb resistance impedes
motion.
gcc
3
g
2
f
2
DD
FF
2
DRg
3
4F
u2
Dc
2
1F
Impelling fluid drag force
Submerged weight of grain
Coulomb resistive force
Threshold of motion: cD FF
FDFc
)z(u
or thus
D
c
2
f
c3
4
RgD
u
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THRESHOLD OF MOTION: TURBULENT ROUGH FLOW, NEARLY FLAT BED
(contd.)
Since ks = nkD and the centroid of the particle is at z = neD, the mean flow velocity
acting on the particle uf is given from the law of the wall as
As long as nku*D/ > 100, B can be set equal to 8.5, so that
In addition, if ufD/ = Fuu*D/ is between 1000 and 100000, cD can be approximated
as 0.45. Setting nk = ne , as an example, it is found that Fu = 8.5 (so uf =8.5 u*).
Further assuming that c = 0.7, the Shields condition for the threshold of motion
becomes
FDFc
)z(u
DunB
Dn
Dnn
u
u
u
uk
k
eDnzf e 1
5.8n
nn5.2F
u
u
k
eu
f
0287.0Fc3
4
RgD
u
c3
4
RgD
u2
uD
cc
2
D
c
2
f
This is not a bad approximation of the asymptotic value of c* from the modified
Shields curve of 0.03 for (RgD)1/2D/. For a theoretical derivation of the full
Shields curve see Wiberg and Smith (1987). [please note we did not include LIFT]
note; uf =u* 8.5
function of
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CASE OF SIGNIFICANT STREAMWISE SLOPE
Let denote the angle of streamwise tilt of the bed, so that
If is sufficiently high then in addition to the drag force FD , there is a direct
tangential gravitational force Fgt impelling the particle downslope.
Force balance:
tanS
)tan
1(cosc
coc
FD
Fgt
Fgn
Fc
gncc
3
gt
3
gn
2
f
2
DD FF,sin2
DRg
3
4F,cos
2
DRg
3
4F,u
2
Dc
2
1F
gncgtD FFF
or reducing,
where c* = the critical
Shields number on the
slope and co* = the value
on a nearly horizontal bed.
note that when = angle of repose c*= 0
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Shields Relation, Streamwise Angle
r = 35 deg
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35 40
deg
co
c
VARIATION OF CRITICAL SHIELDS STRESS WITH STREAMWISE BED SLOPE
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CASE OF SIGNIFICANT TRANSVERSE SLOPE (BUT NEGLIGIBLE
STREAMWISE SLOPE)
Let denote the angle of transverse tilt of the bed
fluid drag
x
y
transverse pull
of gravity
2/1
2
c
2
coc
tan1cos
A general formulation of the threshold of motion for arbitrary bed slope is given in
Seminara et al. (2002). This formulation includes a lift force acting on a
particle, which has been neglected for simplicity in the present analysis.
A formulation similar to that for streamwise tilt yields the result:
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Shields Relation, Transverse Angle
r = 35 deg
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35 40
j deg
x
co
c
deg
VARIATION OF CRITICAL SHIELDS STRESS WITH TRANSVERSE BED SLOPE
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MODES OF TRANSPORT OF SEDIMENT
2/1
pD
f ])(Rec3
4[R
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 in ballistic
trajectory just above bed.
role of turbulence is indirect.
Suspended load:
feels direct dispersive effect of eddies.
may be wafted high into the water column.
typically when u* >> Ws (shear velocity >> settling velocity)
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CHAPTER 7:
RELATIONS FOR 1D BEDLOAD TRANSPORT
Let qb denote the volume bedload transport rate per unit width (sliding, rolling,
saltating). It is reasonable to assume that qb increases with a measure of flow
strength, such as depth-averaged flow velocity U or boundary shear stress b.
A dimensionless Einstein bedload number q* can be defined as follows:
A common and useful approach to the quantification of bedload transport is to
empirically relate qb* with either the Shields stress * or the excess of the Shields
stress * above some appropriately defined “critical” Shields stress c*. As pointed
out in the last chapter, c* can be defined appropriately so as to a) fit the data and
b) provide a useful demarcation of a range below which the bedload transport rate
is too low to be of interest.
The functional relation sought is thus of the form
24
2
32
3
3 /
/
/
/
/)( sm
sm
msm
msm
gD
q
DRgD
s
bbb
)(qqor)(qq cbbbb
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BEDLOAD TRANSPORT RELATION OF MEYER-PETER AND MÜLLER
All the bedload relations in this chapter pertain to a flow condition known as “plane-
bed” transport, i.e. transport in the absence of significant bedforms. The influence
of bedforms on bedload transport rate will be considered in a later chapter.
The “mother of all modern bedload transport relations” is that due to Meyer-Peter
and Müller (1948) (MPM). It takes the form
The relation was derived using flume data pertaining to well-sorted sediment in the
gravel sizes.
Recently Wong (2003) and Wong and Parker (2005) found an error in the analysis
of MPM. A re-analysis of the all the data pertaining to plane-bed transport used by
MPM resulted in the corrected relation
If the exponent of 1.5 is retained, the best-fit relation is
047.0,)(8q c
2/3
cb
047.0,)(93.4q c
6.1
cb
0495.0,)(97.3q c
2/3
cb
BB S7-S10
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Bedload Relation: Modified MPM
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E-02 1.0E-01 1.0E+00
*
qb*
qb* = 3.97 (* - c*)1.50
c* = 0.0495
Data of Meyer-Peter and Muller (5.21 mm,
28.65 mm) and Gilbert (3.17 mm, 4.94 mm,
7.01 mm)
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1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 1.E+14
Grav Brit
Grav Alta
Sand Mult
Sand Sing
Grav Ida
Q
50bf
LIMITATIONS OF MPM
There is nothing intrinsically “wrong” with MPM. In a dimensionless sense,
however, the flume data used to define it correspond to the very high end of the
transport events that normally occur during floods in alluvial gravel-bed streams.
While the relation is important in a historical sense, it is not the best relation to use
with gravel-bed streams.
gravel-bed streams
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0455.0~037.0,7.5q c
5.1
cb
05.0,17q cccb
05.0,7.074.18q cccb
b
b2)/143.0(
2)/143.0(
t
q5.431
q5.43dte
11
2
03.0,12.11q c
5.4
c5.1
b
Fernandez Luque & van Beek (1976)
Ashida & Michiue (1972)
Engelund & Fredsoe (1976)
Einstein (1950)
Parker (1979) fit to
Einstein (1950)
BEDLOAD TRANSPORT RELATIONS FOR UNIFORM SEDIMENT
Some commonly-quoted bedload transport relations with good data bases are given
below.
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1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
0.01 0.1 1
*
qb*
E
AM
EF
FLBSand
P approx E
FLBGrav
PLOTS OF BEDLOAD TRANSPORT RELATIONS
E = Einstein
AM = Ashida-Michiue
EF = Engelund-Fredsoe
P approx E = Parker approx of Einstein
FLBSand = Fernandez Luque-van
Beek, c* = 0.038
FLBGrav = Fernandez Luque-van
Beek, c* = 0.0455
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NOTES ON THE BEDLOAD TRANSPORT RELATIONS
The bedload relation of Einstein (1950) contains no critical Shields number. This
reflects his probabilistic philosophy.
All of the relations except that of Einstein correspond to a relation of the form
In the limit of high Shields number (* >> *c). In dimensioned form this
becomes
where K is a constant; for example in the case of Ashida-Michiue, K = 17. Note
that in this limit the bedload transport rate becomes independent of grain size!!
Some of the scatter between the relations is due to the face that c* should be a
function of Rep.
Some of the scatter is also due to the fact that several of the relations have been
plotted well outside of the data used to derive them.
2/3
b )(~q
Ku
Rgqor
RgD
uK
DRgD
q3
b
2/32
b
simplify D3/2
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SHEET FLOW
• For values of * < a threshold value sheet*, bedload is localized in terms of
rolling, sliding and saltating grains that exchange only with the immediate bed
surface.
• When s* > sheet* the bedload layer devolves into a sliding layer of grains
that can be several grains thick. Sheet flows occur in unidirectional river flows
as well as bidirectional flows in the surf zone.
• Values of sheet* have been variously estimated as 0.5 ~ 1.5. (Horikawa,
1988, Fredsoe and Diegaard, 1994, Dohmen-Jannsen, 1999; Gao, 2003). The
parameter sheet* appears to decrease with increasing Froude number.
• Wilson (1966) has estimated the bedload transport rate in the sheet flow
regime as obeying a relation of the form
All the previously presented bedload relations except that of Einstein also
devolve to a relation of the above form for large *, with K varying between
3.97 and 18.74.
12K,)(Kq 2/3
b
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CALCULATIONS WITH BEDLOAD TRANSPORT RELATIONS
To perform calculations with any of the previous bedload transport relations, it is
necessary to specify:
1) the submerged specific gravity R of the sediment;
2) a representative grain size exposed on the bed surface, e.g. surface geometric
mean size Dsg or surface median size Ds50, to be used as the characteristic size
D in the relation;
3) and a value for the shear velocity of the flow u* (and thus b).
Once these parameters are specified, * = (u*)2/(RgD) is computed, qb* is calculated
from the bedload transport relation, and the volume bedload transport rate per
unit width is computed as qb = (RgD)1/2Dqb*.
The shear velocity u* is computed in the case of normal flow using the Manning-
Strickler resistance relation,
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CHAPTER 4:
RELATIONS FOR THE CONSERVATION OF BED SEDIMENT
This chapter is devoted to the derivation of equations describing the conservation of
bed sediment. Definitions of some relevant parameters are given below.
qb = volume bedload transport rate per unit width [L2T-1]
qs = volume suspended load transport rate per unit width [L2T-1]
qt = qb + qs = volume bed material transport rate per unit width [L2T-1]
gb = sqb = mass bedload transport rate per unit width [ML-1T-1]
(corresponding definitions for gs, gt)
= bed elevation [L]
p = porosity of sediment in bed deposit [1]
(volume fraction of bed sample that is holes rather than sediment: 0.25 ~
0.55 for noncohesive material)
g = acceleration of gravity [L/T2]
x = boundary-attached streamwise coordinate [L]
y = boundary-attached transverse coordinate [L]
z = boundary-attached upward normal (quasi-vertical) coordinate [L]
t = time [T]
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COORDINATE SYSTEM
x = nearly horizontal boundary-attached “streamwise” coordinate [L]
y = nearly horizontal boundary-attached “transverse” coordinate [L]
z = nearly vertical coordinate upward normal from boundary [L]
x
y
z sediment bed
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ILLUSTRATION OF BEDLOAD TRANSPORT
Image of bedload transport of 7 mm gravel in a flume (model river) at St. Anthony
Falls Laboratory, University of Minnesota, from the experiments of Miguel Wong.
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CASE OF 1D, BEDLOAD ONLY, SEDIMENT APPROXIMATED AS UNIFORM
IN SIZE
1qq1gg1x)1(t xxbxbsxxbxbps
or thus
x
qb
-
tp
)1(
This corresponds to the original
form derived by Exner.
bed sediment + pores
water
x
1
x
x +x
qb
qb
net mass flux net volume flux
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2D GENERALIZATION, BEDLOAD ONLY
bp qt
)1(
-
ybyxbxb eqeqq
where
denote unit vectors in the
x and y directions.
yx e,e
x
y
z sediment bed
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CASE OF 1D BEDLOAD + SUSPENDED LOAD
1xED1qq1x)1(t
sssxxbxbsps
Es = volume rate per unit time per unit bed area that sediment is entrained
from the bed into suspension [LT-1].
Ds = volume rate per unit time per unit bed area that sediment is deposited
from the water column onto the bed [LT-1].
ssb
p EDx
q
t)1(
-
or thus
bed sediment + pores
water
x
1
x
x +x
qb
qb
Ds
Es
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EVALUATION OF Ds AND Es
Let denote the volume concentration of sediment c in suspension at
(x, z, t), averaged over turbulence. Here c = (sediment volume)/(water
volume + sediment volume).
In the case of a dilute suspension of non-cohesive material,
Ecvx
q
t)1( bs
bp
-
bss cvD
where cb denotes the near-bed value of c .
Similarly, a dimensionless entrainment rate E can be defined such that
EvE ss
Thus
bc
)t,z,x(c
zbss cvD
bc
c
c
settling velocity
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CHAPTER 8:
FLUVIAL BEDFORMS
The interaction of flow and sediment transport often creates bedforms such as
ripples, dunes, antidunes, and bars. These bedforms in turn can interact with
the flow to modify the rate of sediment transport.
Dunes in the North Loup River, Nebraska, USA; image courtesy D. Mohrig
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TOUR OF BEDFORMS IN RIVERS: RIPPLES
Ripples in the Rum River, Minnesota USA at very low flow; ~ 10 - 20 cm.
Ripples are characteristic of a)
very low transport rates in b)
rivers with sediment size D less
than about 0.6 mm. Typical
wavelengths are on the order of
10’s of cm and and wave heights
are on the order of cm.
Ripples migrate downstream and
are asymmetric with a gentle
stoss (upstream) side and a steep
lee (downstream side). Ripples
do not interact with the water
surface.
flowmigration
View of the Rum River, Minnesota USA
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TOUR OF BEDFORMS IN RIVERS: DUNES
Dunes in the North Loup River, Nebraska USA. Two people are circled for scale. Image courtesy D. Mohrig.
Dunes are the most common bedforms in sand-bed rivers; they can also occur in
gravel-bed rivers. Wavelength can range up to 100’s of m, and wave height
can range up to 5 m or more in large rivers. Dunes are usually asymmetric, with a
gentle stoss (upstream) side and a steep lee (downstream) side. They are characteristic of subcritical flow (Fr
sufficiently below 1). Dunes migrate
downstream. They interact weakly with
the water surface, such that the flow
accelerates over the crests, where water
surface elevation is slightly reduced.
(That is, the water surface is out of phase
with the bed.)
flowmigration
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TOUR OF BEDFORMS IN RIVERS: ANTIDUNES
Trains of surface waves indicating the presence of antidunes in braided channels of the tailings basin of the Hibbing Taconite Mine, Minnesota, USA. Flow is from top to bottom.
Antidunes occur in rivers with
sufficiently high (but not necessarily
supercritical) Froude numbers. They
can occur in sand-bed and gravel-bed
rivers. The most common type of
antidune migrates upstream, and
shows little asymmetry. The water
surface is strongly in phase with the
bed. A train of symmetrical surface
waves is usually indicative of the
presence of antidunes.
flow migration
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TOUR OF BEDFORMS IN RIVERS: ALTERNATE BARS
Alternate bars in the Naka River, an artificially straightened river in Japan. Image courtesy S. Ikeda.
Alternate bars occur in rivers with sufficiently large (> ~ 12), but not too large
width-depth ratio B/H. Alternate bars migrate downstream, and often have
relatively sharp fronts. They are often precursors to meandering. Alternate bars
may coexist with dunes and/or antidunes.
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TOUR OF BEDFORMS IN RIVERS: MULTIPLE-ROW LINGUOID BARS
Plan view of superimposed linguoid bars and dunes in the North Loup River, Nebraska USA. Image courtesy D. Mohrig. Flow is from left to right.
Multiple-row bars (linguoid bars) occur when the width-depth ratio B/H is even
larger than that for alternate bars. These bars migrate downstream. They may co-
exist with dunes or antidunes.
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BEDFORMS IN THE LABORATORY AND FIELD: DUNES
Dunes on an exposed point bar in the meandering Fly River, Papua New Guinea
Dunes in a flume in Tsukuba University, Japan: flow turned off. Image courtesy H. Ikeda.
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Rhine River, Switzerland
BEDFORMS IN THE LABORATORY AND FIELD: ALTERNATE BARS
Alternate bars in a flume in Tsukuba University, Japan: flow turned low.
Image courtesy H. Ikeda.
Alternate bars in the Rhine River between Switzerland and Lichtenstein.
Image courtesy M. Jaeggi.
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BEDFORMS IN THE LABORATORY AND FIELD: MULTIPLE-ROW
(LINGUOID) BARS
Linguoid bars in a flume in Tsukuba University, Japan: flow turned off.
Image courtesy H. Ikeda.
Linguoid bars in the Fuefuki River, Japan. Image courtesy S. Ikeda.
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Ohau River, New Zealand
WHEN THE FLOW IS INSUFFICIENT TO COVER THE BED, THE
RIVER MAY DISPLAY A BRAIDED PLANFORM
Braiding in a flume in Tsukuba University, Japan: flow turned low.
Image courtesy H. Ikeda.
Braiding in the Ohau River, New Zealand. Image courtesy P. Mosley.
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RIPPLES
Ripples are small-scale bedforms that migrate downstream and show a characteristic
asymmetry, with a gentle stoss face and a steep lee face.
Ripples require the existence of a reasonably well-defined viscous sublayer in order
to form. In rivers, a viscous sublayer can exist only when the flow is very slow and
well below flood conditions. Because of the viscous sublayer, ripples do not interact
with the water surface.
Engelund and Hansen (1967) have suggested the following condition for ripple
formation: D v, where v = 11.6 /u* denotes the thickness of the viscous sublayer
(Chapter 6). This relation can be rearranged to yield the threshold condition
flowmigration
2
p
6.11
Re
The above relation can be solved with the modified Brownlie relation of Chapter 6 to
yield a maximum value of Rep for ripple formation. The value so obtained is 91,
corresponding to a grain size of 0.8 mm with = 0.01 cm2/s and R = 1.65. In
practice, ripples are observed only for D < 0.6 mm. Ripples can coexist with dunes.
where
DRgD,
RgDp
b Re
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SHIELDS DIAGRAM WITH CRITERION FOR RIPPLES
0.01
0.1
1
10
1 10 100 1000 10000 100000
Rep
*
motion mod Brownlie
ripples
suspension
2
p
v
6.11orD
Re
2pfs )(orvu ReR
modified Brownlie
no ripples
ripples
no motion
suspension
no suspension
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DEFINITION OF DUNES AND ANTIDUNES
Dunes are 1D (or quasi-1D) bedforms for which the water surface fluctuations are
approximately out of phase with the bed fluctuations. That is, the water surface is
high where the bed is low and vice versa. As is shown below dunes migrate
downstream.
Antidunes are 1D (or quasi-1D) bedforms for which the water surface fluctuations are
approximately in phase with the bed fluctuations. That is, the water surface is high
where the bed is high and vice versa. As shown below, most antidunes migrate
upstream, but there is a regime within which they can migrate downstream.
flowmigration
flow migration
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59 GSA Special Papers 2007 vol. 426 171-188
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John B. Southard, Surface forms, 1978
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0.01
0.1
1
10
1 10 100 1000 10000 100000
Rep
*
motion mod Brownlie
ripples
suspension
dunes C&C
ripples C&C
extrap C&C dunes
2
p
v
6.11orD
Re
2pfs )(orvu ReR
modified Brownlie
C&C ripples/no ripples
C&C no dunes/dunes
extrapolated C&C
no dunes/duneslower regime plane bed
dunes
ripples
no motion
suspension
SHIELDS DIAGRAM INCLUDING RESULTS OF CHABERT AND CHAUVIN (1963)
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This diagram uses the hydraulic
parameters X1 = Fr and X2 =
U/u*. The parameter Rep is not
included, and the diagram is
valid only for sand.
The diagram clearly shows an
extensive range of flow for
which Fr < 1 but antidunes
form. The “plane bed” regime
on the left-hand side of the
diagram is upper-regime plane
bed. Lower-regime plane bed
is not shown in the diagram.
BEDFORM REGIME DIAGRAM
OF ENGELUND AND HANSEN
(1966)
Fr
U/u*
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The total shear velocity u*, shear velocity due to skin friction u*s and shear velocity
due to bedforms u*f, and the associated Shields numbers are defined as
Engelund and Hansen (1967) determined the following empirical relation for lower-
regime form drag due to dune resistance;
or thus
50s
2
ff
50s
2
ss
50s
2
RgD
u,
RgD
u,
RgD
u
2s 4.006.0
Note that bedforms are absent (skin friction only) when s* = *; bedforms are
present when s* < *. The relation is designed to be used with the following
skin friction predictor:
Engelund and Hansen (1967) also present a form drag relation for upper-
regime bedforms (antidunes).
FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967)
bff
bss
b u,u,u
2
sf 4.006.0
s
s2/1
fsk
H11n
1C 65ss D2k
why do I care
about s ? For
the sediment transport
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Engelund-Hansen Bedform Resistance Predictor
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
x
x E-H Relation
No form drag
s
s
f
No form drag
Engelund-Hansen
FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967) contd.
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DEPTH-DISCHARGE PREDICTIONS WITH THE FORM DRAG PREDICTOR OF
ENGELUND AND HANSEN (1967)
Form drag relations allow for a prediction of flow depth H and velocity U as a
function of water discharge per unit width qw. In order to do this with the relation of
Engelund and Hansen (1967) it is necessary to specify the stream slope S, bed
material sizes Ds50 and Ds65, submerged specific gravity of the sediment R. The
computation proceeds as follows for the case of normal flow, for which b = u*2 =
gHS.
Compute ks from Ds65.
Assume a value (a series of values) of Hs.
Assuming normal flow, compute u*s = (gHsS)1/2 and s* =u*s2/(RgDs50).
Compute * from s* according to Engelund-Hansen.
Again assuming normal flow, * = (HS)/(RDs50) so that H = RDs50*/S.
Compute Czs = Cfs-1/2 from Hs/ks and the skin friction predictor.
Compute the velocity U from the relation U/u*s = Czs.
Compute the water discharge per unit width qw = UH.
Plot H versus qw.
Recommended