1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 6:
THRESHOLD OF MOTION AND SUSPENSION
Rock scree face in Iceland.
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
ANGLE OF REPOSE
A pile of sediment at resting at the angle of
repose r represents a threshold condition;
any slight disturbance causes a failure. Here
the pile of sediment is under water. Consider
the indicated grain. The net downslope
gravitational force acting on the grain
(gravitational force – buoyancy force) is
3
Fc
Fgn
3
4
4
D
D
Fgt  s g  sin r  g  sin r 
3
3
2
2
3
4
D
Rg  sin r
3
2
, R
r
s
1

The net normal force is
3
4
D
Fgn  Rg  cos r
3
2
The net Coulomb resistive force to motion is
3
Fc   c
4
D
Rg  cos r
3
2
Force balance requires that
Fgt  Fc  0
Fgt
or thus:
tan r  c
which is how c is measured (note
that it is dimensionless). For
natural sediments, r ~ 30 ~ 40
and c ~ 0.58 ~ 0.84.
2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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THRESHOLD OF MOTION
The Shields number * is defined as
b
u2*
 

RgD RgD

Shields (1936) determined experimentally that a minimum, or critical Shields
number c is required to initiate motion of the grains of a bed composed of noncohesive particles. Brownlie (1981) fitted a curve to the experimental line of
Shields and obtained the following fit:

c
 0 .6
p
  0.22 Re
 0.06  10
RgD D
, Rep 

( 7.7 Rep 0.6 )
Based on information contained in Neill (1968), Parker et al. (2003) amended the
above relation to

c
0.6
p
  0.5 [0.22 Re
( 7.7 Rep0.6 )
 0.06  10
]

In the limit of sufficiently large Rep (fully rough flow), then, c becomes equal to
3
0.03.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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MODIFIED SHIELDS DIAGRAM
0.1
0.09
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.
0.08
0.07
 c*
0.06
0.05
0.04
0.03
0.02
silt
0.01
sand
gravel
0
1
10
100
1000
Rep
10000
100000
1000000
4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
LAW OF THE WALL FOR TURBULENT FLOWS
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:
u 1 z
 uk s 
 n   B

u   k s 
  
where u 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( z )
z
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
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B AS A FUNCTION OF ROUGHNESS REYNOLDS NUMBER
12
u 1 z
u k 
 n   B  s 
u   k s 
  
B
10
8
6
4
1
10
100
u*ks/
1000
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
ROUGH, SMOOTH AND TRANSITIONAL REGIMES
Logarithmic form of law of the wall:
u 1 z
u k 
 n   B  s 
u   k s 
  
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:
B  8.5 ,
u 1 z
 n   8.5
u   k s 
for
ks
 8.62 or
v
uk s
 100

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:
B  5.5 
1  uk s 
n
 ,
   
u 1  uz 
 n
  5.5
u    
for
ks
 0.26
v
For 0.26 < ks/v < 8.62 the near-wall flow is transitional between the
hydraulically smooth and hydraulically rough regimes.
or
uk s
3

7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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B AS A FUNCTION OF ROUGHNESS REYNOLDS NUMBER: REGIMES
12
u 1 z
u k 
 n   B  s 
u   k s 
  
1 u k 
B  n  s   5.5
   
B
10
8
B  8.5
6
smooth
transitional
rough
4
1
10
100
u*ks/
1000
8
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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
Drag Curve for Sphere
2
1
D 2
FD  c D   u f
2
2
10000
1000
Note the existence of an
“inertial range” (1000 <
ufD/ < 100000) where cD
is between 0.4 and 0.5.
100
cD
where the drag coefficient
cD is a function of the
Reynolds number (ufD)/,
as given in the diagram to
the right.
10
1
0.1
0.1
1
10
100
1000
10000
100000
1000000
ufD/
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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THRESHOLD OF MOTION: TURBULENT ROUGH FLOW, NEARLY FLAT BED
This is a brief and partial sketch: more detailed analyses can be found in Ikeda (1982)
and Wiberg and Smith (1987). 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.
2
1
D
FD  c D   u2f
2
2
4
D
Fg  Rg 
3
2
Fc   cFg
Impelling fluid drag force
u( z )
3
Threshold of motion:
Submerged weight of grain
Fc
FD
Coulomb resistive force
FD  Fc
or thus
u2f
4 c

RgD 3 c D
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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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
u( z )
uf u z neD 1  neD 
 nkuD 
  B

 n

u
u
  nkD 



As long as nku*D/ > 100, B can be set equal to 8.5, so that
Fc
FD
n 
uf
 Fu  2.5n e   8.5
u
 nk 
In addition, if ufD/ = Fuu*D/ is between 1000 and 100000, cD can be approximated
as 0.45. Setting nk = ne = 2 as an example, it is found that Fu = 8.5. Further
assuming that c = 0.7, the Shields condition for the threshold of motion becomes
u2f
4 c
u2
4 c


 c 
 0.0287
2
RgD 3 c D
RgD
3 c DFu
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).
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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CASE OF SIGNIFICANT STREAMWISE SLOPE
Let  denote the angle of streamwise tilt of the bed, so that
S  tan 
If  is sufficiently high then in addition to the drag force FD , there is a direct
tangential gravitational force Fgt impelling the particle downslope.
2
1
D
FD  c D   u2f
2
2
3
4
D
, Fgn  Rg  cos 
3
2
3
4
D
, Fgt  Rg  sin 
3
2
, Fc   cFgn
Force balance:
FD  Fgt   cFgn
tan 
   cos (1 
)
c

c
or reducing,
Fc
Fgt

Fgn
FD

co
where c* = the critical
Shields number on the
slope and co* = the value
on a nearly horizontal bed.
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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VARIATION OF CRITICAL SHIELDS STRESS WITH STREAMWISE BED SLOPE
Shields Relation, Streamwise Angle
r = 35 deg
1.2
1
c
co
0.8
0.6
0.4
0.2
0
0
5
10
15
20
 deg
25
30
35
40
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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CASE OF SIGNIFICANT TRANSVERSE SLOPE (BUT NEGLIGIBLE
STREAMWISE SLOPE)
Let  denote the angle of transverse tilt of the bed
y
x
transverse pull
of gravity

fluid drag
A formulation similar to that for streamwise tilt yields the result:
1/ 2
 tan 2  



c  co cos 1 
2
c 

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
14
particle, which has been neglected for simplicity in the present analysis.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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VARIATION OF CRITICAL SHIELDS STRESS WITH TRANSVERSE BED SLOPE
Shields Relation, Transverse Angle
r = 35 deg
1.2
1
0.8


x

c
 0.6
co
0.4
0.2
0
0
5
10
15
20
 deg
25
30
35
40
j deg
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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SEDIMENT MIXTURES: COARSER GRAINS ARE HARDER TO MOVE

c
bc  RgD
0.1
0.09
0.08
0.07
0.06
 c*
In the limiting case of coarse gravel
of uniform size D, the modified
Brownlie relation of Slides 3 and 4

predicts a critical Shields number c
of 0.03, so that the boundary shear
stress bc at which the gravel moves
is given by the relation
0.05
0.04
0.03
0.02
silt
0.01
sand
gravel
0
1
10
100
1000
10000
100000
1000000
Rep
Now this uniform coarse gravel is replaced with a mixture of gravel sizes Di such
that the geometric mean size of the surface layer (i.e. the layer exposed to the
flow) Dsg is identical to the size D of the uniform gravel. It is further assumed (for
the sake of simplicity) that each gravel Di in the mixture is coarse enough so that
the critical Shields stress of uniform sediment with size Di would also be 0.03.
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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SEDIMENT MIXTURES: COARSER GRAINS ARE HARDER TO MOVE contd.
If every grain in the mixture acted as though it were
surrounded by grains of the

same size as itself, the critical Shields number  sci for all sizes Di in the surface
layer exposed to the flow would be constant at 0.03, which would also be the

critical Shields number scg for the surface geometric mean size Dsg. Thus the
following relation would hold;


sci


scg
 0.03
or
sci
1

scg
If this were true the critical shear stresses bsci and bscg for sizes Di and Dg,
respectively, in the surface layer would be given as

i sci
bsci  RgD 

sg scg
, bscg  RgD 
or thus
bsci
Di

bscg Dsg
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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SEDIMENT MIXTURES: COARSER GRAINS ARE HARDER TO MOVE contd.
So if every grain in the mixture acted as though it were surrounded by grains of the
same size as itself, the critical Shields number sci would be the same for all grains
in the surface, and the boundary shear stress bsgi required to move grain size Di
would increase linearly with size Di. That is, if each grain acted as if it were not
surrounded by grains of different sizes (grain independence), a grain that is twice
the size of another grain would require twice the boundary shear stress to
move.
sci
1 ,

scg
bsci
D
 i
bscg Dsg
The above grain-independent behavior is mediated by differing grain mass. That
18
is, larger grains are harder to move because they have more mass.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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SEDIMENT MIXTURES: HIDING EFFECTS
Grains in a mixture do not act as though they are surrounded by grains of the
same size. As Einstein (1950) first pointed out, on the average coarser grains
exposed on the surface protrude more into the flow, and thus feel a preferentially
larger drag force. Finer grains can hide behind and between coarse grains, so
feeling a preferentially smaller drag force. These exposure effects are grouped
together as hiding effects.
Hiding effects reduce the difference in boundary shear stress required to move
different grain sizes in a mixture. As a result, the relation for bsci is amended to
1 
bsci  Di 

bscg  Dsg 
where in general 0   < 1.
19
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SEDIMENT MIXTURES: HIDING EFFECTS contd.
1 


The relation bsci   Di 
bscg
D 
 sg 
combined with the definitions
yield the result
bsci  RgDisci

sci

scg


 Di 


D 
 sg 
, bscg  RgDsgscg

The value  = 0 yields the case of grain independence: there are no hiding
effects, the critical Shields number is the same for all grain sizes, and the boundary
shear stress required to move a grain in a mixture increases linearly with grain
size. The value  = 1 yields the equal threshold condition: hiding is so effective
that it completely counterbalances mass effects, all grains in a mixture move at the
same critical boundary shear stress, and critical Shields number increases as grain
size Di to the – 1 power.
In actual rivers the prevailing condition is somewhere between grain
20
independence and the equal threshold condition, although it tends to be
somewhat biased toward the latter.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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CRITICAL SHIELDS STRESS FOR SEDIMENT MIXTURES
Consider a bed consisting of a mixture of many sizes. The larger grains in the
mixture are heavier, and thus harder to move than smaller grains. But the larger
grains also protrude out into the flow more, and the smaller grains tend to hide in
between them, so rendering larger grains easier to move than smaller grains.
Mass effects make coarser grains harder to move than finer grains. Hiding effects
make coarser grains easier to move than finer grains. The net residual is a mild
tendency for coarser grains to be harder to move than finer grains, as first
shown by Egiazaroff (1965).
Let Dsg (Ds50) be a surface geometric mean
(surface median) size, and let scg* (sc50*)
denote the critical Shields number needed to larger grains protrude more
move that size. The critical Shields number
smaller grains hide
sci* need to move size Di on the surface can
between larger ones
be related to grain size Di as follows:

sci

scg


 Di 


D 
 sg 

or

sci

sc 50


 D 
  i 
 Ds50 
u

where  varies from about 0.65 to 0.90 (Parker, in press).
21
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CRITICAL SHIELDS STRESS FOR SEDIMENT MIXTURES contd.
Now let bscg (bsc50) denote the (dimensioned) critical shear stress needed to move
size Dsg (Ds50), and bsci denote the (dimensioned) critical shear stress needed to
move size Di in the mixture exposed on the bed surface. By definition, then,


scg
bscg

 RgD sg
,


sc 50

bscg
 RgD s50
,


sci
bsci

 RgDi
Reducing the above relations with the relations of the previous slide, i.e.
sci  Di 


scg  Dsg 

or
sci  Di 

 

sc50  Ds50 

larger grains protrude more
u
smaller grains hide
between larger ones
it is found that:
1 
bsci  Di 

bscg  Dsg 
1 
or
bsci  Di 

 
bsc50  Ds50 
22
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LIMITS OF EQUAL THRESHOLD AND GRAIN INDEPENDENCE
Consider the sci
relations
scg
D 
 i 
D 
 sg 

or
sci  Di 


sc50  Ds50 
If  = 1 then all surface grains
sci
move at the same absolute

value of the boundary shear scg
stress (equal threshold):
D 
 i 
D 
 sg 
1 

or
1
sci  Di 


sc50  Ds50 
or
If  = 0 then all surface grains move
independently of each other, as if they did
not feel the effects of their neighbors (grain
independence):
sci
 1 or
scg
sci
1
sc50
or
bsci
bscg
bcsi  Di 

bcsg  Dsg 
1
or
1 
or
bcsi  Di 

 
bcs50  Ds50 
bsci
 1 or
bscg
larger grains protrude more
bsci
1
bsc50
u
smaller grains hide
between larger ones
1
D 
 i 
D 
 sg 
1
or
bsci  Di 


bsc50  Ds50 
In most gravel-bed rivers coarser surface grains are harder to move than finer
surface grains, but only mildly so ( is closer to 1 than 0, but still < 1).
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CRITICAL SHIELDS NUMBER FOR SEDIMENT MIXTURES
10
xx
sci
scg
equal threshold
grain independence
gamma = 0.85
1
0.1
0.1
1
Di/Dsg
10
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CRITICAL BOUNDARY SHEAR STRESS FOR SEDIMENT MIXTURES
10
xx
bsci
bscg
equal threshold
grain independence
gamma = 0.85
1
0.1
0.1
1
Di/Dsg
10
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SIMPLE POWER LAW MAY NOT BE SUFFICIENT
Several researchers (e.g. Proffitt and Sutherland, 1983) have found that a simple power law is
not sufficient to represent the relation for the critical Shields stress for mixtures. More
specifically, it is found that the curve flattens out as Di/Dsg becomes large. That is, the very
coarsest grains in a mixture approach the condition of grain independence, with a critical
Shields number near 0.015 ~ 0.02 (e.g. Ramette and Heuzel, 1962). The bedload transport
relation of Wilcock and Crowe (2003), presented in Chapter 7, captures the essence of this
trend; large amounts of sand render gravel easier to move.
10
x
sci
scg
1
0.1
0.1
1
Di / Dsg
x
10
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONDITION FOR “SIGNIFICANT” SUSPENSION OF SEDIMENT
In order for sediment to be maintained in suspension to any significant degree,
some measure of the characteristic velocity of the turbulent fluctuations of the flow
must be at least of the same order of magnitude as the fall velocity vs of the
sediment itself. Let urms denote a characteristic near-bed velocity of the turbulence
(“rms” stands for “root-mean-squared”), defined as
2
rms
u

1 2
 u  v 2  w 2
3

z b
where u’, v’ and w’ denote turbulent velocity fluctuations in the streamwise,
transverse and upward normal directions and z = b denotes a near-bed elevation
such that b/H << 1, where H denotes depth. A loose criterion for the onset of
significant turbulence is then
urms ~ v s
In the case of a rough turbulent flow, the shear velocity near the bed can be
evaluated as
u2   uw 
z b
(e.g. Tennekes and Lumley, 1972; Nezu and Nakagawa, 1993).
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONDITION FOR “SIGNIFICANT” SUSPENSION OF SEDIMENT contd.
Since turbulence tends to be well-correlated (Tennekes and Lumley, 1972), the
following order-of magnitude estimate holds;
 uw 
z b
~

1 2
u  v 2  w 2
3

z b
from which it can be concluded that
urms ~ u
Based on these arguments, Bagnold (1966) proposed the following approximate
criterion for the critical shear velocity u*sus for the onset of significant suspension
(see also van Rijn, 1984);
usus  v s
Dividing both sides by (RgD)1/2, the following criterion is obtained for the onset of
significant suspension:
Rf = Rf(Rep) defines the
functional relationship for fall
u2*sus
v 2s

2
sus 

 R f (Rep )
velocity given in Chapter 2
RgD
RgD
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SHIELDS DIAGRAM WITH CRITERION FOR SIGNIFICANT SUSPENSION
10
bedload and suspended load transport
u  v s
negligible suspension
1
suspension
bfc50
motion
0.1
bedload transport
no motion
silt
0.01
1.E+00
sand
1.E+01
gravel
1.E+02
1.E+03
Rep
1.E+04
1.E+05
1.E+06
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A NOTE ON THE MEANING OF THE “THRESHOLD OF MOTION”
There is no such thing as a precise “threshold of motion” for a granular bed
subjected to the flow of a turbulent fluid. Both grain placement and the turbulence
of the flow have elements of randomness. For example, Paintal (1971) conducted
experiments lasting weeks, and found extremely low rates of sediment transport at
values of the Shields number that are below all reasonable estimates of the “critical”
Shields number. In particular, he found that at low Shields numbers
 
qb  6.56 x 1018 
where
16
qb
q 
RgD D

b
denotes a dimensionless bedload transport rate (Einstein number, discussed in
more detail in the next chapter) and qb denotes the volume bedload transport rate
per unit width. Strictly speaking, then, the concept of a threshold of motion is
invalid. It is nevertheless a very useful concept for the following reason. If the
sediment transport rate is so low that an order-one deviation from the rate would
cause negligible morphologic change over a given time span of interest (e.g. 50
years for an engineering problem or 50,000 year for a geological problem),
30
the flow conditions can be effectively treated as below the threshold of motion.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LOG-LOG PLOT OF PAINTAL BEDLOAD TRANSPORT RELATION
1.E-05
1.E-06
1.E-07
1.E-08
qb*
1.E-09
1.E-10
1.E-11
1.E-12
No critical Shields
number is evident!
1.E-13
1.E-14
1.E-15
0.001
0.01
*
0.1
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
LINEAR-LINEAR PLOT OF PAINTAL BEDLOAD TRANSPORT RELATION
3.E-06
3.E-06
qb*
2.E-06
2.E-06
For practical purposes
c* might be set near
0.023.
1.E-06
5.E-07
0.E+00
0
0.005
0.01
0.015
0.02
0.025
*
0.03
0.035
0.04
0.045
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 6
Bagnold, R. A., 1966, An approach to the sediment transport problem from general physics, US
Geol. Survey Prof. Paper 422-I, Washington, D.C.
Brownlie, W. R., 1981, Prediction of flow depth and sediment discharge in open channels, Report
No. KH-R-43A, W. M. Keck Laboratory of Hydraulics and Water Resources, California
Institute of Technology, Pasadena, California, USA, 232 p.
Egiazaroff, I. V., 1965, Calculation of nonuniform sediment concentrations, Journal of Hydraulic
Engineering, 91(4), 225-247.
Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel
Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.
Ikeda, S., 1982, Incipient motion of sand particles on side slopes, Journal of Hydraulic
Engineering, 108(1), 95-114.
Neill, C. R., 1968, A reexamination of the beginning of movement for coarse granular bed
materials, Report INT 68, Hydraulics Research Station, Wallingford, England.
Nezu, I. and Nakagawa, H., 1993, Turbulence in Open-Channel Flows, Balkema, Rotterdam, 281
p.
Paintal, A. S., 1971, Concept of critical shear stress in loose boundary open channels, Journal of
Hydraulic Research, 9(1), 91-113.
Parker, G., Toro-Escobar, C. M., Ramey, M. and S. Beck, 2003, The effect of floodwater
extraction on the morphology of mountain streams, Journal of Hydraulic Engineering,
129(11), 885-895.
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 6 contd.
Parker, G., in press, Transport of gravel and sediment mixtures, ASCE Manual 54, Sediment
Enginering, ASCE, Chapter 3, downloadable from
http://cee.uiuc.edu/people/parkerg/manual_54.htm .
Proffitt, G. T. and A. J. Sutherland, 1983, Transport of non-uniform sediments, Journal of
Hydraulic Research, 21(1), 33-43.
Ramette, M. and Heuzel, M, 1962, A study of pebble movements in the Rhone by means of
tracers, La Houille Blanche, Spécial A, 389-398 (in French).
van Rijn, L., 1984, Sediment transport, Part II: Suspended load transport, Journal of Hydraulic
Engineering, 110(11), 1613-1641.
Schlichting, H., 1968, Boundary-Layer Theory, 6th edition. McGraw-Hill, New York, 748 p.
Seminara, G., Solari, L. and Parker, G., 2002, Bedload at low Shields stress on arbitrarily sloping
beds: failure of the Bagnold hypothesis, Water Resources Research, 38(11), 1249,
doi:10.1029/2001WR000681.
Shields, I. A., 1936, Anwendung der ahnlichkeitmechanik und der turbulenzforschung auf die
gescheibebewegung, Mitt. Preuss Ver.-Anst., 26, Berlin, Germany.
Tennekes, H., and Lumley, J. L., 1972, A First Course in Turbulence, MIT Press, Cambridge,
USA, 300 p.
Wiberg, P. L. and Smith, J. D., 1987, Calculations of the critical shear stress for motion of uniform
and heterogeneous sediments, Water Resources Research, 23(8), 1471-1480.
Wilcock, P. R., and Crowe, J. C., 2003, Surface-based transport model for mixed-size sediment,
34
Journal of Hydraulic Engineering, 129(2), 120-128.