1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: RIPPLES
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.
flow
View of the Rum River, Minnesota USA
migration
Ripples in the Rum River, Minnesota
2
USA at very low flow;  ~ 10
- 20 cm.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: DUNES
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.)
flow
migration
Dunes in the North Loup River,
Nebraska USA. Two people are circled
3
for scale. Image courtesy
D. Mohrig.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: DUNE MIGRATION
Double-click on the image to see the video; video courtesy D. Mohrig.
4
rte-bookmohrigloup.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: ANTIDUNES
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
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.
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: CYCLIC STEPS (CHUTE AND POOL TRAINS)
Trains of cyclic steps occur in very
steep flows with supercritical Froude
numbers. They are long-wave
relatives of antidunes. The steps are
delineated by hydraulic jumps
(immediately downstream of which
the flow is locally subcritical). The
steps migrate upstream. These
features are also called chute-andpool topography.
hydraulic jump
flow
Train of cyclic steps in a small
laboratory channel at St. Anthony
Falls Laboratory. The water has been
dyed to aid visualization; two hydraulic
jumps can be seen in the figure.
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: CYCLIC STEPS (contd.)
Cyclic steps form in the field when slopes are steep, the flow is supercritical and
there is a plethora of sediment.
jumps
flow
Trains of cyclic steps in a coastal outflow channel on a beach in Calais,
France. Image courtesy H. Capart.
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: ALTERNATE BARS
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.
Alternate bars in the Naka River, an artificially
straightened river in Japan. Image courtesy S. Ikeda.
8
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TOUR OF BEDFORMS IN RIVERS: MULTIPLE-ROW LINGUOID BARS
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 coexist with dunes or antidunes.
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.
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORMS IN THE LABORATORY AND FIELD: DUNES
Dunes in a flume in Tsukuba
University, Japan: flow turned
off. Image courtesy H. Ikeda.
Dunes on an exposed point bar in the
meandering Fly River, Papua New Guinea
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
Rhine River, Switzerland
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications Ohau
to
River, New Zealand
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
2
 11.6 

RgD D
 where   b
 
,
Re

p
 Re 
RgD

p 

flow
migration
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
14
practice, ripples are observed only for D < 0.6 mm. Ripples can coexist with dunes.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SHIELDS DIAGRAM WITH CRITERION FOR RIPPLES
10
u  v s or   R f (Rep )
2
suspension
1
*
no suspension
 11 .6 

D   v or   
 Re 
p 

motion mod Brownlie
ripples
suspension
2

0.1
no ripples
ripples
no motion
modified Brownlie
0.01
1
10
100
1000
Rep
10000
100000
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
flow
migration
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.
flow
migration
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESPONSE OF FLOW TO BED UNDULATIONS:
INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS
Steady, uniform flow over a flat erodible bed (base flow; no bedforms) has flow depth
Ho and flow velocity Uo = qw/Ho. Unperturbed bed elevation is at  = 0. The bed is
then given a slight wavy perturbation of the form
 2x 
   sin

  
where ’ << Ho denotes the amplitude of the perturbation and  denotes the
wavelength of the perturbation. How does the flow and water surface respond to
such a perturbation?
flow
bed and water surface of base flow
Ho

H
bed and water surface of perturbed flow

'
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESPONSE OF FLOW TO BED UNDULATIONS:
INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS contd.
Consider inviscid (frictionless) steady 1D shallow water flow over an undulating bed.
The St. Venant shallow water equations simplify as follows:
H UH

0
t
x
 UH  qw  U 
UH U2H
1 H2


 g
 gH  Cf U2
t
x
2 x
x

qw
H

q2w  dH
d
 g  3 
 g
H  dx
dx

The equation in the box can be made dimensionless using the depth Ho of the base
flow:

   


H  HoH ,   Ho , x  Hox ,   H    Ho ,   H   


d
H
d

(1  Fr 2 )    
dx
dx
 3
, Fr  Fr H
2
2
o
q2w
, Fr 
gH3o
2
o
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESPONSE OF FLOW TO BED UNDULATIONS:
LINEAR INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS
Solving for the variation in flow depth,


dH
1
d
 

dx
(1  Fr 2 ) dx
 3
, Fr  Fr H
2
2
o
The variation in water surface elevation is given as

d d  
   (  H) 
dx dx


d
1
d


dx (1  Fr 2 ) dx
q2w
, Fr 
gH3o
2
o

Fr 2 d


(1  Fr 2 ) dx
Now the bed perturbation can be represented in dimensionless form as follows:


2H0

 

  Ho ,    sin(kx ) ,  
 1 , k 
Ho


Here  denotes the dimensionless amplitude of the bed perturbation and k denotes
the dimensionless wavenumber of the bed perturbation. We further write the
response of the depth and water surface elevation to the perturbation as



H  1  h  1  h sin(kx ) ,
  

    H  1   sin(kx ) ,
  
    h

h
where  denotes the
 dimensionless amplitude of the response of depth to the bed
19
perturbation, and  denotes the corresponding dimensionless response in
water surface elevation.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESPONSE OF FLOW TO BED UNDULATIONS:
LINEAR INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS contd.

Now as long as h << 1,
 3


 3

2
2
Fr  Fr H  Fro [1  h sin(kx)]  Fro [1  3h sin(kx)]  Fro2
2
2
o
With this approximation, substituting




H  1  h  1  h sin(kx ) ,
  

 
    H  1  (  h) sin(kx )
into


dH
1
d
 

dx
(1  Fr 2 ) dx
and


d
Fr 2 d
 

2
dx
(1  Fr ) dx
gives the results

h  
1



2
(1  Fro )

  
Fro2 

2
(1  Fro )
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SHALLOW-WATER RESPONSE OF WATER SURFACE TO BED PERTURBATION





 
   sin(kx ) ,   1   sin(kx ) ,   
 
When Fro < 1,    0 and the water
surface perturbation is out of phase with
the bed
is

 perturbation: the water surface
low (  0) where the bed is high (  0)
and the water surface is high where the
bed is low. According to long wave theory,
then, dunes can occur in subcritical flow
(Fro < 1)
flow
Fro2 

2
(1  Fro )


 
When Fr0 > 1,    0 and the depth
perturbation is in phase with the bed

perturbation: the water surface is high (  0)

where the bed is high (  0) and the water
surface is low where the bed is low.
According to long wave theory, then,
antidunes can occur in supercritical flow
(Fro > 1).
flow


21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PREDICTIONS OF LINEAR INVISCID SHALLOW-WATER THEORY FOR DUNES
AND ANTIDUNES
5
antidunes; water surface
responds strongly to
bed
4
3


Fro2
 

(1  Fro2 )
dunes; water surface
responds weakly to bed
2
1
 
ˆ ˆ
0
-1
Fro  1; resonant conditions,
high-amplitude standing waves
-2
-3
-4
-5
0
0.5
1
1.5
Fro
2
2.5
3
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
BEYOND THE SHALLOW-WATER APPROXIMATION:
POTENTIAL FLOW FORMULATION
The shallow-water theory of bedforms is not entirely accurate. This is because the
wavelength  of dunes and antidunes usually scales as a multiple of the flow depth H,
and so the condition H/ << 1 is usually not satisfied. In more precise terms, the
wavenumber of the bedforms k = 2Ho/ does not usually satisfy the condition k << 1.
A better view of bedforms is obtained by solving for the linearized potential flow over
a wavy bed. This formulation includes the vertical coordinate z as well as the
horizontal coordinate x, and describes the vertical as well as the horizontal structure
of the response of the flow to bed. Such a solution was first implemented by
Anderson (1953) and extended by Kennedy (1963).
condition of vanishing pressure at water surface
Let  = velocity potential function:
u, w     ,   ,  2  0
 x z 
flow
z

x

23
condition of vanishing normal velocity at bed
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
POTENTIAL FLOW FORMULATION contd.
In general, subcritical flow is a flow for which the water surface perturbation is
approximately out of phase with a bed perturbation, and supercritical flow is a flow for
which the water surface is approximately in phase with a bed perturbation. Potential
flow theory indicates that the border between subcritical and supercritical flow is a
function of both Froude number Fro and wavenumber k = 2Ho/ as follows:
Fro2 
tanh( k )
k
Now as k  0, Fro  1, indicating that for long bedforms the division between
subcritical and supercritical flow is given by the long wave (shallow-water) limit of 1.
If e.g. the bedform has a wavelength  equal to 5 Ho (a reasonable guess for many
dunes and antidunes), k = 1.26 and the borderline between subcritical and
supercritical flow is Fro = 0.82. That is, the zone of supercritical response
extends somewhat into the range Fro < 1, and antidunes can occur in flows for
which Fro < 1.
In general, lower-regime flow refers to truly subcritical flow in the sense Fro <
[tanh(k)/k]1/2, and upper-regime flow refers to truly supercritical flow in the
sense Fro > [tanh(k)/k]1/2. It is important to realize that part of the zone
of upper-regime flow is subcritical in the long-wave sense.
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
POTENTIAL FLOW FORMULATION contd.
In addition to the criterion
Fro 
tanh( k )
k
dividing subcritical from supercritical response, potential flow reveals another criterion
Fro 
1
k tanh( k )
further dividing the regime of supercritical flow. When Fro < [tanh(k)/k]1/2 both the
water surface and depth are out of phase with the bed, and the flow accelerates over
bed crests and decelerates over bed troughs. This gives rise to downstreammigrating dunes.
When Fro > [tanh(k)/k]1/2 and Fro < [k tanh(k)]-1/2, both the water surface and the
depth are in phase with the bed, and the flow decelerates over crests and accelerates
over troughs. This gives rise to upstream-migrating antidunes.
When Fro > [tanh(k)/k]1/2 and Fro > [k tanh(k)]-1/2 the water surface is in phase with the
bed, so the bedforms are antidunes, but the depth is out of phase with the bed, and
the flow accelerates over the crests and decelerates over the troughs. These
antidunes (which cannot be obtained from the St. Venant formulation) thus
25
migrate downstream.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
FLOW IN THE DUNE REGIME
Fro < [tanh(k)/k]1/2
Water surface is out of phase with the bed.
Depth variation is out of phase with the bed
Flow accelerates from trough to crest.
Sediment transport increases from trough to crest.
Bedform migrates downstream.
Bedform becomes asymmetric.
flow
sediment
transport
deposits
erodes
downstream
upstream
flow accelerates over crest:
dune migrates downstream
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
FLOW IN THE UPSTREAM-MIGRATING ANTIDUNE REGIME
[tanh(k)/k]1/2 < Fro < [k tanh(k)]-1/2
Water surface is in phase with the bed.
Depth variation is in phase with the bed
Flow decelerates from trough to crest.
Sediment transport decreases from trough to crest.
Bedform migrates upstream (or hardly at all).
Bedform stays symmetric.
sediment
transport
flow
deposits
upstream
erodes
downstream
flow decelerates over crest:
antidune migrates upstream
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
FLOW IN THE DOWNSTREAM-MIGRATING ANTIDUNE REGIME
[k tanh(k)]-1/2 < Fro
Water surface is in phase with the bed.
Depth variation is out of phase with the bed.
Flow accelerates from trough to crest.
Sediment transport increases from trough to crest.
Bedform migrates downstream.
Bedform becomes asymmetric.
These are antidunes that look like dunes: not too common, but they are observed.
flow
sediment
transport
erodes
upstream
deposits
downstream
flow accelerates over crest:
antidune migrates downstream
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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PHASE DIAGRAM FOR DUNES AND ANTIDUNES BASED ON LINEAR
POTENTIAL THEORY OVER A WAVY BED
3
2.5
Fro 
1
k tanh( k )
downstream-migrating
2
Fro
long wave limit
1.5
supercritical response
(antidunes possible)
upstream-migrating
1
Fro 
0.5
tanh( k )
k
subcritical response
(dunes possible)
downstream-migrating
0
0
0.5
1
k
1.5
2
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
LOWER- AND UPPER-REGIME PLANE BED
Dunes usually do not form in gravel-bed streams. This is because such streams
usually fall into a regime known as lower-regime plane bed, for which the flow is
subcritical and neither dunes nor ripples form. Chabert and Chauvin (1963) have
described this regime experimentally, and Engelund (1970) and Fredsoe (1974) have
developed stability analyses for bedforms which describe this regime.
In sand-bed streams, there is a second regime in the vicinity of the line Fro =
[tanh(k)/k]1/2 within which neither dunes nor antidunes form. This regime is known as
upper-regime plane bed. Engelund (1970) and Fredsoe (1974) have explained this
region as one of competition between the effects of bedload and suspended load.
The former favors the formation of dunes, and the latter favors the formation of
antidunes. Within the regime of upper-regime plane bed, the two effects cancel each
other, and a plane bed prevails.
A rough sketch of the zones for lower-regime plane bed, dunes, upper regime plane
bed, upstream-migrating antidunes and downstream-migrating antidunes is given in
the following diagram based on potential flow. It should be pointed out, however, that
the analyses of Engelund (1970) and Fredsoe (1974) result in somewhat modified
30
criteria for the divisions between supcritical and supercritical flow, and
upstream and downstream migrating antidunes. (See Engelund and Fredsoe, 1982).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROXIMATE PHASE DIAGRAM FOR 1D BEDFORMS
2
Fro 
1
k tanh( k )
1.5
Fro
cyclic
steps
antidunes
1
upper-regime plane-bed
Fro 
tanh( k )
k
dunes
0.5
lower-regime plane-bed
0
0
0.5
1
k
1.5
2
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXPERIMENTAL RESULTS OF CHABERT AND CHAUVIN (1963)
Chabert and Chauvin (1963) report on experiments which yield a threshold conditions
for ripples that is very similar to that proposed by Engelund and Hansen (1967).
In addition, they obtain a criterion for the threshold between lower-regime plane bed
and dunes that can be approximated as
  2.72 c
where c* is given by the modified Brownlie relation,

c
0.6
p
  0.5 [0.22 Re
( 7.7 Rep0.6 )
 0.06  10
]
Thus in the limit of coarse material (Rep >> 1, gravel-bed streams) dunes should not
form until * exceeds 0.0816. It was seen from Slide 21 of Chapter 3, however, that
this condition is not common for gravel-bed streams at bankfull flow.
Dunes can form in gravel-bed streams if the conditions are right; e.g. see Dinehart
(1992).
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SHIELDS DIAGRAM INCLUDING RESULTS OF CHABERT AND CHAUVIN (1963)
10
 11 .6 

D   v or    
 Re 
p 

u  v s or   R f (Rep )
2
2
suspension
1
motion mod Brownlie
ripples
suspension
dunes C&C
ripples C&C
extrap C&C dunes
*
C&C ripples/no ripples
C&C no dunes/dunes
dunes
0.1
ripples
lower regime plane bed
no motion
extrapolated C&C
no dunes/dunes
modified Brownlie
0.01
1
10
100
1000
Rep
10000
100000
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CYCLIC STEPS (CHUTE-AND-POOL TOPOGRAPHY)
Trains of cyclic steps occur in very
steep flows with supercritical Froude
numbers. They are long-wave relatives
of antidunes (Winterwerp et al., 1992;
Taki and Parker, in press). The steps
are delineated by hydraulic jumps
(immediately downstream of which the
flow is locally subcritical). The steps
migrate upstream. These features are
also called chute-and-pool topography
(Simons et al., 1965). Their regime of
formation is schematized in the
previous Slide 30.
hydraulic jump
flow
Train of cyclic steps in a small
laboratory channel at St. Anthony
Falls Laboratory. The water has been
dyed to aid visualization; two hydraulic
jumps can be seen in the figure.
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
1D BEDFORM REGIME DIAGRAMS
A number of diagrams have been proposed to characterize bedform regime. The
most useful of these are dimensionless. Following the analysis of Vanoni (1974)
and Parker and Anderson (1978), the following general relation can be posited:
bedform type  fn( X1, X2 ,Rep ,R,  g )
Here Rep, R and g have their standard meanings of explicit particle Reynolds
number, sediment submerged specific gravity and geometric standard deviation of
bed sediment. In addition, X1 and X2 are two dimensionless parameters describing
the flow which must be independent from each other. Suitable choices include
Shields number * = u*2/(RgDs50), Froude number Fr = qw/[(gH)1/2H], bed slope S,
dimensionless depth Ĥ = H/Ds50, dimensionless unit stream power US/vs etc.
Vanoni (1974) has provided a relatively complete set of bedform diagrams for sand,
including dunes, antidunes, ripples, flat (by which he means lower-regime flat bed),
transition (by which he means upper-regime flat bed) and chute-and-pool
topography (in which cyclic steps should be included. Vanoni chooses X1 = Fr and
X2 = Ĥ , In addition, he assumes that R is constant at 1.65, and he neglects the
effect of g (i.e. assumes uniform material), so that
bedform type  fn(Fr, Ĥ, Rep )
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORM REGIME DIAGRAM 1 OF VANONI (1974)
Fr
Ĥ
Bedform Chart for D50 = 0.011 mm and 0.088 – 0.15 mm (Rep = 0.11 and 2.4 – 5.4)
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORM REGIME DIAGRAM 2 OF VANONI (1974)
Fr
Ĥ
Bedform Chart for D50 = 0.12 – 0.20 mm (Rep = 3.9 – 8.3)
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORM REGIME DIAGRAM 3 OF VANONI (1974)
Fr
Ĥ
Bedform Chart for D50 = 0.15 – 0.32 mm (Rep = 5.4 – 16.8)
38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORM REGIME DIAGRAM 4 OF VANONI (1974)
Fr
Ĥ
Bedform Chart for D50 = 0.23 – 0.45 mm (Rep = 10.2 – 28.0)
39
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORM REGIME DIAGRAM 5 OF VANONI (1974)
Fr
Ĥ
Bedform Chart for D50 = 0.4 – 0.6 mm (Rep = 23.5 – 43.1)
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORM REGIME DIAGRAM 6 OF VANONI (1974)
1.0
Fr
0.1
102
Ĥ
103
Bedform Chart for D50 = 0.93, 1.20 and 1.35 mm (Rep = 83.3, 122, 146)
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDFORM REGIME DIAGRAM
OF ENGELUND AND HANSEN
(1966)
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.
U/u*
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.
Fr
42
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 8
Anderson, A. G., 1953, The characteristics of sediment waves formed by flow in open channels,
Proceedings, 3rd Midwest Conference on Fluid Mechanics, University of Minnesota.
Chabert, J. and Chauvin, J. L., 1963, Formation des dunes et de rides dans les modeles fluviaux,
Bulletin C.R.E.C., No. 4.
Dinehart, R. L., 1992, Evolution of coarse gravel bed forms; field measurements at flood stage,
Water Res. Res., 28(10), 2667-2689.
Engelund, F., 1970, Instability of erodible beds, J. Fluid Mech., 42(2).
Engelund, F. and Fredsoe, J., 1982, Sediment ripples and dunes, Annual Review of Fluid
Mechanics, 14, 13-37.
Engelund, F. and Hansen, E., 1966, Hydraulic resistance in alluvial streams, Acta Polytechnica
Scandanavica, V. Ci-35.
Engelund, F. and Hansen, E., 1967, A Monograph on Sediment Transport, Technisk Forlag,
Copenhagen, Denmark.
Fredsoe, J., 1974, On the development of dunes in erodible channels, J. Fluid Mech., 64(1), 116.
Kennedy, J. F., 1963, The mechanics of dunes and antidunes in erodible bed channels, J, Fluid
Mech., 16(4).
Parker, G. and Anderson, A., 1977, Basic principles of river hydraulics, J. Hydr. Engrg., 103(9),
1077-1087.
43
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 8 contd.
Simons, D. B., Richardson, E. V. and Nordin, C. F., 1965, Sedimentary structures
generated by flow in alluvial channels, Special Pub. No. 12, Am. Assoc. Petrol. Geologists.
Taki, K.. And Parker, G., 2005, Transportational cyclic steps created by flow over an erodible bed.
Part 1. Experiments, J. Hydr. Res., in press, downloadable from
http://cee.uiuc.edu/people/parkerg/preprints.htm .
Vanoni, V., 1974, Factors determining bed forms of alluvial streams, Journal of Hydraulic
Engineering, 100(3), 363-377.
Winterwerp, J. C., Bakker, W. T., Mastbergen, D. R., and Van Rossum, H, 1992,
Hyperconcentrated sand-water mixture flows over erodible bed, J. Hydr. Engrg., 119(11),
1508-1525.
44