CHAPTER 2: CHARACTERIZATION OF SEDIMENT AND GRAIN...

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




2

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.




3

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




4

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




5

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




6

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




7

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)




8

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.




9




10

CHAPTER 6:

THRESHOLD OF MOTION AND SUSPENSION

Rock scree face in Iceland.




11

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




12

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




13

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.




14

)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




15

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




16

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




17

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




18

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




19

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




20

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




21




22

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




23

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




24

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:




25

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




26




27

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)




28

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

qq

s

bbb

)(qqor)(qq cbbbb




29

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




30

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)




31

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




32

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.




33

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




34

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




35

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




36

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,




37




38

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]




39

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




40

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.




41

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




42

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




43

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




44

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




45




46

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




47

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




48

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




49

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




50

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.




51

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.




52

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.




53

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.




54

BEDFORMS IN THE LABORATORY AND FIELD: MULTIPLE-ROW

(LINGUOID) BARS

Linguoid bars in a flume in Tsukuba University, Japan: flow turned off.


Linguoid bars in the Fuefuki River, Japan. Image courtesy S. Ikeda.




55

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.


Braiding in the Ohau River, New Zealand. Image courtesy P. Mosley.




56

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




57

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




58

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




59 GSA Special Papers 2007 vol. 426 171-188




60

John B. Southard, Surface forms, 1978




61

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)




62

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*




63

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




64

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.




65

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.

Date post:	21-May-2020
Category:	Documents
Upload:	others
View:	13 times
Download:	0 times

CHAPTER 2: CHARACTERIZATION OF SEDIMENT AND GRAIN...

Documents