Ch 10 Suspended Sediment

8/12/2019 Ch 10 Suspended Sediment

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1D SEDIMENT TRANSPORT MORPHODYNAMICS

with applications to

RIVERS AND TURBIDITY CURRENTS Gary Parker November, 2004

1

CHAPTER 10:

RELATIONS FOR THE ENTRAINMENT AND 1D TRANSPORT OFSUSPENDED SEDIMENT

Dredging mine-derived of sand carried down predominantly bysuspension in the Ok Tedi, Papua New Guinea


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2

THE STRATEGY

Consider the case of an equilibrium suspension in an equilibrium (normal) 1D open

channel flow. Returning to the equation of conservation of suspended sediment

from Chapter 4,

x

z

b

c

u

p

bcE

ss

H

0v

xqdzc

t

bcE

Under equilibrium conditions the dimensionless entrainment rate E is equal to

the near-bed average concentration of suspended sediment! We can:

Obtain empirical relation for E versus boundary shear stress for equilibrium

conditions.

With luck, the relation can be applied to conditions that are not too strongly

disequilibrium.


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3

THE STRATEGY contd.

For equilibrium open-channel suspensions,

1. Determine a position z = b near the bed and measure the volume concentration

of suspended sediment averaged over turbulence there. Note that the

definition of b is peculiar to each researcher, but in general b/H


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4

Smith and McLean (1977) offer the following entrainment relation.

The reference height is evaluated at what the authors describe as the top of the

bedload layer; where ksdenotes the Nikuradse roughness height,

The authors give no guidance for the choice of bc. It is suggested here that it

might be computed as bc=RgDc*, where c* is given by the Brownlie (1981) fit

to the Shields relation:

ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL

Garcia and Parker (1991) reviewed seven entrainment relations andrecommended three of these; Smith and McLean (1977), van Rijn (1984) and

(surprise surprise) Garcia and Parker (1991).

RgD

,0024.0,11165.0 bssobc

bso

bc

bso

E

bcbs

s

bs

bcbs

s

,

Rgk

3.261

,1

k

b

DgD

,1006.022.0 p)7.7(6.0

pc

6.0p

R

ReRe

Re


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5

ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL contd.

The entrainment relation of van Rijn (1984) takes the form

The reference level b is set as follows:

b = 0.5 b, where b= average bedform height, when known;

b = the larger of the Nikuradse roughness height ksor 0.01 H

when bedforms are absent or bedform height is notknown.

The critical Shields number can be evaluated with the Brownlie (1981) fit to

the Shields curve:

DRgD

,1b

D015.0 p

2.0

p

5.1

c

s50 ReReE

DgD

,1006.022.0 p)7.7(

6.0pc

6.0

p

RReRe

Re


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6

Garcia and Parker (1991) use a reference height b = 0.05 H;

Wright and Parker (2004) found that the relation of Garcia and Parker

(1991) performs well for laboratory flumes and small to medium sand-bed

streams, but does not perform well for large, low-slope streams. Wright

and Parker (2004) have thus amended the relationship to cover this latter

range as well Again the reference height b = 0.05 H. This corrects Garcia

and Parker to cover large, low-slope streams:

7bss

6.0

p

s

su

5

u

5

u 10x3.1A,u,v

uZ,

Z

3.0

A1

AZ

ReE

707.06.0

p

s

su

5

u

5

u 10x7.5A,Sv

uZ,

Z

3.0

A1

AZ

ReE

ENTRAINMENT RELATIONS FOR UNIFORM MATERIAL contd.


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7

Garcia and Parker (1991) generalized their relation to sediment mixtures. Therelation for mixtures takes the form

where Fidenotes the fractions in the surface layer and denotes the arithmetic

standard deviation of the bed sediment on the scale. The reference height b is

again equal to 0.05 H.

Wright and Parker (2004) amended the above relation so as to apply to large,

low-slope sand bed rivers as well as the types previously considered by Garcia

and Parker (1991). The relation is the same as that of Garcia and Parker (1991)

except for the following amendments:

ENTRAINMENT RELATIONS FOR SEDIMENT MIXTURES

7m

ii

pi

2.0

50

i6.0pi

si

smui

5

ui

5

ui

i

iui

10x3.1A,298.01

DRgD,

D

D

v

uZ,

Z3.0

A1

AZ

F

EE

ReRe

2.0

50

i08.06.0

pisi

s

mui D

D

Sv

u

Z

Re

7

10x8.7A


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8

McLean (1992; see also 1991) offers the following entrainment formulation forsediment mixtures. Let ETdenote the volume entrainment rate per unit bed area

summed over all grain sizes, pidenote the fractions in the ith grain size range in the

bedload transport and psbi= Ei/ETdenote the fractions in the ith grain size range in

the sediment entrained from the bed. Then where pdenotes bed porosity,

ENTRAINMENT RELATIONS FOR SEDIMENT MIXTURES contd.

004.0,1111E obc

bso

bc

bsopT

bc

cbs

ssis

csi

cs

sis

iN

1i

ii

iisbi u,u,1v/ufor

uv

uu

1v/ufor1

,

p

pp

1A1

1A

D)D(,)D(a

Damaxb

bc

bs2

bc

bs1

8484B

84Bo

84D0709.0)nD(022.0)nD(0204.0A

68.0A,056.0a,12.0a

84

2

842

10D

The critical boundary shear stress bcis evaluated using bed material D50;

again the Brownlie (1981) fit to the Shields curve is suggested here.


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9

yxc)vwyxc)vw

zxvczxvczyuczyuczyxct

zzsszss

yysysxxsxss

LOCAL EQUATION OF CONSERVATION OF SUSPENDED SEDIMENT

0z

c)vw(

y

vc

x

uc

t

c s

x x+x

y+yy

z+z

z

Once entrained, suspended sediment can be carried about by the turbulent flow.

Let c denote the instantaneous concentration of suspended sediment, and (u, v,

w) denote the instantaneous flow velocity vector. The instantaneous velocity

vector of suspended particles is assumed to be simply (u, v, w - vs) where vs

denotes the terminal fall velocity of the particles in still water. Mass balance of

suspended sediment in the illustrated control volume can be stated as

or thus


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10

AVERAGING OVER TURBULENCE

www,vvv,uuu,ccc

0wvuc

t

A

t

A,BABA

In a turbulent flow, u, v, w and c all show fluctuations in

time and space. To represent this, they are decomposed

into average values (which may vary in time and space at

scales larger than those characteristic of the turbulence)

and fluctuations about these average values.

By definition, then,

t

u

u

u

The equation of conservation of suspended sediment mass is now averaged over

turbulence, using the following properties of ensemble averages: a) the averageof the sum = the sum of the average and b) the average of the derivative = the

derivative of the average, or


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12

The convective flux of any quantity is the quantity per unit volume times the velocity itis being fluxed. So, for example, the convective flux of streamwise momentum in the

upward direction is wu = wu. The viscous shear stress acting in the x (streamwise)

direction on a face normal to the z (upward) direction is

LOCAL STREAMWISE MOMENTUM CONSERVATION

z

u

x

w

z

uzx

zSyxgzypzyp

yxyxyxwuyxwu

zxvuzxvuzyuuzyuuzyxut

xxx

zzxzzzxzzz

yyyxxx

The balance of streamwise momentum in the control

volume requires that:

(streamwise momentum)/t = net convective inflow

of momentum + net shear force + net pressure force

+ downslope force of gravity

x x+x

y+yyz+z

z


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13

A reduction yields the relation

Averaging over turbulence in the same way as before yields the result

where

Here denotes the z-x component of the Reynolds stressgenerated by

the turbulence; the term is known as the Reynolds fluxof streamwise

momentum in the upward direction. For fully turbulent flow, the Reynolds stressRzxis usually far in excess of the viscous stress , which can be dropped.

LOCAL STREAMWISE MOMENTUM CONSERVATION contd.

gSz

u

x

p1

z

uw

y

uv

x

u

t

u2

22

gSzz

1

x

p1

z

wu

y

vu

x

u

t

u Rzxzx2

wu,zu

Rzxzx

wuRzx wu

zx


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14

The shear Reynolds stress Rzxis

abbreviated as ; its value at the bed is

b.. When the flow is steady and uniform

in the x and y directions, streamwise

momentum balance becomes

LOCAL STREAMWISE MOMENTUM CONSERVATION FOR NORMAL FLOW

gSz

1

x

p1

z

wu

y

vu

x

u

t

u 2

x

z

b

c

u

p

or thus

Integrating this equation under the condition of vanishing shear stress at the

water surface z = H yields the result

gSdz

d

gHS,H

z

1 bb



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15

REYNOLDS FLUX OF SUSPENDED SEDIMENT

The terms denote convective Reynolds fluxesof suspended

sediment. They characterize the tendency of turbulence to mix suspended

sediment from zones of high concentration to zones of low concentration, i.e.

down the gradient of mean concentration. In the case illustrated below

concentration declines in the positive z direction; turbulence acts to mix the

sediment from the zone of high concentration (low z) to the zone of low

concentration (high z).

cwandcv,cu

0cw0w,0c

0cw0w,0c

z



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16

REYNOLDS FLUX OF STREAMWISE MOMENTUM

The shear stress , or equivalently the Reynolds flux ofstreamwise (x) momentum in the upward (z) direction characterizes the tendency of

turbulence to transport streamwise momentum from high concentration to low. In the

case of open channel flow, the source for streamwise momentum is the downstream

gravity force term gS. This momentum must be fluxed downwardtoward the bed

and exited from the system (where the loss of momentum is manifested as a

resistive force balancing the downstream pull of gravity) in order

wuRzx wu

low streamwise

momentum u:u'0, v'


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17

REPRESENTATION OF REYNOLDS FLUX WITH AN EDDY DIFFUSIVITY

The concentration of any quantity in a flow is the quantity per unit volume. Thusthe concentration of streamwise momentum in the flow is u and the volume

concentration of suspended sediment is c. The tendency for turbulence to mix

any quantity down its concentration gradient (from high concentration to low

concentration) can be represented in terms of a kinematic eddy diffusivity:

Reynolds flux of suspended sediment in the z direction:

Reynolds flux of streamwise momentum in the z direction:

In the above relations stis the kinematic eddy diffusivity of suspended sediment

[L2/T] and tis the kinematic eddy diffusivity (eddy viscosity) of momentum.

z

cwc st

zuwu t

z)z(c

0dzcdcw0

dzcd

st



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18

EDDY VISCOSITY FOR TURBULENT OPEN CHANNEL FLOW

The standard equilibrium velocity profile for hydraulically rough turbulent open-channel flow is the logarithmic profile;

where = 0.4 and u*= (gHS)1/2. The eddy diffusivity of momentum can be back-

calculated from this equation;

Solving for t

, a parabolic form is obtained;

or

ss k

z30n

15.8

k

zn

1

u

u

H

z1u

z

u

dz

udwu 2tt

H

z1zut

H

z,1

Hu

t

Hut



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19

EQUILIBRIUM VERTICAL DISTRIBUTION OF SUSPENDED SEDIMENT

According to the Reynolds analogy, turbulence transfers any quantity, whether itbe momentum, heat, energy, sediment mass, etc. in the same fundamental way.

While it is an approximation, it is a good one over a relatively wide range of

conditions. As a result, the following estimate is made for the eddy diffusivity of

sediment:

For steady flows that are uniform in the x and z directions maintaining a

suspension that is similarly steady and uniform, the equation of conservation of

suspended sediment reduces to

Hz1zutst

x

z

b

c

u

p

z

cw

y

cv

x

cu

z

cv

z

cw

y

cv

x

cu

t

cs



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20

EQUILIBRIUM SUSPENSIONS contd.

The balance equation of suspended sediment thus becomes

This equation can be integrated under the condition of vanishing net sediment

flux in the z direction at the water surface to yield the result

i.e. the upward flux of suspended driven by turbulence from high concentration

(near the bed) to low concentration (near the water surface) is perfectly balanced

by the downward flux of suspended sediment under its own fall velocity. The

Reynolds flux F can be related to the gradient of the mean concentration as

The balance equation thus reduces to:

H

z1zu,

dz

cdcwF stst

cwF,0dz

cdv

dz

dFs

0cvF s

0cvdz

cd

H

z

1zu s



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21

SOLUTION FOR THE ROUSE-VANONI PROFILE

The balance equation is:

The boundary condition on this equation is a specified upward flux, or

entrainment rate of sediment into suspension at the bed:

Rouse (1939) solved this problem and obtained the following result,

which is traditionally referred to as the Rouse-Vanoni profile.

Evdz

cd

H

z1zuF s

bz

bz

0c

H

z1zu

v

dz

cd s

H

b,

H

z,Ec,

/)1(

/1

c

cbb

u

v

bbb

s



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22

REFERENCE LEVEL

The reference level cannot be taken as zero. This is because turbulence cannot

persist all the way down to a solid wall (or sediment bed). No matter whether the

boundary is hydraulically rough or smooth, essentially laminar effects must

dominate right near the wall (bed).

It is for this reason that the logarithmic velocity law

yields a value for of - at z = 0. The point of vanishing velocity is reached at z

= ks/30. Since the eddy diffusivity from which the profile of suspended sediment

is computed was obtained from the logarithmic profile, it follows that cannot be

computed down to z = 0 either. The entrainment boundary condition must be

applied at z = b ks/30.

ss k

z30n

15.8

k

zn

1

u

u

u

c



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23

AND NOW ITS TIME FOR SPREADSHEET FUN!!

Go toRTe-bookRouseSpreadsheetFun.xlsRouse-Vanoni Equilibrium Suspended Sediment Profile Calculator

Input

b/H 0.05

vs 3 cm/s

u 0.2 m/s

c/cb z/H ref u/vs 6.6667 1 0.05

0.756 0.1

0.635 0.15

0.557 0.2

Sample Fall Velocities, 0.5 0.25

R = 1.65, = 0.01 cm2/s 0.455 0.30.418 0.35

vs D 0.386 0.4

cm/s m 0.357 0.450.0000421 1.0 0.331 0.5

0.0002031 2.0 0.307 0.55

0.0010048 4.0 0.285 0.60.0048709 8.0 0.263 0.65

0.0356491 20.0 0.241 0.7

0.0816579 30.0 0.22 0.75

0.1798665 45.0 0.197 0.8

0.3256999 62.0 0.173 0.85

0.5117601 80.0 0.145 0.9

0.7484697 100.0 0.11 0.95

1.0785878 125.0 0.077 0.98

1.4376162 150.0 0.046 0.995

2.2156534 200.0 0 1

3.04447576 250 0 0.05

5.65004674 400 1 0.05

The above values were computed from the Dietrich (1982) fall velocity relation

Rouse-Vanoni Profile of Suspended Sediment

Concentration

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.2 0 0.2 0.4 0.6 0.8 1

bc

c

H

z

Ec,

H

b,

H

z,

/)1(

/1

c

cbb

u

v

bbb

s

This spreadsheetallows calculation

of the suspended

sediment profile

from specified

values of b/H, vs

and u*using theRouse-Vanoni

profile.



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24

1D SUSPENDED SEDIMENT TRANSPORT RATE FROM EQUILIBRIUM SOLUTION

H

b

H

0s dzcudzcuq

H

b,

H

z,

/)1(

/1Ec b

u

v

bb

s

5.8k

zn

1u

k

z30n

uu

cc

In order to perform the calculation, however, it is necessary to know the velocityprofile over a bed which may include bedforms. This velocity profile may be

specified as

)Cz(c

c

2/1f

e

H11k

k

H11n

1CCz

The volume suspended sediment transport rate per unit width is qscomputed as

)z(u

where kcis a composite roughness height. If bedforms are absent, k

c= k

s=

nkDs90. If bedforms are present, the total friction coefficient Cf= Cfs+ Cffmay be

evaluated (using a resistance predictor for bedforms if necessary) and kcmay be

back-calculated from the relation



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25

b

cs

s ,

k

H,

v

uEHuq

1D SUSPENDED SEDIMENT TRANSPORT RATE FROM EQUILIBRIUM SOLUTION

It follows that qsis given by the relations

1

c

u

v

bbb

cs b

s

dk

H

30n/)1(

/)1(

,k

H

,v

u

The integral is evaluated easily enough using a spreadsheet. This is done in the

next chapter.



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26

CLASSICAL CASE OF DISEQUILIBRIUM SUSPENSION: THE 1D PICKUP

PROBLEM

Consider a case where sediment-free equilibrium open-channel flow over a rough,

non-erodible bed impinges on an erodible bed offering the same roughness.

H

ri id bed erodible bed

cu

The flow can be considered quasi-steady over time spans shorter than that by

which significant bed degradation occurs.

The flow but not the suspended sediment profile can be considered to be at

equilibrium.



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27

THE 1D PICKUP PROBLEM contd.

H

rigid bed erodible bed

cu

z

F

z

cvw

y

cv

x

cu

t

cs

z

c

zz

cv

x

cu ts

x

z

0c,Evz

c,0

z

ccv

0xs

bz

t

Hz

ts

xas)z(c)x,z(c equilSolution yields

the result that

Can be used to find

adaptation length Lsrfor

suspended sediment

Governing equation

Boundary conditions

A method for estimating Lsris given in Chapter 21.



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28

Should the formulation be

with E computed based on local flow conditions, or

with qscomputed from the quasi-equilibrium relation

applied to local flow conditions?

The answer depends on the characteristic length L of the phenomenon of interest

(one meander wavelength, length of alluvial fan etc.) compared to the adaptation

length Lsrequired for the flow to reach a quasi-equilibrium suspension. If L < Lsthe former formulation should be used. If L > Lsthe latter formulation can be used.

WHICH VERSION OF THE EXNER EQUATION OF BED SEDIMENT

CONTINUITY SHOULD BE USED FOR A MORPHODYNAMIC PROBLEM

CONTROLLED BY SUSPENDED SEDIMENT?

Ecvxt

)1( bsp

bq

-

bcs

s ,k

H

,v

uEHu

q

xxxt)1( p

tsb qqq

---

Selenga Delta, Lake Baikal,Russia: image from NASAhttps://zulu.ssc.nasa.gov/mrsid/mrsid.pl



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29

SELF-STRATIFICATION OF THE FLOW DUE TO SUSPENDED SEDIMENT

A flow is stably stratifiedif heavier fluid lies below lighter fluid. The densitydifference suppresses turbulent mixing.

The city of Phoenix, Arizona, USA

during an atmospheric inversion

Well, somewhere down there

Sediment-laden flows are self-stratifying

Rc

)Rc1(c)c1(

e

susp

ssusp

c

lighter up here

heavier down here

Here susp= density of the suspension and e=

fractional excess density due to the presence ofsuspended sediment.



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30

FLUX AND GRADIENT RICHARDSON NUMBERS

The damping of turbulence due to stable stratification is controlled by the flux

Richardson number Rif.

[Rate of expenditure of turbulent kinetic

energy in holding the (heavy) sediment in

suspension]/[Rate of generation of

turbulent kinetic energy by the flow]

dz

udwu

wcRg

dz

udwu

wg ef

Turbulence is not suppressedat all for Rif= 0. Turbulence is killed completely

when Rifreaches a value near 0.2 (e.g. Mellor and Yamada, 1974)

dz

cd

wc,dz

ud

wu tt Now let

Thenf2

dz

ud

dz

cdRg

where Ridenotes the gradient

Richardson Number



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31

SUSPENSION WITH SELF-STRATIFICATION:

SMITH-MCLEAN FORMULATION

These relations may be solved iteratively for concentration and velocity profiles in

the presence of stratification.

2

tot

dz

ud

dz

cdRg

7.41H

z1zu7.41

0cv

dz

cdst

H

z1u

dz

ud 2t

Ev

dz

cds

bz

t

c

bz

k

b30ln

1

u

u

The balance equations and boundary conditions take the forms:

Smith and McLean (1977), for example, propose the following relation for damping

of mixing due to self-stratification:



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32

SUSPENSION WITH SELF-STRATIFICATION:

GELFENBAUM-SMITH FORMULATION

These relations may be solved iteratively for concentration and velocity profiles in

the presence of stratification. The workbook RTe-bookSuspSedDensityStrat.xlsprovides a numerical implementation.

2

tot

dzud

dz

cdRg

,1.351

35.1X,

X101

RiRi

Ri

0cv

dz

cdst

H

z1u

dz

ud 2t

Ev

dz

cds

bz

t

c

bz

k

b30ln

1

u

u

It also uses the specification b = 0.05 H. The balance equations and boundary

conditions take the forms:

The workbook RTe-bookSuspSedDensityStrat.xls implements the formulation for

stratification-mediated suppression of mixing due to Gelfenbaum and Smith (1986);



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33

ITERATION SCHEME

The governing equations for flow velocity and suspended sediment concentration

can be integrated to give the forms

z

bt

sb

z

bt

2

b dzv

expcc,dzH

z1

uuu

where

Ec,kb30lnuu b

cb

The relations of the previous slide can be rearranged to give 2

t

2

t

s

H

z1

u

cv

Rg

Ri

The iteration scheme is commenced with the logarithmic velocity profilefor velocity and the Rouse-Vanoni profile for suspended sediment:

,

H

b,

H

z,

/)1(

/1cc,

k

z30n

uu b

u

v

bb

b

)0(

c

)0(

s

H

z1zu)0(t

where the superscript (0) denotes the 0thiteration (base solution).



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34

ITERATION SCHEME contd.

The iteration then proceeds as

,

H

z1

u

cv

Rg

2

)n(t

2

)n(

)n(t

s

1)(nRi

z

b )1n(t

sb

)1n(z

b )1n(t

2

b

)1n( dzv

expcc,dzH

z1

uuu

1)(n

1)(n)0(

t)1n(

t1.351

35.1X,

X101

Ri

Ri

)n(u )1n(c Iteration continues until is tolerably close to and is tolerably close to.

A dimensionless version of the above scheme is implemented in the workbook Rte-

bookSuspSedDensityStrat.xls. Moredetails about the formulation are provided in the

document Rte-bookSuspSedStrat.doc.

)1n(u )n(c



35/38



35

INPUT VARIABLES FOR Rte-bookSuspSedDensityStrat.xls

The first step in using the workbook is to input the parameters R+1 (sedimentspecific gravity), D (grain size), H (flow depth), kc(composite roughness height

including effect of bedforms, if any), u(shear velocity) and (kinematic viscosity

of water). When bedforms are absent, the composite roughness height kcis

equal to the grain roughness ks. In the presence of bedforms, kcis predicted from

one of the relations of Chapter 9 and the equations

The user must then click a button to clear any old output. After this step, the user

is presented with a choice. Either the near-bed concentration of suspended

sediment can be specified by the user, or it can be calculated from the Garcia-

Parker (1991) entrainment relation. In the former case, a value for must be

input. In the latter case, a value for the shear velocity due to skin friction usmustbe input. It follows that in the latter case uscan be predicted using one of the

relations of Chapter 9.

Once either of these options are selected and the appropriate data input, a click

of a button performs the iterative calculation for concentration and velocity

profiles. Note: the iterative scheme may not always converge!

bc

2/1

f)Cz(c CCz,

eH11k

bc



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36

Dimensionless Velocity Profiles versus Normalized Depth

1

10

100

0.01 0.1 1

uno(

nos

tratification),un

(stratification

uno

un

Dimensionless Concentration Profiles versus NormalizedDepth

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

cno(

nos

t

ratification),cn

(stra

tification

cno

cn

SAMPLE CALCULATION (a) with Garcia-Parker entrainment relation

bc

c

u

u

H

z

Hz

Stratification included

qswith stratification = 0.72 x

qswithout stratificationStratification neglected


Stratification neglected000115.0cb

R 1.65D 0.2 mm

H 5 m

kc 50 mm

u* 4 cm/s

u*s 2 cm/s

0.01cm2/s



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37

Dimensionless Velocity Profiles versus Normalized Depth

1

10

100

0.01 0.1 1

uno(

nos

tratif

ication),un

(stratific

ation

unoun

Dimensionless Concentration Profiles versus NormalizedDepth

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

cno(

nos

tratification),cn

(stratification

cno

cn

SAMPLE CALCULATION (b) with Garcia-Parker entrainment relation

bc

c

uu

H

z

H

z


Stratification neglected

Stratification neglected


qswith stratification = 0.39 x

qswithout stratification

00363.0cb

R 1.65D 0.2 mm

H 5 m

kc 50 mm

u* 6 cm/s

u*s 4 cm/s

0.01cm2/s



38/38



38

REFERENCES FOR CHAPTER 10

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.

Garca, M., and G. Parker, 1991, Entrainment of bed sediment into suspension, Journal of

Hydraulic Engineering, 117(4): 414-435.

Gelfenbaum, G. and Smith, J. D., 1986, Experimental evaluation of a generalized suspended-

sediment transport theory, in Shelf and Sandstones, Canadian Society of Petroleum

Geologists Memoir II, Knight, R. J. and McLean, J. R., eds., 133

144.McLean, S. R., 1991, Depth-integrated suspended-load calculations, Journal of Hydraulic

Engineering, 117(11): 1440-1458.

McLean, S. R., 1992, On the calculation of suspended load for non-cohesive sediments, 1992,

Journal of Geophysical Research, 97(C4), 1-14.

Mellor, G. and Yamada, T., 1974, A hierarchy of turbulence closure models for planetary

boundary layers: Journal of Atmospheric Science, v.31, 1791-1806.

van Rijn, L. C., 1984, Sediment transport. II: Suspended load transport Journal of HydraulicEngineering, 110(11), 1431-1456.

Rouse, H., 1939, Experiments on the mechanics of sediment suspension, Proceedings 5th

International Congress on Applied Mechanics, Cambridge, Mass,, 550-554.

Smith, J. D. and S. R. McLean, 1977, Spatially averaged flow over a wavy surface, Journal of

Geophysical Research, 82(12): 1735-1746.

W i ht S d G P k 2004 Fl i t d d d l d i d b d

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Ch 10 Suspended Sediment