Ch9 Sediment Transport

7/30/2019 Ch9 Sediment Transport

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Chapter 9

Sediment transport

1 IntroductionThe Airy (Airy G. B., 1845) small amplitude theory (also named linear wave

theory), described in chapter 3, provides a useful first approximation to the wave

kinematics and is often extended to describe also processes related with

nonlinear phenomena. However, waves are usually not small in amplitude, and

larger waves produce the largest forces and greatest sediment movement, so

nonlinear waves have to be studied for the analysis of some coastal processes.

Sea level variations (wave setup and setdown) can be explained using some

concepts of the linear theory; but the observed decrease and increase in the mean

water level usually involve the wave height to the second power, which results in

nonlinear quantities. These processes are explained with the radiation stress

concept, that is the mean value of horizontal momentum across unit area of a

vertical plane with respect to time, minus the mean flux in the absence of waves.

Gradients in this quantity therefore correspond to a net addition or loss of

momentum to a water column, i.e. a net force, arising from the processes of wave

shoaling and breaking.

Another nonlinear quantity is the mean transport of water toward theshoreline, the mass transport, which is not predicted by the linear Airy theory,

which assumes that each water particle under a waveform is travelling in a

closed elliptical orbit. We define the mass transport as:

2

1

),(1

12

t

t d

dzdtzxutt

M (9.1)

where the time interval between t1 and t2 is a sufficiently long (many wave

periods for irregular waves; one wave period for periodic waves). If we integrate

over the depth from the bottom to the mean water surface, z=0, rather than the

instantaneous water surface , we obtainMequal to zero, as predicted by linear

theory. If we continue the integration up to , the mass transport becomes:


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178 AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

C

EM (9.2)

which shows that there is a nonlinear transport of water in the wave direction dueto the larger forward transport of water under the wave crest because the total

depth is greater when compared with the backward transport under the trough.From this formula, it follows that the mass transport is larger for more energetic

waves.This mass transport has momentum associated with it, which means that

forces will be generated whenever this momentum changes magnitude ordirection by Newtons second law. To determine this momentum, we integrate

the momentum flux from the bottom to the surface as follows:

udzdtuttM

t

t df

2

1)(

1

12(9.3)

This quantity has a first approximation f=MCg=En, which indicates that the

flux of momentum is described by the mass transport times the group velocity.

The sediment transport is usually divided into bed load, suspended load and

swash load, as shown in the scheme of Figure 1.7. The bed load transport is

either in sheet flow or rolled along the bottom, the suspended load is carried up

within the fluid column and moved by currents, the swash load is moved on the

beach face by the swash.

The bed load transport is initiated when the resisting force of the sand

particle on the bottom becomes smaller than a wave force on it. The depth of thispoint on the beach is called a critical depth for sediment movement and the

critical velocity is defined as the water particle velocity at this depth.

In the shallow water region where incident waves break, suspended load is

generated by a great deal of sediment brought into suspension by the turbulence

caused by breaking waves. Sediment materials are suspended and transported

offshore by undertow currents. A large trough is formed and becomes deeper

until the energy is completely exhausted. Offshore a bar is formed, localized

between the breaking point and the deepest zone reached by vorticity.

Theswash load is generated in the swash zone, when fluid motion depends

on the fluctuation of coastline associated with the frequency of oscillations: atlow frequencies, if the beach is permeable and the sand is not saturated, the water

percolates through the substrate and the backrush decreases, with accumulation

of sediment. At high frequencies (storm waves), if the substrate is saturated, the

water cannot percolate through the sand and the backrush current cannot

decrease, with erosion.


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SEDIMENT TRANSPORT 179

2 Basic concepts of sediment transport2.1

Critical bed shear stress

The sediment on the sea bed is transported when it is exposed to large enoughforces, or shear stresses, by the water movements. These movements can be

caused by the current or by the wave orbital velocities or by a combination of

both. In fact, while on deep water there is no orbital motion of particles on thebottom, when the water depth roughly reaches one half of the wave length, the

wave particle orbits begin to interact with the bottom, causing bed shear.If a steady flow over a bed composed of cohesionless grains is considered,

these grains will not move at very small flow velocities, but when the flow

velocity becomes large enough, the driving forces on the sediment particles will

exceed the stabilizing forces. This flow velocity is called the critical flowvelocity.

A now classical solution to the problem based on dimensional analysis was

offered by Shields (1936). The threshold of particle motion is supposed to beattained for a given ratio between driving and stabilizing forces.

2.2 The Shields parameter and modified Shields diagramThe driving force acting on the bottom grains is a function of the second power

of sediment diameterD, given by 2DFb

. On the other hand, the motion is

contrasted by the individual grains tendency to stay on the bottom due to thefriction caused by their submerged weight and to the presence of neighboringgrains. For a non cohesive sediment, the particles submerged weight is defined asfollows:

3)( gDWs

(9.4)

where s is the sediment density, is the fluid density, g is the gravityacceleration andD is the particle diameter. The ratio F/Wis defined as Shields

parameter (Shields, 1936):

gDs

u

gDsgD

b

s

b

)1()1()(

2

*

(9.5)

where s (= s/ ) is the ratio between sediment density ( s) and fluid density ( )

and u* is the friction velocity, defined as /2

*u .

The parameter determining the characteristics of the near-bottom flow andhence the mobilizing force acting on individual sediment grains is thedimensionless boundary Reynolds number:

DuR

e

*

*(9.6)


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where:

=kinematic fluid viscosity= /

=(absolute) dynamic fluid viscosity

Physical studies demonstrated that the condition for motion inception may be

expressed as a critical Shields parameter c which value is a function of theReynolds numberf(Re*):

)()1()1()(

*

2

*e

cc

s

cc Rf

gDs

u

gDsgD(9.7)

So the critical Shields parameter c is the effective Shields parameter ( ) atwhich sediment movement starts. The empirical diagram of c versusRe* is given

in figure 9.1. Typical c values for sand in water are of the order 0.05.

Figure 9.1 Shields diagram for initiation of motion in steady turbulent flow(after Raudkivi, 1976).

The Modified Shields diagramMadsen and Grant (1976) introduced the sediment-fluid parameter S*, whichgives the relation between the critical Shields parameter and the sediment.


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Figure 9.2 - Modified Shield diagram (after Madsen and Grant, 1976).

From the definition of c we obtain:

cc gDsu )1(* (9.8)

which can be introduced in the definition ofRe* to obtain the S* expression:

c

eRgDs

v

DS

4)1(

4

**

(9.9)

The sediment fluid parameter is often used to give a modified representation

of the traditional Shields diagram, in which the values of S*are represented on

the abscissa instead of the Reynolds parameter values.

2.3 Sediment fall velocityIn order to describe suspended sediment transport, it is important to understand

the behavior of suspended sand grains in different type of flow.

The simplest case is when the fluid accelerations are negligible compared to

the acceleration of gravity. In this case the relative velocity between sand and

water is everywhere equal to the settling velocity wf.

The rate at which a particle settles depends on grain and fluid properties (i.e.

grain size, shape and density, water density, flow viscosity rate and turbulence).

Assuming a spherical sediment grain, the force balance of submerged weight and

fluid drag on a grain falling through an otherwise quiescent fluid gives:

223

42

1

6)( fDs wDCDg (9.10)


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D

f

CgDs

w

3

4

)1((9.11)

Where: wf= sediment fall velocity; D = grain diameter; s= density of the

particle; = density of the fluid; s= s/ . The drag coefficient CD is a function of

the Reynolds numberReD =Dwf/v which is a function ofS*, the sediment-fluid

parameter defined by equation (9.9). From the empirical relationship of CD

versusReD, CD is obtained for a specified value ofReD.

eDDRC /364.1 (9.12)

With this value ofCD the dimensionless fall velocity is obtained, and the value is

used with the specified value ofReD to obtain the corresponding value ofS*. In

this manner (Madsen and Grant 1976), the graph of nondimensional fall velocityas a function of the sediment fluid parameter, shown in figure 9.3 is obtained.

The sediment fall velocity can be calculated as a function of the sediment

fluid parameter:

82.1)1( gDs

wf

forS* >300 (9.13)

For small values ofS*, e.g. for quartz grains ofD


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Bed load

Suspended load

Figure 9.3 - Nondimensional fall velocity for spherical particles versus the

sediment fluid parameter (Madsen and Grant, 1976).

The basic idea of this distinction is that the bed loadis defined as the part of

the total load that is in more or less continuous contact with the bed during thetransport, so it involves that part of total load which is supported by intergranular

forces (Bagnold, 1966). Thus the bed load must be determined almost

exclusively by the effective bed shear acting directly on the sand surface.

The suspended load is the part of the total load that is moving without

continuous contact with the bed as a result of the agitation of fluid turbulence.

2.4.1 Bed-load and shear stressMaking use of the Bagnolds definition of bed-load, it is fairly easy to estimate

the weight of material which will be moved as bed-load under a certain effective

stress. The bed-load must, due to this immersed weight, deliver an effective

normal stress l(M/LT2) onto the top most of the immobile bed:

0

)()1( dzzcgs Bl (9.17)

where cB is the volumetric concentration of bed-load in vol/vol, and s are the

sediment density and porosity, respectively, and g is the acceleration due to

gravity.

Assuming that the yield criterion for the top layer of immobile grains is

slc

tanmax

(9.18)

the amount of bed-load which is in equilibrium with is given by:


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s

cB

gsdzzc

tan)1(

')(

0

(9.19)

Here it is convenient to introduce the maximum concentration cmax, which is thevolumetric concentration of solid sediment in the immobile bed. In terms ofcmax,

the vertical scale of the bed-load distribution is then defined by

0max

)(1

dzzcc

L BB (9.20)

Introducing this expression forLB into equation (9.18), we see that the

vertical distribution scale measured in grain diameters is:

s

CB

cd

L

tan

'

max (9.21)

Bagnold (1966) gave tan s =0.63 as a typical value for fairly rounded grains

corresponding to a maximum concentration of the same value, i.e. cmax=0.63

[vol/vol], and he noted that the product cmax tan s =0.63 is fairly constant at

about 0.4 for different grain shapes. Hence, as rules of thumb we have:

dL cB )'(5.2 (9.22)

maxmax )'(5.2 dcLc cB (9.23)

LB is the equivalent thickness at rest of the bed-load, and cmaxLB is thecorresponding solids volume per unit area of the bed.

2.4.2 Steady bed load in sheet flow transportThe bed load transport rate can be expressed as :

BBsBb ULcdzzuzcq max0

)()( (9.24)

us is the sediment velocity distribution and cmax is the maximum concentration of

solid sediment in the immobile bed. We can predict qb values empirically with

reasonable confidence for steady flow because it was measured directly in a large

number of experiments.

One of the first theoretical approaches to the problem of predicting the rate of

bed load transport was presented by Einstein (1950). One of the most important

innovations in his analysis was the application of the theory of probability to

account for the statistical variation of the agitating forces on bed particles caused

by turbulence.

Based on experimental observations, Einstein assumed that the mean distance

traveled by a sand particle between erosion and subsequent deposition, is simply

proportional to the grain diameter and independent of the hydraulic conditionsand the amount of sediment in motion.


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The principle in Einsteins analysis is as follows: the number of particles,

deposited in a unit area, depends on the number of particles in motion and on the

probability that the dynamical forces permit the particles deposit the number ofparticles eroded from the same unit area depends on the number of particles

within the area and on the probability that the hydrodynamic forces on these

grains are sufficiently strong to move them. For equilibrium conditions the

number of grains deposited must equal the number of particles eroded. In this

way, a functional relation (bed load function) is derived between the two non-

dimensional quantities.

2)1( gds

qB (9.25)

where qB is the rate of bed load transport in volume of material per unit time andwidth.

The general trend of can be also expressed by the Meyer-Peter and Muller(1948):

5.1)'(8 c (9.26)

Another expression is given in Nielsen (1992)

')'(c

8 (9.27)

The representation of the functions is given in figure 9.4.

2.4.3 Basics of suspended load transport formulationUnder the assumption of an uniform flow, the relation between the averagevelocity (v), the water level slope (i) the water depth (h) and the bed shear

friction coefficient (C) is given by the Chezy formula:

dC

vidiCv

2

2

(9.28)

In that case, the shear stress is given by:

22 / Cgvc (9.29)


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Figure 9.4 - Representation of the functions

for such flow the vertical velocity gradient dzzdv )( can be written as:

fzdzzdv )()( (9.30)

where f is the diffusion coefficient and is the shear stress at height zfrom thebed.

The diffusion coefficient is given by the mixing length theory:

dz

zdv

lf)(2

(9.31)

where lis the mixing length, given by:

l=kz (near the bed) (9.32)

dzkzl /1 (for the entire water column) (9.33)

so the shear stress varies linearly with the height above the bed:

)/1()(

)/1()()(

2

2 dz

dz

zdvdzkzz c (9.34)


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the vertical gradient dzzdv )( is:

kzdzzdvc

/)( (9.35)

The solution of the above differential equation is :

0

*

0

lnln1

)(z

z

k

u

z

z

kzv C (9.36)

where u*is the shear stress velocity, which is the velocity occurring at a certain

elevation above the bed, assuming a logarithmic velocity profile. A physically

important quantity is the velocity which marks the change of the turbulent flow

of the logarithmic velocity profile to a much less turbulent or even laminar sub-

layer close to the bed: the velocity distribution near the bed is usually assumedbe linear and tangent to the logarithmic distribution at a heightztabove the bed.

From ttt zvdzdv // follows that zt=ez0. The velocity at the height zt is

found :

vkC

g

k

vvt

* (9.37)

with equation (9.35) a new formulation for the bed shear stress c is found:

22

tc vk (9.38)

This expression relates the bed shear stress to the velocity near the bed for the

combination of the two fundamentally different velocity profiles of a uniform

flow and the orbital velocity due to the waves. The value of z0 is related to the

apparent bottom roughness . Experimentally, Nikuradse found:

z0/r=1/33 (9.39)

The above given information is required to calculate the concentration of the

material in suspension. This material is kept into suspension by the exchange of

upward and downward transport as result of the turbulent diffusion. This upwarddiffusion coefficient for the sediment is related to the turbulent fluid diffusion

coefficient.

Thus, the upward transport due to turbulent diffusion is, in the equilibrium

situation, equal to the downward motion of the sediment due to the fall velocity:

dz

zdczzwc s

)()()( (9.40)

where z is the height above the bed; w is the fall velocity of the sedimentparticles in still water; c(z) is the average concentration at heightzabove the bed;

s(z)is the diffusion coefficient for the sediment at heightz.


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d

zd

d

zz ss max4)( (9.41)

this results in a concentration distribution given by :

0

0)(

Z

ad

a

z

zdczc (9.42)

wheremax** 4// sdwkvwz ; c0 is the reference concentration at level z=a

above the bed. Einstein calculated the value of c0 at a height a of only some graindiameters from the bed (Einstein, 1950). For a rippled or ondulated bed this

assumption is not realistic. Bijker assumed therefore that a would be equal to thebed roughness r. The concentration ca at the top of this layer is calculated underthe assumption that the bed load is transported in this layer by the averagevelocity and that the concentration is constant (in the same layer). The suspendedload can be calculated as:

qb = rvbedlayerca (9.43)

Over the height z0 the velocity distribution is linear, from z0 to r the velocity

distribution is logarithmic. This results in the following formula for the average

velocity v in the bottom layer:

r

zbedlayer dzz

z

k

v

zk

v

rv 0 0

*

0

*

ln2

11(9.44)

or :

*34.6 vvbedlayer (9.45)

2.5 The bottom boundary layer and the bed roughnessThe bottom boundary layer is the zone in the immediate vicinity of the bed

where the fluid motion is significantly influenced by the frictional resistance of

the bottom. The velocity in the boundary layer grades from zero at the bed to thevelocity of the free stream at some distance above the bed. The thickness of the

wave boundary layer ( ) depends on the wave period and is approximately

(Madsen and Grant, 1976):

wku*,(9.46)

where k is the von Karmans constant (=0.4), u*,w is the wave-induced shear

velocity and is the wave radian frequency (2 /T). The thickness of the waveboundary layer is typically in the order of a few centimeters. While an oscillatory

boundary layer more or less breaks down and reforms twice every wave cycle, acurrent boundary layer is significantly thicker, as this layer has ample time to


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develop. The most important physical aspect of the boundary layer is the shear

stress ( b) , i.e. the force which the fluid motion exerts on the bed. The shear

stress can be defined as

2

*ub and determinates the velocity gradient close tothe bed and the mobilising forces applied to the sediment grains on the bottom.This quantity is of fundamental importance to sediment entrainment andtransport. Under unidirectional currents, the bed shear stress can be computed

from the velocity profile. In the lower 1-2 m of the water this profile is generallywell described by the von Karman-Prandtl equation:

)/ln( 0* zz

k

uuz (9.47)

where u* is the mean velocity at elevation z above the bed, u* is the shear

velocity associated with the current and z0 is the zero intercept of the velocityprofile. However, sediment entrainment and concentration in surf zone aremainly determined by the shear stress due to waves. This is because the wave

boundary layer is much thinner than the current boundary layer. Therefore,velocity gradients (the shear velocity) due to the waves are significantly larger

than for the mean current. An appropriate way of determining shear velocity andbed shear stress is through the use of the definition of bed shear stress underwaves:

22*,,

2

1ufu wwwb (9.48)

where is the density of the water, u is the free stream wave orbital velocity and

fw is a wave friction factor, i.e. a coefficient of proportionality describing therelation between shear velocity and the free stream velocity. A commonly

accepted expression for the non-dimensional wave-friction factor is due to Swart(1974):

)977.5)/(213.5exp( 194.0Akf sw 63.0/Aks (9.49)

where ks is the bed roughness (or hydraulic roughness) andA is the water particle

semi-excursion (i.e. orbital amplitude, A=umT/2 ) . Forks/A>0.63,fw=0.30. The

bed roughness is usually defined as: (Nielsen, 1992):

ks=30z0 (9.50)

Hence, the rougher the bed the steeper the velocity gradients and the more

stress exerted by the fluid motion. However, in sediment transport modeling, the

specification ofks remains a problem. When waves and currents coexist, the bed

roughness is mainly linked to the wave motion. Total bed roughness is composed

of the grain roughness (roughness due to the individual particles on the bed kd),

roughness exerted by bedforms (form drag roughness, kr) and due to sediment

movement (km). The latter two terms are called moveable bed roughness. Thus:

ks=kd+kr+km (9.51)


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For a flat fixed bed, ks=kd=2.5D50 (where D50 is the mean grain size).However, in the surf zone the bed is rarely immobile and for a moveable bedunder waves, Nielsen (1992) proposed

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2

)05.0'(1708

Dkr

rs (9.52)

where r is the bedform height, r is the bedform wavelength and is the skinfriction Shields parameter.

2.6 Bed load and suspended load: a simple parametrical modelBed loadBagnold (1966) pointed out one of the shortcomings in Einsteins formulation by

stating the following paradox. Consider the ideal case of fluid flow over a bed ofuniform, perfectly piled spheres in a plane bed, so that all particles are equallyexposed. Statistical variations due to turbulence are neglected. When the tractivestress exceeds the critical value, all particles in the upper layer are peeled offsimultaneously and are dispersed. Hence the next layer of particles is exposed tothe flow and should consequently also be peeled off. The result is that all thesubsequent underlying layers are also eroded, so that a stable bed could not existat all when the shear stress exceeds the critical value.

Bagnold explained the paradox by assuming that in a water-sediment mixturethe total shear stress would be separated in two parts:

= F+ G (9.53)where F is the shear stress transmitted by the intergranular fluid, while G is theshear stress transmitted because of the interchange of momentum caused by theencounters of solid particles, i.e. a tangential dispersive stress. The existence ofsuch dispersive stresses was confirmed by his experiments.

Bagnold argues that when a layer of spheres is peeled off, some of thespheres may go into suspension, while others will be transported as bed load.Thus a dispersive pressure on the next layer of spheres will develop and act as astabilizing agency. Hence, a certain part of the total bed shear stress istransmitted as a grain shear stress G, and a correspondingly minor part as fluid

stress = F+ G.Continuing this argument, it is understood that exactly so many layers ofspheres will be eroded that the residual fluid stress F on the first immovablelayer is equal to (or smaller than) the critical tractive stress G . The mechanismin transmission of a tractive shear stress greater than the critical is then thefollowing: c is transferred directly from the fluid to the immovable bed, whilethe residual stress - c is transferred to the moving particles and further fromthese to the fixed bed as a dispersive stress.

The effective bottom shear stress ( 'c) is given by:

2

50

'

5.212log

06.0

2

1

UDdc(9.54)


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Bijker (1986) presented a method for the calculation of sediment transport in

combined wave and current motion. The mean bed shear stress (wc

) by Bijker

(1986) in this situation is given by:

max,2

1wcwc

(9.55)

c is the bed shear stress by current alone, and w,max is the maximum bed shear

stress by wave alone:

22

12log

06.0

2

1

2

1U

kdU

s

c(9.56)

2

max, 2

1

mcw Uf(9.57)

3.65.5exp

2.0

w

sw

A

kf (9.58)

where:

d=water depth [m]; ks=bed roughness [m]; U=average velocity of current [m/s];

Aw=amplitude of the water particle on the bottom [m]; Um=maximum horizontal

velocity of the water particle on the bottom [m/s].

The bed load transport is given by:

upstirring

wcr

ngtransporti

cB

gDsDq 5050

)1(27.0exp2 (9.59)

where: qB= bed load transport [m3/m s]; D50=median sediment grain size [m];

g=acceleration of gravity [m/s2]; r=ripple factor=

c/ wc [-]; 0.27=experimental

coefficient.

Suspended load

The suspended load is defined as the part of the total load which is movingwithout continuous contact with the bed as the result of the agitation of the fluid

turbulence. The appearance of ripples will increase the bed shear stress (flow

resistance). On the other hand, more grains will be suspended due to the flow

separation on the lee side of the ripples, thus the suspended load is related to the

total bed shear stress.

Einstein-Bijker formula for suspended sediment transport is:

21033.0

ln83.1 Ik

dIqq

s

BS(9.60)

whereI1 andI2 are the Einsten integrals given by:


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dBB

B1

A)(1

A0.216I

*

*

*z

1

Az

1)(z

1 (9.61)

lnBdBB

B1

A)(1

A0.216I

*

*

*z

1

Az

1)(z

2 (9.62)

where A=ks/d; B=z/d; z*=wf/( u*,C); /*, ccu friction velocity. is theVon Karman constant ( =0.40, dimensionless) and qB is the bed load transport

under combined wave and current [m3/m s].The values of Integrals I1 and I2 can be used to calculate the Einstein Total

Integral Q as follows:

])033.0/ln([ 21 IkdIQ s (9.63)

For given values ks, dandz* , the Einstein Total Integral Q can be also calculatedusing the table 9.1 or figure 9.5, which gives the representation of 1.83Q (= qS

/qB) versus A(=ks//h). The suspended load qS is a function of the bed loadtransport and the Einstein Total Integral:

BSQqq 83.1 (9.64)

This indicates that the suspended load transport is directly and linearly

proportional to the bed load. The total transport QTcan now be written as:

)83.11( QqqqQBSBT

(9.65)

2.7 Case studyExample 1

Calculate the sediment fall velocity in sea water ( =1,025Kg/m3 ;

=10-6 m2/s) for a quartz sediment sand ( s =2,650 Kg/m3) with

diameterD=0.15 mm.

Solution:

The sediment fall velocity can be calculated using the sediment fluid-parameterS* :

s= s/ =2.59

g=acceleration due to gravity=9.8 m/s;D=0.15mm=1.5 10-4m.

81.1105.18.9)159.2(104

105.1)1(

4

4

6

4

*gDs

v

DS

The figure 9.5 in correspondence of S*=1.81 the dimensionless fall velocity can

be calculated by the following expression:


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3.0)1(

*SgDs

wf

so we have:

scmsmgDsSwf /45.1/0145.0)1(*

Example 2

In a coastal region, (sea water density =1025 kg/m3; kinematic

viscosity v=10-6 m2/s) the flow speed is U=1 m/s at 2 m waterdepth; Wave parameters are: H=0.5 m and T=8 s (L=35 m); thesediment density is 2650 kg/m3 and D50=0.15 mm. The bed

roughness is ks=2 cm. Calculate the sediment transport undercurrent and under combined wave and current.

Solution:

The effective bottom shear stress is given by:

22

2

50

' /33.15.212log

06.0

2

1mNU

Ddc

The total bottom shear stress is:

22

2/24.3

12log06.0

21 mNU

kd sc

The ripple factor is:

c

cr

'

0.41

The bed load transport is:

wcr

c

B

gDsDq 50

50

)1(27.0exp2 1.0395 10-5

m

m3

The relative density of the sediment is:

59.2/s

s

The fall velocity is:

8.2

36)1(5.7

36

50

50

2

50D

DsD

wf

0.012 m/s

The friction velocity is:


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/*, ccu 0.056 m/s

the Q values can be calculated by numerical integration or using figure 9.5 or

using table 9.1. By numerical integration:

dBB

B

A

AI

z

Az

z *1

*

)1*(

1

1

)1(216.0 2.62

BdBB

B

A

AI

z

Az

z

ln1

)1(216.0

*1

*

)1*(

2-4.83

The suspended sediment transport is:

21033.0

ln83.183.1 Ik

hIqQqqs

BBS3.0755 10-4

m

m3

Figure 9.5 - Suspended sediment transport parameters.


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The total sediment transport is:

SBT

qqQ 3.1795 10-4

m

m3

If we consider the combined wave and current, by linear wave theory theamplitude of water particle on the bottom is:

)/2sinh(

1

2 Lh

HA

w0.68

the maximum horizontal velocity of water particle on the bottom is:

TAAU

wwm

20.53 m/s

the wave friction coefficient is:

3.65.5exp

2.0

w

s

wA

kf 0.028

the maximum bottom shear stress by wave is:

2

max,2

1mcw

Uf =4.07 N/m2

The mean bottom stress under combined wave and current is:

max,2

1wcwc

=5.28 N/m2

the bed load transport is:

wcr

c

B

gDsDq 50

50

)1(27.0exp2 = 1.2532 10-5

m

m 3

the friction velocity is:

Wc

WCu*, =0.072 m/s

The suspended sediment transport parameters are:

A=ks/d=0.01 z*=wf/( u*,wC)=0.42

the Q values can be calculated by numerical integration or using figure 9.5 orusing table 9.1 . By numerical integration:

dBB

B

A

AI

z

Az

z *1

*

)1*(

1

1

)1(216.0 3.8


20/28


BdBB

B

A

AI

z

Az

z

ln1

)1(216.0

*1

*

)1*(

2-6.3


21/28

EINSTEININ

TEGRALFACTOR

Q

.

ks/h

z*=0.00

z*

=0.20

z*0.40

z*

=0.60

z*

=

0.80

z*

=1.00

z*

=1.50

z*

=2.00

z*

=3.00

z*

=4.00

0.00001

303000

32

800

3880

527

88

20.0

2.33

0.973

0.432

0.276

0.00002

144000

17

900

2430

377

71.6

17.9

2.31

0.973

0.432

0.276

0.00005

53600

7

980

1300

239

53.6

14.4

2.28

0.967

0.432

0.276

0.0001

25300

4

320

803

169

42.7

13.6

2.25

0.967

0.432

0.276

0.0002

11900

2

330

496

119

33.9

11.9

2.21

0.967

0.431

0.275

0.0005

4360

1

020

260

74.3

24.6

9.8

2.13

0.962

0.431

0.275

0.001

2030

545

158

51.2

19.1

8.4

2.05

0.951

0.430

0.275

0.002

940

289

95.6

35.1

14.6

7.0

1.96

0.940

0.428

0.274

0.005

336

123

48.5

20.8

10.0

5.4

1.78

0.907

0.424

0.273

0.01

153

63.9

28.6

13.8

7.3

4.3

1.62

0.869

0.417

0.270

0.02

68.9

32.8

16.5

8.9

5.2

3.3

1.42

0.809

0.404

0.264

0.05

23.2

13.1

7.7

4.8

3.1

2.2

1.10

0.694

0.374

0.249

0.1

9.8

6.3

4.1

2.8

2.0

1.5

0.84

0.568

0.339

0.236

0.2

3.9

2.8

2.0

1.5

1.2

0.9

0.55

0.414

0.317

0.5

0.8

0.7

0.6

0.5

0.4

0.3

0.17

1

0

0

0

0

0

0

0

Table9.1-

EinsteinIntegralfactorQ.

SEDIMENTTRANSPORT 197


22/28


The suspended sediment transport is:

21033.0ln83.183.1 Ik

d

IqQqqs

BBS 5.6481 10

-4

m

m3

The total sediment transport is:

SBTqqQ 5.7734 10-4

m

m3

3 Basic shore processesWhen waves approach a sloping beach and break, nearshore currents are

generated, which action depends on the beach characteristics and the waveconditions. Beach morphology is strongly controlled by nearshore currents

because of sediment movements; water fluxes between the coast and the offshore

zone contribute to renew the coastal waters.Nearshore current patterns are a combination of longshore currents, rip

currents and undertow. For large incident wave angle, alongshore momentumgenerated by the wave breaking process sets up strong longshore currents. The

forward flow of the water particles in the breaking process sets up longshorecurrents. Smaller incident wave angle generate weaker longshore currents. The

forward flow of the water particles in the breaking waves also pumps water

across the breaking zone, increasing the water level there. The onshoremomentum of the waves holds some of this water close to shore, causing ashoreward elevated water level (wave set-up).

This phenomenon can be explained by the concept of radiation stress,introduced by Longuet-Higgins and Stewart (1964) and described in chapter 3.

3.1 Nearshore circulationThe explaination for the generation of the cell circulation was developed

following the introduction of the concept of radiation stress by Longuet-Higgins

and Stewart (1964), defined as the excess of flow of momentum due to the

presence of waves. The shoreward component of the radiation stress produces aset-down immediately offshore of the breakers and a set-up within the surf zone.

In a two-dimensional case, the wave crests are always parallel to the

shoreline. Averaging over one wave period, continuity of mass must be satisfied

at every cross section. This necessitates a vertical distribution of mass transport

velocity: forward flow at the surface and near the bottom, return flow near

middepth (Ippen, 1966).

The forward flow at the surface transports the water in surface rollers toward

the coast,and the wave drift is also directed toward the coast. These contributions

are concentrated near the surface. As the net flow must be zero, they are

compensated by a return flow in the offshore direction, which is concentratednear the bed (undertow).


23/28


In a three-dimensional case, a cellular circulation takes place, which isconstituted by longshore currents and rip currents.

Figure 9.6-Nearshore circulation pattern. Two dimensional case (after Ippen,

1966).The longshore current is generated by the shore-parallel component of the

radiation stress associated with the breaking process for obliquely incomingwaves. This current, which is parallel to the shoreline, carries the sedimentsalongshore and it is approximately proportional to the square root of the waveheight and to sin(2 b), where b is the wave incidence angle at breaking. Themovement of beach sediment along the coast is referred to as littoral transport orlongshore sediment transport, whereas the actual volumes of sand involved in thetransport are termed the littoral drift. This longshore movement of beachsediments is of particular importance because the transport can either be

interrupted by the construction of jetties and breakwaters (structures which blockall or a portion of the longshore sediment transport), or can be captured by inletsand submarine canyons. In the case of a jetty, the result is a buildup of the beachalong the updrift side of the structure and an erosion of the beach downdrift ofthe structure (CEM, 2001).

The rip currents are part of cellular circulations fed by longshore currentswithin the surf zone that increase from zero at a point between two neighboringrips, reaching a maximum just before turning seaward to form the rip (see figure9.6).

The longshore currents are in turn fed by the slow shoreward transport ofwater into the surf zone from breaking waves. A nearshore circulation cell thus

consists of longshore currents feeding the rips, the seaward flowing rip currentsthat extend through the breaking zone and spread out into rip heads, and a returnonshore flow to replace the water moving offshore through rips. (Shepard andInman, 1950). The cell circulation results from alongshore variation in waveheights, which in turn produce a longshore variation in set-up elevations. Theset-up results from the Sxx onshore component of the radiation stress being

balanced by the pressure gradient of the seaward-sloping water surface in thenearshore. Balancing those forces yields:

x

d

x 23

81 (9.66)


24/28


For the cross-shore slope of the set-up denoted by . There is a direct

proportionality with the beach slope xdS /0 , but not a direct dependence on

the

Figure 9.7 - The nearshore cell circulation consists of (1) feeder longshorecurrents, (2) seaward-flowing rip currents, and (3) a return flow of water from

the offshore zone into the surf zone (after Komar, 1988).

wave height. However, the larger waves break in deeper water than smaller

waves, and the set up therefore begins farther seaward at longshore locationswhere the larger waves occur. Inside the surf zone, the mean water level is highershoreward of the larger breakers than it is shoreward of the small waves. A

longshore pressure gradient therefore exists, which will drive a longshore currentfrom positions of high waves and set-up to adjacent position of low-waves. In

addition of the Sxx component of the radiation stress, there is a Syy component, amoment flux acting parallel to the wave crest, in this case parallel to the

shoreline. This component is given by:

)2sinh(8

1 2

kd

kdgHSyy (9.67)

since the wave height varies alongshore, Syy will similarly vary and there willexist a longshore gradient:

y

H

kd

kdgH

y

Syy

)2sinh(4

1(9.68)

This longshore gradient produces a flow of water away from the regions of

high waves and toward position of low waves. The flow then turns seaward as arip current where the waves and the set-up are lowest and the longshore currents

converge.


25/28


The cell circulation therefore depends on the existence of variations in wave

heights along the shore. The most obvious way to produce this variation is by

wave refraction, which can concentrate the wave rays in one area of a beach,

causing high waves, and at the same time spread the rays in the adjacent area of

the beach and then produce low waves. The position of rip currents and the

overall cell circulation will then be governed by wave reflection and hence by

the offshore topography.

Headlands, breakwaters and jetties can affect the incident waves by the

partial sheltering of the shore and thereby produce significant longshore

variations in wave height and set-up. Wave reflection and diffraction produce

alongshore gradients with lower waves and set-up in the lee of the headland or

breakwater, which in turn generate longshore currents flowing inward toward the

sheltered region. In some situations this process can account for the developmentof strong rip currents adjacent to jetties and breakwaters.

An example of rip current acting on bottom topography is the phenomenon of

rip channels. On a barred profile the wave breaking on the bar will induce a wave

set-up, causing an increase of water level inshore of the bar. A bar will be in

many cases interrupted by holes (rip channels) found at more or less regular

intervals. The wave breaking is less intensive in the rip-channels due to the larger

depth and because the wave refraction may concentrate the wave energy on the

bars at the sides of the channel.

Wave-current interaction may affect the development of rip-currents. In fact,

the weak currents generated by a gentle alongshore variation of the wave fieldcan cause significant refractive effect on the waves as to change the structure of

the forcing which drives the currents and the instability of the cellular

circulation. When currents are weak compared to the wave group velocity, their

effects on waves are small, but such effects are sometimes not negligible. This is

the case of rip currents produced by alongshore topographic variations on

otherwise alongshore uniform beaches.

These alongshore variations in the topography, like gentle rip channels,

produce longshore variations in the radiation stress and provides the source of

vorticity and of horizontal circulations, which interact with the waves, so wave

radiation stress will be modified.Such changes of course are small relative to the effect due to wave breaking,

but can be comparable to the variations caused by the topography.

When this is the case, the circulations of interest can be significantly affected

by the wave-current interaction. The interaction of the narrow offshore directed

rip currents and incident waves produces a forcing effect opposite to that due to

topography, hence it reduces the strength of the currents and restricts their

offshore extent. The two physical processes due to refraction by currents, behind

the wave rays and changes in the wave energy, both contribute to this negative

feedback, on the wave forcing (Yu and Slinn, 2003).


26/28


Figure 9.8 - The circulation current generated by normally incident waves on a

barred coast with rip-channels (after Fredse and Deigaard, 1992).

3.2 Wave run-up in the swash zoneThe wave run-up is defined as the maximum elevation of the wave uprush above

the mean sea level. The uprush is given by the sum of two components: the

positive elevation of the mean water level caused by wave action and the

fluctuations above the mean water level (swash). The concept of wave run-up is

frequently used to describe the beach profile processes. The wave run-up

parameter calculation depends on the processes of wave transformation, such as

the wave reflection, the interaction between the bottom and the waves and the

sediments properties (e.g. porosity and permeability).

Actually, some formulations of wave run-up are based on the empirical

studies carried out by Hunt (1979) for regular waves and for irregular waves.

Regular waves

For regular breaking waves, the run-up is a function of the beach slope, incident

wave height and wave steepness. According the Hunt (1979) formulation:

0

0H

Rfor 0.1< 0


27/28


The maximum run-up is an important parameter to describe the active portion ofthe beach profile. Mase (1988) describes the expressions (Table 9.2) for the

maximum run-up (Rmax) and other run-up parameters valid for flat and

impermeable beaches and for 5/1tan30/1 , 007.000

LH .

3.3 Bar formation by cross-shore flow mechanismsThe classic response of a planar beach to storm waves consists of erosion of the

beachface and inner surf zone and deposition around the wave breakpoint

resulting in the formation of a storm bar. The first explanation of bar formation

was given as early as 1863 by Hagan (in Komar, 1976) who explained a bar

formation by a seaward going undertow meeting the shoreward moving waves.

This intuitive idea has now been formalised where the undertow is represented

by the bed return flow and the intuitive expression shoreward moving wavesrefers to the shoreward flow asymmetry associated with the highly non-linear

shoaling waves. Thus, offshore of the breakpoint, the residual flow close to the

bed is shoreward, whereas onshore of the breakpoint, the near-bed flow is

seaward. This results in a region of flow convergence and hence sediment

accumulation close to the breakpoint and the formation of a bar (Dyhr-Nielsen

and Sorensen, 1970).

For bars formed according to the cross-shore flow mechanism, the distance

from the shoreline to the bar crest (xbar) is described by (Holman and Sallenger,

1993)

tantan

bbbar

Hdx (9.71)

where db is the depth at the break point, tan( ) is the nearshore gradient, is the

breaker criterion andHb is the breaker height.

The cross-shore flow mechanism of bar formation does not account for the

formation of multiple bars because only one breakpoint or breaker zone will

occur. However, several explanations can be given for the formation of multiple

bars according to the cross-shore flow mechanism. The first involves the

presence of multiple breakpoints. Once waves break on an outer bar of a gentlysloping beach they may recover and reform as they travel across the deeper

shoreward trough. These reformed waves may break for a second or perhaps a

third time before they eventually reach the shoreline, resulting in multiple bar

morphology. The presence of several distinct wave climates, each one

responsible for a single bar, may result in multiple bar morphology (Evans,

1939). Another mechanism may involve varying water levels and breaker

position due to tides (Komar, 1976).


28/28


Table 9.2 - Mase (1988) expressions for run-up parameters.

Symbol Formulation

Maximum Run-up Rmax 77.00

0

max 32.2H

R

Run-up exceeded by 2% of the

crests

R2% 71.00

0

%2 86.1H

R

Average of the highest 1/10 of therun-ups

R1/10 71.00

0

10/1 70.1H

R

Average of the highest 1/3 of therun-ups

R1/3 70.00

0

3/1 38.1H

R

Mean run-up R 69.0

0

0

88.0H

R

Documents

Ch9 Sediment Transport