SEDIMENTATION AND EROSION HYDRAULICS.pdf

6.1 INTRODUCTION

Since the beginning of mankind, sedimentation processes have affected water supplies,irrigation, agricultural practices, flood control, river migration, hydroelectric projects,navigation, fisheries, and aquatic habitat. In the last few years, sediment also has beenfound to play an important role in the transport and fate of pollutants; thus, sedimentationcontrol has become an important issue in water quality management. Toxic chemicals canbecome attached to, or adsorbed by, sediment particles and then be transported to anddeposited in other areas. By studying the quantity, quality, and characteristics of sedimentin rivers and streams, scientists and engineers can determine the sources of the sedimentand evaluate the impact of pollutants on the aquatic environment. In the United States,sedimentation control is a multibillion-dollar issue. For example, approximately $500 mil-lion are spent every year to dredge waterways and harbors for navigation purposes. Mostof the dredged sediment is the result of substantial soil erosion in watersheds. Estimatesby the U.S. Department of Agriculture indicate that annual offside costs of sedimentderived from copland erosion are on the order of $2 billion to $6 billion, with an addi-tional $1 billion arising from loss in compared productivity.

The sediment cycle starts with the process of erosion, where by particles or fragmentsare weathered from rock material. Action by water, wind, glaciers, and plant and animalactivities all contribute to the erosion of the earth’s surface. Fluvial sediment is the termused to describe the case where water is the key agent for erosion. Natural, or geologic,erosion takes place slowly, over centuries or millennia. Erosion that occurs as a result ofhuman activity may take place much faster. It is important to understand the role of eachcause when studying sediment transport.

Any material that can be dislodged is ready to be transported. The transportationprocess is initiated on the land surface when raindrops result in sheet erosion. Rills, gul-lies, streams, and rivers then act as conduits for the movement of sediment. The greaterthe discharge, or rate of flow, the higher the capacity for sediment transport.

The final process in the cycle is deposition. When there is not enough energy to transportthe sediment, it comes to rest. Sinks, or depositional areas, can be visible as newly deposit-ed material on a floodplain, on bars and islands in a channel, and on deltas. Considerabledeposition occurs that may not be apparent, as on lake and river beds. A knowledge of sed-iment dynamics is an integral part of understanding the aquatic ecosystem.

This chapter presents fundamental aspects of the erosion, transport, and deposition ofsediment in the environment. The emphasis is on the hydraulics of bedload and suspend-

CHAPTER 6SEDIMENTATION AND EROSION HYDRAULICS

Marcelo H. GarcíaDepartment of Civil and Environmental Engineering

University of Illinois at Urbana-ChampaignUrbana, IL

6.1

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Source: HYDRAULIC DESIGN HANDBOOK

ed load transport in rivers, with the goal of establishing the background needed for sedi-mentation engineering. Because of their relevance, the hydraulics of both reservoir sedi-mentation and turbidity currents also is considered. Emphasis is placed on noncohesivesediment transport, where the material involved can be silt, sand, or gravel. When possi-ble, the behavior of both uniform-sized material and sediment mixtures is analyzed.Although such topics as cohesive sediment transport, debris and mud flows, alluvial fans,river meandering, and sediment transport by wave action are not discussed here, it ishoped that the material covered in this chapter will provide a firm foundation to tackleproblems in those.

For more information on sediment transport and sedimentation engineering, readersare referred to Allen (1985), Ashworth et al. (1996), Bogardi (1974), Bouvard (1992),Carling and Dawson (1996), Chang (1988), Coussot (1997), Fredsøe and Deigaard(1992), Garde and Ranga Raju (1985), Graf (1971), Jansen et al. (1979), Julien (1992),Mehta (1986), Mehta et al. (1989a, 1989b), Morris and Fan (1998), Nakato and Ettema(1996), National Research Council (1996), Nielsen (1992), National Research council(1996), Parker and Ikeda (1989), Raudkivi (1990, 1993), Renard et al. (1997), Sieben(1997), Simons and Senturk (1992), Sloff (1997), van Rijn (1997), Yalin (1972, 1992),Yang (1996), and Wan and Wang (1994).

6.2 HYDRAULICS FOR SEDIMENT TRANSPORT

6.2.1 Flow Velocity Distribution

Consider a steady, turbulent, uniform, open-channel flow having a mean depth H and amean flow velocity U (Fig. 6.1). The channel is extremely wide and its bottom has a meanslope S and a surface roughness that can be characterized by an effective height ks

(Brownlie, 1981b). When the bottom of the channel is covered with sediment having amean size or diameter D, the roughness height ks will be proportional to that diameter.Because of the weight of the water, the flow exerts on the bottom a tangential force perunit bed area known as the bed shear stress τb, which can be expressed as:

τb ρgHS (6.1)

where ρ is the water density and g is the gravitational acceleration. With the help of theboundary shear stress, it is possible to define the shear velocity u* as

6.2 Chapter Six

FIGURE 6.1 Definition diagram for open-channel flow over an erodible bed.



SEDIMENTATION AND EROSION HYDRAULICS

u* τb/ρ (6.2)

The shear velocity, and thus the boundary shear stress, provides a direct measure of theintensity of flow and its ability to entrain and transport sediment particles. The size of thesediment particles on the bottom determines the surface roughness, which in turn affectsthe flow velocity distribution and its sediment transport capacity. Since flow resistanceand sediment transport rates are interrelated, the ability to determine the role played bythe bottom roughness is important.

Research has shown (Schlichting, 1979) that the flow velocity distribution is well rep-resented by:

uu

* κ

1 lnz const. (6.3)

where u is the time-averaged flow velocity at distance z above the bed and κ is known asVon Karman’s constant and is equal to 0.4. For obvious reasons, the above law is knownas the logarithmic law of the wall. It strictly applies only in a thin layer near the bed. It isempirically found to apply as a reasonable approximation throughout most of the flow inmany rivers.

If the bottom boundary is sufficiently smooth (a condition rarely satisfied in rivers),turbulence will be drastically suppressed in an extremely thin layer near the bed. In thisregion, a linear velocity profile will hold:

uu

*

uv*z (6.4)

where ν is the kinematic viscosity of water. This law merges with the logarithmic law nearz δv, where

δv 11.6 uν

* (6.5)

denotes the height of the viscous sublayer. In the logarithmic region, the constant of inte-gration introduced above has been evaluated from data to yield

uu

* κ

1 ln

u

ν* z

5.5 (6.6)

Most boundaries in river flow are rough. Let ks denote an effective roughness height.If ks/δv 1, then no viscous sublayer will exist. The corresponding logarithmic velocityprofile is given by

uu

* κ

1 ln

kzs

8.5 κ1

ln30 k

zs

(6.7)

As noted above, this relation often holds as a first approximation throughout the flow in ariver. It is by no means exact.

The conditions ks/δν » 1 for rough turbulent flow and ks/δν « 1 for smooth turbulentflow can be rewritten to indicate that u*ks/ν should be much larger than 11.6 for turbulentrough flow and much smaller than 11.6 for turbulent smooth flow. A composite form thatrepresents both ranges, as well as the transitional range between them, can be written as

uu

* κ

1 ln

kzs

Bs (6.8)

with Bs as a function of Re* u*ks/ν, which can be estimated with

Sedimentation and Erosion Hydraulics 6.3




Bs 8.5 [2.5 ln(Re*) 3]e0.127[ln(Re*)]2 (6.9)

as proposed by Yalin (1992).

6.2.2 Relations for Channel Resistance

Most river flows are indeed hydraulically rough. Equation (6.7) can be used to obtain anapproximate expression for depth-averaged velocity U that is reasonably accurate formany flows. Using the following integral:

U H1

H

0udz (6.10)

but changing the lower limit slightly to avoid the fact that the logarithmic law is singularat z 0, the following result is obtained:

uU

* H

1

H

ks

κ1

ln

kzs

8.5

dz (6.11)

or, performing the integration

uU

* κ

1 ln

Hks

6 κ1

ln11

Hks

(6.12)

This relation is known as Keulegan's resistance relation for rough flow.An approximation to Keulegan's relation is the Manning-Strickler power form

uU

* 8

Hks

1/6(6.13)

Between Eqs. (6.2) and (6.12), a resistance relation can be found for bed shear stress:

τb ρCf U 2 (6.14)

where the friction coefficient Cf is given by

Cf

κ1

ln11

Hks

–2(6.15)

If Eq. (6.13) is used instead of Eq. (6.12), the friction coefficient takes the form

Cf 8

Hks

1/6

–2

(6.16)

It is useful to show the relationship between the friction coefficient Cf and the rough-ness parameters in open-channel flow relations commonly used in practice. Between Eqs.(6.1) and (6.14), the following form of Chezy's law can be derived:

U CcH1/2S1/2 (6.17)

where the Chezy coefficient Cc is given by the relation

6.4 Chapter Six




Cc

Cg

f

1/2(6.18)

A specific evaluation of Chezy's coefficient can be obtained by substituting Eq. (6.15) intoEq. (6.18). It is seen that the coefficient is not constant but varies as the logarithm of H/ks.A logarithmic dependence is typically a weak one, partially justifying the commonassumption that Chezy's coefficient in Eq. (6.17) is a constant. Substituting Eq. (6.16) intoEqs. (6.17) and (6.18), Manning's law is obtained:

U = 1n H2/3S1/2 (6.19)

where Manning's n is given by

n = 8kgs1

1

/6

/2 (6.20)

The above relation is often called the Manning-Strickler form of Manning's n.

6.2.3 Fixed-Bed and Movable-Bed Roughness

It is clear that to use the above relations for channel flow resistance, a criterion for evalu-ating ks is necessary. Nikuradse (1933) proposed the following criterion: Suppose a roughsurface is subjected to a flow. The equivalent roughness ks of that surface is equal to thediameter of sand grains that, when glued uniformly to a completely smooth wall and thensubjected to the same external conditions, yields the same velocity profile. Nikuradse usedsand glued to the inside of pipes to conduct this evaluation. Extending Nikuradse's con-cept of equivalent grain roughness to the case of rivers and streams, ks can be assumed tobe proportional to a representative sediment size Dx,

ks = αsDx (6.21)

Suggested values of αs, which have appeared in the literature, are listed in Table 6.1 (Yen,1992). Different sizes of sediment have been suggested for Dx in Eq. (6.21). Statistically, D50

(the grain size for which 50% of the bed material is finer) is most readily available andmeaningful. Physically, a representative size larger than D50 is more meaningful to estimate


TABLE 6.1 Ratio of Nikuradse Equivalent Roughness Size and Sediment Size for Rivers.

Investigator Measure of Sediment Size, Dx αs = ks /Dx

Ackers and White (1973) D35 1.23

Strickler (1923) D50 3.3

Keulegan (1938) D50 1

Meyer-Peter and Muller (1948) D50 1

Thompson and Campbell (1979) D50 2.0

Hammond et al. (1984) D50 6.6

Einstein and Barbarossa (1952) D65 1

Irmay (1949) D65 1.5

Engelund and Hansen (1967) D65 2.0

Lane and Carlson (1953) D75 3.2




flow resistance because of the dominant effect by large sediment particles.In flow over a geometrically smooth, fixed boundary, the apparent roughness of the

bed ks can be computed using Nikuradse's approach. However, once the transport of bedmaterial has been instigated, the characteristic grain diameter and the thickness of the vis-cous sublayer no longer provide the relevant length scales. The characteristic length scalein this situation is the thickness of the layer where the sediment particles are being trans-ported by the flow, usually referred to as the bedload layer.

Once the bed shear stress τb exceeds the critical shear stress for particle motion τc, theapparent bed roughness ka can be estimated as follows (Smith and McLean, 1997):

ka α0 ((ρτ

s

b

ρτ)c)g ks (6.22)

where α0 26.3, ks is Nikuradse's fixed-bed roughness, and ρs is the bed sediment densi-ty. This approach is particularly suitable for sand bed rivers.

Under intense sediment transport conditions, bedforms, such as dunes, can develop. Inthis situation, the apparent roughness also will be influenced by the form drag caused bythe presence of bedforms. Nikuradse's approach is valid only for grain-induced roughness.Methods for flow resistance in the presence of both bedforms and grain roughness are pre-sented later.

6.3 SEDIMENT PROPERTIES

6.3.1 Rock Types

The solid phase of the problem embodied in sediment transport can be any granular sub-stance. In engineering applications, however, the granular substance in question typicallyconsists of fragments ultimately derived from rocks–hence the name sediment transport.The properties of these rock-derived fragments, taken singly or in groups of many parti-cles, all play a role in determining the transportability of the grains under fluid action. The

6.6 Chapter Six

TABLE 6.1. (Continued)

Investigator Measure of Sediment Size, Dx αs = ks /Dx

Gladki (1979) D80 2.5

Leopold et al. (1964) D84 3.9

Limerinos (1970) D84 2.8

Mahmood (1971) D84 5.1

Hey (1979), Bray (1979) D84 3.5

Ikeda (1983) D84 1.5

Colosimo et al. (1986) D84 3 6

Whiting and Dietrich (1990) D84 2.95

Simons and Richardson (1966) D85 1

Kamphuis (1974) D90 2.0

van Rijn (1982) D90 3.0

SOURCE: Adapted from Yen (1992)





important properties of groups of particles include porosity and size distribution. The mostcommon rock type one is likely to encounter in the river or coastal environment is quartz.Quartz is a highly resistant rock and can travel long distances or remain in place for longperiods without losing its integrity. Another highly resistant rock type that is often foundtogether with quartz is feldspar. Other common rock types include limestone, basalt, gran-ite, and more esoteric types, such as magnetite. Limestone is not a resistant rock; it tendsto abrade to silt rather easily. Silt-sized limestone particles are susceptible to solutionunless the water is buffered sufficiently. As a result, limestone typically is not a majorcomponent of sediments at locations distant from its source. On the other hand, it oftencan be the dominant rock type in mountain environments.

Basaltic rocks tend to be heavier than most rocks composing the earth’s crust and typ-ically are brought to the surface by volcanic activity. Basaltic gravels are relatively com-mon in rivers that derive their sediment supply from areas subjected to vulcanism in recentgeologic history. Basaltic sands are much less common. Regions of weathered graniteoften provide copious supplies of sediment. Although the particles produced by weather-ing are often in the granule size, they often break down quickly to sand size.

Sediments in the fluvial or coastal environment in the size range of silt, or coarser, aregenerally produced by mechanical means, including fracture or abrasion. The clay miner-als, on the other hand, are produced by chemical action. As a result, they are fundamen-tally different from other sediments in many ways. Their ability to absorb water meansthat the porosity of clay deposits can vary greatly over time. Clays also display cohesivi-ty, which renders them more resistant to erosion.

6.3.2 Specific Gravity

The specific gravity of sediment is defined as the ratio between the sediment density ρs

and the density of water ρ. Some typical specific gravities for various natural and artifi-cial sediments are listed in Table 6.2.

6.3.3 Size

Herein, the notation D is used to denote sediment size, the typical units of which aremillimeters (mm) for sand and coarser material or microns (µ) for clay and silt.Another standard way of classifying grain sizes is the sedimentological Φ scale,according to which

TABLE 6.2 Specific Gravity of Rock Typesand Artificial Material

Rock type or Specific gravitymaterial ρs /ρ

quartz 2.60 2.70limestone 2.60 2.80basalt 2.70 2.90magnetite 3.20 3.50plastic 1.00 1.50coal 1.30 1.50walnut shells 1.30 1.40




6.8 Chapter Six

D 2Φ (6.23)

Taking the logarithm of both sides, it is seen that

Φ log2(D) 11nn((D2)

) (6.24)

Note that the size Φ 0 corresponds to D 1 mm. The usefulness of the Φ scale willbecome apparent upon a consideration of grain size distributions. The minus sign has beeninserted in Eq. (6.24) simply as a matter of convenience to sedimentologists, who are moreaccustomed to working with material finer than 1 mm than they are with coarser materi-al. The reader should always recall that larger Φ implies finer material. The Φ scale pro-vides a simple way of classifying grain sizes into the following size ranges in descendingorder: boulders, cobbles, gravel, sand, silt, and clay. (Table 6.3).

Note that the definition of clay according to size (D 2) does not always correspondto the definition of clay according to mineral. That is, some clay-mineral particles can becoarser than this limit, and some silt-sized particles produced by grinding can be finer thanthat. In general, however, the effect of viscosity makes it difficult to grind up particles inwater to sizes finer than 2.

In practical terms, there are several ways to determine grain size. The most popularway for grains ranging from Φ 4 to Φ 4 (0.0625 to 16 mm) is with the use ofsieves. Each sieve has a square mesh, the gap size of which corresponds to the diameterof the largest sphere that would fit through it. Thus, the grain size D so measured corre-sponds exactly to the diameter only in the case of a sphere. In general, the sieve size Dcorresponds to the smallest sieve gap size through which a given grain can be fitted.

For coarser grain sizes, it is customary to approximate the grain as an ellipsoid. Threelengths can be defined. The length along the major (longest) axis is denoted as a, thelength along the intermediate axis is denoted as b, and the length along the minor (small-est) axis is denoted as c. These lengths are typically measured with a caliper. The value bis then equated to grain size D.

For grains in the silt and clay sizes, many methods (hydrometer, sedigraph, and soforth) are based on the concept of equivalent fall diameter. That is, the terminal fall veloc-ity vs of a grain in water at a standard temperature is measured. The equivalent fall diam-eter D is the diameter of the sphere having exactly the same fall velocity under the sameconditions. Sediment fall velocity is discussed in more detail below.

A variety of other more recent methods for sizing fine particles rely on blockage oflight beams. The blocked area can be used to determine the diameter of the equivalent cir-cle: i.e., the projection of the equivalent sphere. It can be seen that all the above methodscan be expected to operate consistently as long as grains shape does not deviate too great-ly from a sphere. In general, this turns out to be the case. There are some important excep-tions, however. At the fine end of the spectrum, mica particles tend to be platelike; thesame is true of shale grains at the coarser end. Comparison with a sphere is not necessar-ily an especially useful way to characterize grain size for such materials.

6.3.4 Size Distribution

Any sample of sediment normally contains a range of sizes. An appropriate way to char-acterize these samples is by grain size distribution. Consider a large bulk sample of sedi-ment of given weight. Let pf(D)—or pf(Φ)—denote the fraction by weight of material inthe sample of material finer than size D(Φ). The customary engineering representation of




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6.10 Chapter Six

the grain size distribution consists of a plot of pf 100 (percentage finer) versus log10(D):that is, a semilogarithmic plot is used. The same size distribution plotted in sedimento-logical form would involve plotting pf100 versus Φ on a linear plot.

The size distribution pf(Φ) and size density p(Φ) by weight can be used to extract use-ful statistics concerning the sediment in question. Let x denote some percentage, say 50%;the grain size Φx denotes the size such that x percent of the weight of the sample is com-posed of finer grains. That is, Φx is defined such that

pf (x) 10

x0

(6.25)

It follows that the corresponding grain size of equivalent diameter is given by Dx, where

Dx 2 Φx (6.26)

The most commonly used grain sizes of this type are the median size D50 and the sizeD90: i.e., 90% of the sample by weight consists of finer grains. The latter size is especial-ly useful for characterizing bed roughness.

The density p(Φ) can be used to extract statistical moments. Of these, the most usefulare the mean size Φm and the standard deviation σ. These are given by the relations.

Φm Φp(Φ)dΦ; σ2 (Φ Φm)2p(Φ)DΦ (6.27a, b)

The corresponding geometric mean diameter Dg and geometric standard deviation σg

are given as

Dg 2Φm; σg 2σ (6,28a,b)

Note that for a perfectly uniform material, σ 0 and σg 1. As a practical matter, a sed-iment mixture with a value of σg less than 1.3 is often termed well sorted and can be treat-ed as a uniform material. When the geometric standard deviation exceeds 1.6, the mater-ial can be said to be poorly sorted (Diplas and Sutherland, 1988).

In fact, one never has the continuous function p(Φ) with which to compute themoments of Eqs. (6.27a, and b). Instead, one must rely on a discretization. To this end, thesize range covered by a given sample of sediment is discretized using n intervals bound-ed by n 1 grain sizes Φ1, Φ2,…, Φn 1 in ascending order of Φ. The following defini-tions are made from i 1 to n:

Φi 12(Φi Φi1) (6.29a)

pi pf(Φi) pf (Φi1) (6.29b)

Eqs. (6.27a and b) now discretize to

Φm n

i1

Φi pi σ2 n

i1

(Φi Φm)2pi (6.30)

In some cases, especially when the material in question is sand, the size distributioncan be approximated as gaussian on the Φ scale (i.e., log-normal in D). For a perfectlyGaussian distribution, the mean and median sizes coincide:

Φm Φ50 12(Φ84 Φ16) (6.31)





Furthermore, it can be demonstrated from a standard table of the Gauss distribution thatthe size Φ displaced one standard deviation larger that Φm is accurately given by Φ84; bysymmetry, the corresponding size that is one standard deviation smaller than Φm is Φ16.The following relations thus hold:

12(Φ84 Φ16) (6.32a)

Φm 12(Φ84 Φ16) (6.32b)

Rearranging the above relations with the aid of Eqs. (6.28a and b) and Eqs. (6.31 and6.32a),

σg DD

8

1

4

61/2

(6.33a)

Dg (D84D16)1/2 (6.33b)

It must be emphasized that the above relations are exact only for a gaussian distributionin Φ. This is not often the case in nature. As a result, it is strongly recommended that Dg

and σg be computed from the full size distribution via Eqs. (6.30a and b) and (6.28a andb) rather than the approximate form embodied in the above relations.

6.3.5 Porosity

The porosity p quantifies the fraction of a given volume of sediment that is composed ofvoid space. That is,

p

If a given mass of sediment of known density is deposited, the volume of the depositmust be computed, assuming that at least part of it will consist of voids. In the case ofwell-sorted sand, the porosity often can take values between 0.3 and 0.4. Gravels tend tobe more poorly sorted. In this case, finer particles can occupy the spaces between coarserparticles, thus reducing the void ratio to as low as 0.2. Because so-called open-work grav-els are essentially devoid of sand and finer material in their interstices, they may haveporosities similar to sand. Freshly deposited clays are notorious for having high porosi-ties. As time passes, the clay deposit tends to consolidate under its own weight so thatporosity slowly decreases.

The issue of porosity becomes of practical importance with regard to salmon spawn-ing grounds in gravel-bed rivers, for example (Diplas and Parker, 1985). The percentageof sand and silt contained in the sediment is often referred to as the percentage of fines inthe gravel deposit. When this fraction rises above 20 or 26 percent by weight, the depositis often rendered unsuitable for spawning. Salmon bury their eggs within the gravel, anda high fines content implies a low porosity and thus reduced permeability. The flow ofgroundwater necessary to carry oxygen to the eggs and remove metabolic waste productsis impeded. In addition, newly hatched fry may encounter difficulty in finding enoughpore space through which to emerge to the surface. All the above factors dictate loweredsurvival rates. Chief causes of elevated fines in gravel rivers include road building andclear-cutting of timber in the basin.

volume of voidsvolume of total space




6.3.6 Shape

Grain shape can be classified in a number of ways. One of these, the Zingg classificationscheme, is illustrated here (Vanoni, 1975). According to the definitions introduced earlier,a simple way to characterize the shape of an irregular clast (stone) is by lengths a, b, andc of the major, intermediate, and minor axes, respectively. If the three lengths are equal,the grain can be said to be close to a sphere in shape. If a and b are equal but c is muchlarger, the grain should be rodlike. Finally, if c is much smaller than b, which in turn, ismuch larger than a, the resulting shape should be bladelike.

6.3.7 Fall Velocity

A fundamental property of sediment particles is their fall velocity. The relation for termi-nal fall velocity in quiescent fluid vs can be presented as

Rf 43 CD

1(Rp)1/2

(6.34)

where

Rf R

vs

gD (6.35a)

Rp vs

vD (6.35b)

and the functional relation CD CD(Rp) denotes the drag curve for spheres. This relationis not particularly useful because it is not explicit in vs; one must compute fall velocity bytrial and error. One can use the equation for CD given below

CD 2R4p

(1 0.152Rp1/2 0.0151Rp) (6.36)

and the definition

Rep Rg

D D (6.37)

to obtain an explicit relation for fall velocity in the form of Rf versus Rep. In Fig. 6.2, theranges for silt, sand, and gravel are plotted for 0.01 cm2/s (clear water at 20ºC) and R 1.65 (quartz). A good summary of relations for terminal fall velocity for the case ofnonspherical (natural) particles can be found in Dietrich (1982), who also proposed thefollowing useful fit:

Rf expb1 b2ln(Rep) b3[ln(Rep)]2 b4[ln(Rep)]3 b5[ln(Rep)]4 (6.38)

where b1 2.891394, b2 0.95296, b3 0.056835, b4 0.002892, and b5 0.000245

6.3.8 Relation Between Size Distribution and Stream Morphology

The study of sediment properties and, in particular, size distribution is most relevant to thecontext of stream morphology. The following discussion points out some of the moreinteresting issues.

6.12 Chapter Six





FIG

UR

E 6

.2Se

dim

ent f

all v

eloc

itydi

agra

m




6.14 Chapter Six

FIGURE 6.3 Particle size distribution of bed materials in Kankakee River, Illinois.(Bhowmik et al., 1980)

In Fig. 6.3, several size distributions from the sand-bed Kankakee River in Illinois, areshown (Bhowmik et al., 1980). The characteristic S shape suggests that these distributionsmight be approximated by a gaussian curve. The median size D50 falls near 0.3 to 0.4 mm.The distributions are tight, with a near absence of either gravel or silt. For practical pur-poses, the material can be approximated as uniform.

In Fig. 6.4, several size distributions pertaining to the gravel-bed Oak Creek in Oregon,are shown (Milhous, 1973). In gravel-bed streams, the surface layer (“armor” or “pave-ment”) tends to be coarser than the substrate (identified as “subpavement” in the figure).Whether the surface or substrate is considered, it is apparent that the distribution rangesover a much wider range of grain sizes than is the case in Fig. 6.3. More specifically, in





FIG

UR

E 6

.4Si

ze d

istr

ibut

ion

of b

ed m

ater

ial s

ampl

es in

Oak

Cre

ek. O

rego

n. S

ourc

e:(M

ilhou

s,19

73)




6.16 Chapter Six

the distributions of the sand-bed Kankakee River, Φ varies from about 0 to about 3, where-as in Oak Creek, Φ varies from about 8 to about 3. In addition, the distribution of Fig.6.4 is upward-concave almost everywhere and thus deviates strongly from the gaussiandistribution.

These two examples provide a window toward generalization. A river can be looselyclassified as sand-bed or gravel-bed according to whether the median size D50 of the sur-face material or substrate is less than or greater than 2 mm. The size distributions of sand-bed streams tend to be relatively narrow and also tend to be S shaped. The size distribu-tions of gravel-bed streams tend to be much broader and to display an upward-concaveshape. Of course, there are many exceptions to this behavior, but it is sufficiently generalto warrant emphasis.

More evidence for this behavior is provided in Fig. 6.5. Here, the grain size distribu-tions for a variety of stream reaches have been normalized using the median size D50.Four sand-bed reaches are included with three gravel-bed reaches. All the sand-bed distributions are S shaped, and all have a lower spread than the gravel-bed distributions.The standard deviation is seen to increase systematically with increasing D50(White et al.,1973).

The three gravel-bed size distributions differ systematically from the sand-bed distrib-utions in a fashion that accurately reflects Oak Creek (Fig. 6.4). The standard deviation inall cases is markedly larger than any of the sand-bed distributions, and the distributions

FIGURE 6.5 Dimensionless grain-size distribution for different rivers (White et al., 1973)

are upward-concave except perhaps near the coarsest sizes.

6.4 THRESHOLD CONDITION FOR SEDIMENT MOVEMENT





When a granular bed is subjected to a turbulent flow, virtually no motion of the grains isobserved at some flows, but the bed is mobilized noticeably at other flows. Factors thataffect the mobility of grains subjected to a flow are summarized below:

randomness

grain placement

turbulence

forces on grain

fluid

liftmean & turbulent

drag

gravity

In the presence of turbulent flow, random fluctuations typically prevent the clear defini-tion of a critical, or threshold condition for motion: The probability for the movement ofa grain is never precisely zero (Lavelle and Mofjeld, 1987). Nevertheless, it is possible todefine a condition below which movement can be neglected for many practical purposes.

6.4.1 Granular Sediment on a Stream Bed

Figure 6.6 is a diagram showing the forces acting on a grain in a bed of other grains. Whencritical conditions exist and the grain is on the verge of moving, the moment caused bythe critical shear stress τc about the point of support is just equal to that of the weight ofthe grain. Equating these moments gives (Vanoni, 1975):

FIGURE 6.6 Forces acting on a sediment particle on an inclined bed




6.18 Chapter Six

τc cc

1

2

aa

1

2 (γs γ) Dcos φ(tan θ tanφ) (6.39)

in which γs specific weight of sediment grains, γ specific weight of water, D diam-eter of grains, is the slope angle of the stream, the angle of repose of the sediment, c1

and c2 are dimensionless constants, and a1 and a2 are lengths shown in Fig. 6.6. Any consis-tent set of units can be used in Eq. (6.39). For a horizontal bed, Eq. (6.39) reduces to

τc cc

1

2

aa

1

2 (γs γ)D tan θ (6.40)

For an adverse slope (i.e., 0),

τc cc

1

2

aa

1

2 (γs γ)D cos φ(tan θ tan φ) (6.41)

Equations (6.39), (6.40), and (6.41) cannot be used to give τc because the factors c1, c2,a1, and a2 are not known. Therefore, the relation between the pertinent quantities isexpressed by dimensional analysis, and the actual relation is determined from experimen-tal data. Figure 6.7 is such a relation, first presented by Shields (1936) and carries hisname. The curve is expressed by dimensionless combinations of critical shear stress τc,sediment and water specific weights γs and γ, sediment size D, critical shear velocity u*c

τc/ρ and kinematic viscosity of water ν.These quantities can be expressed in any consistent set of units. Dimensional analysis

yields,

τc* (γs

τc

γ)D fu*

νcD

(6.42)

The Shields values of τc* are commonly used to denote conditions under which bed

sediments are stable but on the verge of being entrained. Not all workers agree with theresults given by the Shields curve. For example, some workers give τc

* 0.047 for thedimensionless critical shear stress for values of R* u*D/ν in excess of 500 instead of0.06, as shown in Fig. 6.7. Taylor and Vanoni (1972) reported that small but finite amountsof sediment were transported in flows with values of τc

* given by the Shields curve.The value of τc to be used in design depends on the particular case at hand. If the sit-

uation is such that grains that are moved can be replaced by others moving from upstream,some motion can be tolerated, and the Shields values can be used. On the other hand, ifgrains removed cannot be replaced, as on a stream bank, the Shields value of τc are toolarge and should be reduced.

The Shields diagram is not especially useful in the form of Fig. 6.7 because to find τc,one must know u* τc/ρ. The relation can be cast in explicit form by plotting τc

* ver-sus Rep, noting the internal relation

u*

D

Ru*

gD (τ*)1/2Rep (6.43)

where R ρs

ρρ

is the submerged specific gravity of the sediment. A useful fit is given

by Brownlie (1981a):

τ*c 0.22Rep

0.6 0.06 exp(17.77Rep0.06) (6.44)

RgD D





FIG

UR

E 6

.7Sh

ield

s di

agra

m f

or in

itiat

ion

of m

otio

n. S

ourc

e V

anon

i (19

75)




6.20 Chapter Six

FIGURE 6.8 Angle of repose of granular material. (Lane,1955)

With this relation, the value of τc* can be computed readily when the properties of the

water and the sediment are given.The value of bed-shear stress τb for a wide rectangular channel is given by τb γHS,

as shown earlier. The average bed-shear stress for any channel is given by τb γRhS, inwhich Rh the hydraulic radius of the channel cross section.

6.4.2 Granular Sediment on a Bank

A sediment grain on a bank is less stable than one on the bed because the gravity forcetends to move it downward (Ikeda, 1982). The ratio of the critical shear stress τwc for a par-ticle on a bank to that for the same particle on the bed τc is (Lane, 1955)

ττw

c

c = cos φ1 1

ttaann φ

φ1

2 (6.45)

where φ1 is the slope of the bank and θ is the angle of repose for the sediment. Values of θ are





given in Fig. 6.8 after Lane (1955) and also can be found in Simons and Senturk (1976).

6.4.3 Granular Sediment on a Sloping Bed

Equation (6.39) shows that τc diminishes as the slope angle φ increases. For extremelysmall φ’s, τc is given by Eq. (6.40). Taking the ratio between Eqs. (6.39) and (6.40) yields

ττ

c

c

ο

φ cos φ

1

ttaann

φθ

(6.46)

τcφ is the critical shear stress for sediment on a bed with a slope angle φ, and τco is the crit-ical shear stress for a bed with an extremely small slope. The value of τco can be foundfrom the Shields diagram or with Eq. (6.44). Equation (6.46) is for positive φ, which ispositive for downward sloping beds. For beds with adverse slope, φ is negative and theterm tan φ/tan θ in Eq. (6.46) is positive.

6.4.4 Sediment Mixtures

Several authors have offered empirical or quasi-theoretical extensions of the above rela-tions to the case of mixtures (e.g., Wilcock, 1988). Let Di denote the characteristic grainsize of the ith size range in a mixture. Furthermore, let Dsg denote the geometric mean sizeof the surface (exchange, active) layer. Most of the generalizations can be written in thefollowing form (Parker, 1990):

τ*ci τ*

cg

DD

s

i

g

β(6.47)

Here

τ*ci ρR

τb

gc

Di

i (6.48a)

and

τ*cg ρR

τb

gc

Dsg

sg (6.48b)

where τbci and τbcsg denote the values of the dimensioned critical shear stress required tomove sediment of sizes Di and Dsg in the mixture, respectively, and β is an exponent tak-ing a value given below;

β 0.9 (6.49)

Figure 6.9 shows the similarity between four different published expressions havingthe general form given by Eq. (6.47), which is of interest because it includes the effect ofhiding. For uniform material, the critical Shields stress is defined by Eq. (6.44). Considertwo flumes, one with uniform size Da and the other with uniform size Db. For sufficient-ly coarse material (u*D/ν » 1 or Rep » 1), the critical Shields stress must be the same forboth sizes (Fig. 6.7). It follows from Eq. (6.42) that where τbca and τbcb denote the dimen-sioned boundary shear stresses for cases a and b respectively,

τbcb τbca

DD

b

a

(6.50)




6.22 Chapter Six

For the case of mixtures, on the other hand, it is seen from Eqs. (6.47) and (6.48) that

τbci τbcsg

DD

s

i

g

1β τbcsg

DD

s

i

g

0.1(6.51)

Comparing Eqs. (6.50) and (6.51), it is seen that a finer particle (Db Da, or alternative-ly, Di Dsg) is more mobile than a coarser particle. For example, suppose that one grainsize is four times coarser than another. If two uniform sediments are being compared, itfollows from Eq. (6.50) that the critical shear stress for the coarser material is four timesthat of the finer material. In the case of a mixture, however, the critical shear stress for thecoarser material is only about 40.1, or 1.15 times that for the finer material.

A finer particle in a mixture is thus seen to be only a little more mobile than its coars-er-sized brethren, where uniform beds of fine material are much more mobile than are uni-form beds of coarser material. The reason is that finer particles in a mixture are relativelyless exposed to the flow; they tend to hide in the lee of coarser particles. By the same token,a particle is relatively more exposed to the flow when most of its neighbors are finer.

A method to calculate the critical shear stress for motion of uniform and heterogeneoussediments was proposed by Wiberg and Smith (1987) on the basis of the fluid mechanicsof initiation of motion, which takes into account both roughness and hiding effects.

6.5 SEDIMENT TRANSPORT

FIGURE 6.9 Critical shear stress for sediment mixture (Source: Misri etal., 1983)





6.5.1 Sediment Transport Modes

The most common modes of sediment transport in rivers are bedload and suspended load.In the case of bedload, the particles roll, slide, or saltate over each other, never deviatingtoo far above the bed. In the case of suspended load, the fluid turbulence comes into playcarrying the particles well up into the water column. In both cases, the driving force forsediment transport is the action of gravity on the fluid phase; this force is transmitted tothe particles via drag.

The same phenomena of bedload and suspended load transport occur in a variety ofother geophysical contexts. Sediment transport is accomplished in the near-shore lake andoceanic environment by wave action. Turbidity currents carry sediment into lakes, reser-voirs, and the deep sea.

The phenomenon of sediment transport can sometimes be disguised in rather esotericphenomena. When water is supercooled, large quantities of particulate frazil ice can form.As this water moves under a frozen ice cover, the phenomenon of sediment transport inrivers is stood on its head. The frazil ice particles float rather than sink and thus tend toaccumulate on the bottom side of the ice cover rather than on the river bed. Turbulencetends to suspend the particles downward rather than upward.

In the case of a powder snow avalanche, the fluid phase is air and the solid phase con-sists of snow particles. The dominant mode of transport is suspension. These flows areclose analogies of turbidity currents, insofar as the driving force for the flow is the actionof gravity on the solid phase rather than the fluid phase. That is, if all the particles dropout of suspension, the flow ceases. In the case of sediment transport in rivers, it is accu-rate to say that the fluid phase drags the solid phase along. In the case of turbidity currentsand powder snow avalanches, the solid phase drags the fluid phase along.

Desert sand dunes provide an example for which the fluid phase is air, but the domi-nant mode of transport is saltation rather than suspension. Because air is so much lighterthan water, quartz sand particles saltate in long, high trajectories, relatively unaffected bythe direct action of turbulent fluctuations. The dunes themselves are created by the effectof the fluid phase acting on the solid phase. They, in turn, affect the fluid phase by chang-ing the resistance.

Among the most interesting sediment–transport phenomena are debris flows, slurries,and hyperconcentrated flows. In all these cases, the solid and fluid phases are present insimilar quantities. A debris flow typically carries a heterogeneous mixture of grain sizesranging from boulders to clay. Slurries and hyperconcentrated flows are generally restrict-ed to finer grain sizes. In most cases, it is useful to think of such flows as consisting of asingle phase, the mechanics of which are highly non-Newtonian.

The study of the movement of grains under the influence of fluid drag and gravitybecomes even more interesting when one considers the link between sediment transportand morphology. In the laboratory, the phenomenon can be studied in the context of a vari-ety of containers, such as channel and wave tanks, specified by the experimentalist. In thefield, however, the fluid-sediment mixture constructs its own container. This new degreeof freedom opens up a variety of intriguing possibilities.

Consider the river. Depending on the existence or lack of a viscous sublayer and therelative importance of bedload versus suspended load, a variety of rhythmic structures canform on the river bed. These include ripples, dunes, antidunes, and alternate bars. The firstthree of these can have a profound effect on the resistance to flow offered by the river bed.Thus, they act to control river depth. River banks themselves also can be considered to bea self-formed morphological feature, thus specifying the entire container.

The container itself can deform in plan. Alternate bars cause rivers to erode their banksin a rhythmic pattern, thus allowing for the onset of meandering. Fully developed rivermeandering implies an intricate balance between sediment erosion and deposition. If a




stream is sufficiently wide, it will braid rather than meander, dividing into several inter-twining channels.

Rivers create morphological structures at much larger scales as well. These includecanyons, alluvial fans, and deltas. Turbidity currents create similar structures in the ocean-ic environment. In the coastal environment, the beach profile itself is created by the inter-action of water and sediment. On a larger scale, offshore bars, spits, and capes constituterhythmic features created by wave-current-sediment interaction. The boulder levees oftencreated by debris flows provide another example of a morphologic structure created by asediment-bearing flow.

The floodplains of most sand-bed rivers often contain copious amounts of silt and clayfiner than approximately 50 µ. This material is often called wash load because it movesthrough the river system without being present in the bed in significant quantities.Increased wash load does not cause deposition on the bed, and reduced wash load doesnot cause erosion because it is transported well below capacity. This is not meant to implythat the wash load does not interact with the river system. Wash load in the water columnexchanges with the banks and the floodplain rather than the bed. Greatly increased washload, for example, can lead to thickened floodplain deposits, with a consequent increasein bankfull channel depth.

The emphasis here is the understanding of bedload and suspended load transport inrivers, with the goal of providing the knowledge needed to do sound sedimentation engi-neering, particularly with problems involving stream restoration and naturalization.

6.5.2 Shields Regime Diagram

In the context of rivers, it is useful to have a way to determine what kind of sediment-transport phenomena can be expected for different flow conditions and different charac-teristics of sediment particles. In Fig. 6.10, the ordinates correspond to bed shear stresseswritten in the dimensionless form proposed by Shields

τ* ρgτRb

D RH

DS (6.52)

and the particle Rep, defined by Eq. (6.37) is used for the abscissa values. There are threecurves in the diagram which make it possible to know, for different values of (τ*, Rep), ifthe given bed sediment will go into motion, and if this is the case whether or not the pre-vailing mode of transport will be in suspension or as bedload. The diagram also can be usedto predict what kind of bedforms can be expected. For example, ripples will develop in thepresence of a viscous sublayer and fine-grained sediment. If the viscous sublayer is dis-rupted by coarse sediment particles, then dunes will be the most common type of bedform.

The Shields regime diagram also shows a clear distinction between the conditionsobserved in sand-bed rivers and gravel-bed rivers at bankfull stage. If one wanted to mod-el in the laboratory sediment transport in rivers, the experimental conditions would be dif-ferent, depending on the river system in question. As could be expected, the diagram alsoshows that in gravel-bed rivers, sediment is transported as bedload. In sand-bed rivers, onthe other hand, suspended load and bedload transport coexist most of the time.

The regime diagram is valid for steady, uniform, turbulent flow conditions, where thebed shear stress τb can be estimated with Eq. (6.1). The ranges for silt, sand, and gravelalso are included. In the diagram, the critical Shields stress for motion was plotted withthe help of Eq. (6.44). The critical condition for suspension is given by the following ratio:

uv

*

s 1 (6.53)

6.24 Chapter Six





FIG

UR

E 6

.10

Shie

lds

regi

me

diag

ram

. (So

urce

:Gar

y Pa

rker

)




6.26 Chapter Six

where u* is the shear velocity and vs is the sediment fall velocity. Equation (6.53) can betransformed into

τ∗s R

12f

(6.54)

where

τ∗s

gRu2

*

D (6.55)

and Rf is given by Eq. (6.35a) and can be computed for different values of Rep with thehelp of Eq. (6.38).

Finally, the critical condition for viscous effects (ripples) was obtained with the helpof Eq. (6.5) as follows:

11.6 u*

νD 1 (6.56)

which in dimensionless form can be written as

τ*v

1R1

e

.p

6

2(6.57)

Relations (6.44), (6.54), and (6.57) are the ones plotted in Fig. 6.10. The Shields regimediagram should be useful for studies concerning stream restoration and naturalizationbecause it provides the range of dimensionless shear stresses corresponding to bankfullflow conditions for both gravel- and sand-bed streams.

6.6 BEDLOAD TRANSPORT

6.6.1 The Bed Load Transport Function

Bedload particles roll, slide, or saltate along the bed. The transport thus occurs tangentialto the bed. In a case where all the transport is directed in the streamwise, or s direction,the volume bedload-transport rate per unit width (n direction) is given by q; the units arelength3/length/per time, or length2/time. In general, q is a function of boundary shear stressτb and other parameters; that is,

q q(τb, other parameters) (6.58)

In general, bedload transport is vectorial, with components qs and qn in the s and n direc-tions, respectively.

6.6.2 Erosion Into and Deposition from Suspension

The volume rate of erosion of bed material into suspension per unit time per unit bed areais denoted as E. The units of E are length3/length2/time, or velocity. A dimensionless sed-iment entrainment rate Es can thus be defined with the sediment fall velocity vs:

E vsEs (6.59)




In general, Es can be expected to be a function of boundary shear stress τb and other para-meters. Erosion into suspension can be taken to be directed upward normal: i.e., in thepositive z direction.

Let c denote the volume concentration of suspended sediment (m3 of sediment/m3 ofsediment-water mixture), averaged over turbulence. The streamwise volume transport rateof suspended sediment per unit width is given by

qs H

0c udz (6.60)

In a two-dimensional case, two components, qSs and qSn, result, where

qSs H

0c udz (6.61a)

qSn H

0c vdz (6.61b)

Deposition onto the bed is by means of settling. The rate at which material is fluxedvertically downward onto the bed (volume/area/time) is given by vscb, where cb is a near-bed value of c. The deposition rate D realized at the bed is obtained by computing thecomponent of this flux that is actually directed normal to the bed:

D vscb (6.62)

6.6.3 The Exner Equation of Sediment Mass Conservation forUniform Material

Consider a portion of river bottom, where the bed material is taken to have a (constant)porosity λp. Mass balance of sediment requires the following equation to be satisfied:

∂∂t

[mass of bed material] net mass bedload inflow rate

net mass rate of deposition from suspension.

A datum of constant elevation is located well below the bed level, and the elevation of thebed with respect to such datum is given by η. Then, bed level changes as a result of bed-load transport, sediment entrainment into suspension, and sediment deposition onto thebed can be predicted with the help of

(1 λp) ∂∂ηt

∂∂qs

s

∂∂qn

n vs (cb Es) (6.63)

To solve the Exner equation, it is necessary to have relations to compute bedload transport(i.e., qs and qn), near-bed suspended sediment concentration cb, and sediment entrainmentinto suspension Es. The basic form of Eq. (6.63) was first proposed by Exner (1925).

6.6.4 Bedload Transport Relations

A large number of bedload relations can be expressed in the general form





6.28 Chapter Six

q* q*(τ*, Rep, R) (6.64)

Here, q* is a dimensionless bedload transport rate known as the Einstein number, firstintroduced by H. A. Einstein in 1950 and given by

q* Rg

q

D D (6.65)

The following relations are of interest. In 1972, Ashida and Michiue introduced

q* 17(τ* τ*c) [(τ*)1/2 (τ∗

c)1/2] (6.66)

and recommend a value of τc* of 0.05. It has been verified with uniform material ranging

in size from 0.3 mm to 7 mm. Meyer-Peter and Muller (1948) introduced the following:

q* 8(τ* τ*c)3/2 (6.67)

where τ*c 0.047. This formula is empirical in nature and has been verified with data for

uniform gravel.Engelund and Fredsøe (1976) proposed,

q* 18.74(τ* τ*c) [(t*)1/2 0.7(τ∗

c)1/2] (6.68)

where τ*c 0.05. This formula resembles that of Ashida and Michiue because the deriva-

tion is almost identical.Fernandez Luque and van Beek (1976) developed the following,

q* 5.7(τ* τ*c)3/2 (6.69)

where τ∗c varies from 0.05 for 0.9 mm material to 0.058 for 3.3. mm material. The relation

is empirical in nature.Wilson (1966):

q* 12(τ* τ∗c)3/2 (6.70)

where τ∗c was determined from the Shields diagram. This relation is empirical in nature;

most of the data used to fit it pertain to very high rates of bedload transport.Einstein (1950):

q* q*(τ*) (6.71)

where the functionality is implicitly defined by the relation

1 (0.143/τ*)2

(0.413/τ*)2et2dt 1

434.53q.5

*

q* (6.72)

Note that this relation contains no critical stress. It has been used for uniform sand andgravel.

Yalin (1963):

q* 0.635s(τ*)1/21

1n(1a

2sa2s)

(6.73)

where

a2 2.45(R 1)0.4 (τ∗c)1/2; s

τ*

τ*c

τ*c

(6.74)

1





and τ∗c is evaluated from a standard Shields curve. Two constants in this formula have been

evaluated with the aid of data quoted by Einstein (1950), pertaining to 0.8 mm and 28.6mm material.

Parker (1978):

q* 11.2(τ*

τ0*3

.03)4.5

(6.75)

developed with data sets pertaining to rough mobile-bed flow over gravel.Several of these relations are plotted in Fig. 6.11. They tend to be rather similar in

nature. Scores of similar relations could be quoted.To date, only few research groups have attempted complete derivations of the bedload

function in water. They are Wiberg and Smith (1989), Sekine and Kikkawa (1992), Garcíaand Niño (1992), Niño and García, (1994, 1998), and Niño et al., (1994).

FIGURE 6.11 Bedload transport relations. (Parker, 1990)




6.30 Chapter Six

6.6.5 Bedload Transport Relation for Mixtures.

Relatively few bedload relations have been developed specifically in the context of mix-tures (e.g., Bridge and Bennett, 1992). One of these is presented below as an example.

The relationship of Parker (1990) applies to gravel-bed streams. The data used to fitthe relation are solely from two natural gravel-bed streams: Oak Creek in Oregon and theElbow River in Alberta, Canada. The relation is surface-based; load is specified per unitof fractional content in the surface layer. The surface layer is divided into N size ranges,each with a fractional content Fi by volume, and a mean phi size φi; Di 2φi. The arith-metic mean of the surface size on the phi scale φ and the corresponding arithmetic stan-dard deviation σφ are given by

φ ΣFiφi; σ2φ ΣFi(φi φ)2 (6.76a, b)

The corresponding geometric mean size Dsg and the geometric standard deviation σsg ofthe surface layer are given by

Dsg 2φ σsg 2σφ (6.77a, b)

In the Parker relation, the volume bedload transport per unit width of gravel in the ithsize range is given by the product qiFi (no summation), where qi denotes the transport perunit fraction in the surface layer. The total volume bedload transport rate of gravel per unitwidth is qT, where

qT qiFi (6.78)

The relation does not apply to sand. Thus, before using the relation for a given surfacedistribution, the sand content of the grain-size distribution must be removed and Fi mustbe renormalized so that it sums to unity over all sizes in excess of 2 mm.

If pi denotes the fraction volume content of material in the ith size range in the bed-load, it follows that

pi qqiF

iFi

i (6.79)

The parameter qi is made dimensionless as follows:

W*si (τb

R/ρ

g)q3/

i

2Fi (6.80)

A dimensionless Shields stress based on the surface geometric mean size is defined asfollows:

τ*sg ρR

τg

b

Dsg (6.81)

Let φsgo denote a normalized value of this Shields stress, given by

φsgo ττ

*

*

r

s

s

g

g

o

o (6.82)

where

τ*rsgo 0.0386 (6.83)

corresponds to a “near-critical” value of Shields stress. The Parker relation can then be





expressed in the form

W*si 0.0218 G [ωφsgogo(δi)] (6.84a)

In the above relationship, go denotes a hiding function given by

go(δi) δ0.0951i ; δi D

D

s

i

g (6.84b)

The parameter ω is given by the relationship

ω 1 σσ

φ

φ

o(ωo 1) (6.84c)

where σφo and ωo are specified as functions of φsgo in Fig. 6.12. The function G is speci-fied as

5474(1 0.853/φ)4.5 φ 1.65

G[φ]

exp[14.2(φ 1) 9.28(φ 1)2] 1 φ 1.65

φMo φ 1 (6.85)

and is shown in Fig. 6.13. Here, Mo 14.2 and φ is a dummy variable for the argumentin Eq. (6.84) and is not to be confused with the φ grain-size scale.

An application of Eq. (6.84) to uniform material with size D results in the relation

q* 0.0218(τ*)3/2G0.0

τ3

*

86

(6.86)

where

q* gR

q

D D ; τ* ρg

τRb

D (6.87)

and q denotes the volumetric sediment transport per unit width. In Fig. 6.11, Eq. (6.86) iscompared to several other relations and selected laboratory data for uniform material. Thefigure is adapted from Figs. 6b and 7 in Wiberg and Smith (1989), where reference to thedata and equations can be found. The data pertain to 0.5 mm sand and 28.6 mm gravel.Equation (6.86) shows a reasonable correspondence with the data and with several otherrelations for uniform material.

The Parker relationship (Eq. 6.84) can be used to predict mobile or static armor ingravel streams. Note that there is no formal critical stress in the formulation; instead for φ 1, the transport rates become extremely small. For the computation of bedload trans-port in poorly sorted gravel-bed rivers, the above formulation has been used to implementa series of programs named “ACRONYM” (Parker, 1990). The program “ACRONYM1”provides an implementation of the surface-based bedload transport equation presented inParker (1990). It computes the magnitude and size distribution of bedload transport overa bed surface of given size distribution, on which a given boundary shear stress isimposed. The program “ACRONYM2” inverts the same bedload transport equation,allowing for calculation of the size distribution at a given boundary shear stress. The pro-gram was used to compute mobile and static armor size distributions in Parker (1990) andParker and Sutherland (1990).




6.32 Chapter Six

FIG

UR

E 6

.12

Plot

s of

ω0

and

σφ0

vers

us φ

sg0,

the

asym

ptot

es a

re n

oted

on

the

plot

. (Pa

rker

,199

0)





FIG

UR

E 6

.13

Plot

of

Gan

dG

Tve

rsus

φ50

. (Pa

rker

,199

0)




The program “ACRONYM3” allows for the computation of aggradation or degrada-tion to a specified active or static equilibrium final state. To this end, Parker’s method(1990) is combined with a resistance relation of the Keulegan type. In the program, bothconstant width and water discharge are assumed.

The program “ACRONYM4” is directed toward the wavelike aggravation of self-sim-ilar form discussed in Parker (1991a, 1991b). It uses Parker’s method and a resistancerelation of the Manning-Strickler type to compute downstream fining and slope concavi-ty caused by selective sorting and abrasion.

6.7 BEDFORMS

The formation and behavior of sediment waves produced by moving water are, in equalmeasure, intellectually intriguing and of great engineering importance. Because of thecentral role they play in river hydraulics, fluvial ripples, dunes, and bars have receivedextensive attention from engineers for at least the past two centuries, and even more inten-sive descriptive study from geologists. Such studies can be divided into three categoriesaccording to the approach followed: analytical, empirical, or statistical.

Analytical models for bedforms have been proposed since 1925 (Anderson, 1953;Blondeaux et al., 1985; Colombini et al., 1987; Engelund, 1970; Exner, 1925; Fredsoe,1974, 1982; Gill, 1971; Haque and Mahmood, 1985; Hayashi, 1970; Kennedy, 1963,1969; Parker, 1975; Raudkivi and Witte, 1990; Richards, 1980; Smith, 1970; Tubino andSeminara, 1990).

Empirical methods include the following works (Coleman and Melville, 1994;Colombini et al., 1990; García and Niño, 1993; Garde and Albertson, 1959; Ikeda, 1984;Jaeggi, 1984; Kinoshita and Miwa, 1974; Menduni and Paris, 1986: Ranga Raju and Soni,1976; Raudkivi, 1963; van Rijn, 1984; Yalin, 1964; Yalin and Karahan, 1979).

Statistical models for bedforms have been advanced by the following authorsAnnambhotla et al.,1972; Hino, 1968; Jain and Kennedy, 1974; Nakagawa and Tsujimoto,1984; Nordin and Algert, (1966).

Despite all the research that has been done, there is presently no completely reliablepredictor for the conditions of occurrence and characteristics of the different bed config-urations (ripples, dunes, flat bed, antidunes).

6.7.1 Dunes, Antidunes, Ripples, and Bars

The ripples, dunes, and antidunes illustrated in Fig. 6.14 are the classic bedforms of erodible-bed open-channel flow. On the one hand, they are a product of the flow and sediment transport; on the other hand, they profoundly influence the flow and sedimenttransport. In fact, all the bedload formulas quoted previously are strictly invalid in the presence of bedforms. The adjustments necessary to render them valid are discussedlater.

Ripples, dunes, and antidunes are undular (wavelike) features that have wavelengths Λand wave heights ∆ that scale no larger than on the order of the flow depth H, as definedbelow.

6.7.1.1 Dunes. Well-developed dunes tend to have wave heights D scaling up to aboutone-sixth of the depth: i.e.,

6.34 Chapter Six





H∆

16 (6.88)

Dune wavelengths can vary considerably. A fairly typical range can be quantified asdimensionless wave number k, where

k 2π

ΛH (6.89)

This range is

0.25 k 4.0 (6.90)

Dunes invariably migrate downstream. Typically, they are approximately triangular inshape and usually (but not always) possess a slip face, beyond which the flow is separat-ed for a certain length. A dune progresses forward as bedload accretes on the slip face.Generally, little bedload is able to pass beyond the face without depositing on it, whereasmost of the suspended load is not directly affected by it.

Let c denote the wave speed of the dune. The bedload transport rate can be estimated asthe volume of material transported forward per unit bed area per unit time by a migratingdune. If the dune is approximated as triangular in shape, the following approximation holds:

q 12 ∆∆c(1 λp) (6.91)

Dunes are characteristic of subcritical flow in the Froude sense. In a shallow-water (longwave) model, the Froude criterion (Fr) dividing subcritical and supercritical flow is

FIGURE 6.14 Schematic of different bedforms. (Vanoni, 1975)




6.36 Chapter Six

Fr 1 (6.92)

where

Fr (6.93)

Dunes, however, do not qualify as long waves because their wavelength is of the order ofthe depth. A detailed potential flow analysis over a wavy bed yields the following (wave-number dependent) criterion for critical flow over a bedform (Kennedy, 1963).

Fr2 1k

tanh(k) (6.94)

Note that as k → 0(Λ → ∞) tanh(k) → k, and condition (6.92) is recovered in the long-wave limit. For dunes to occur, then, the following condition must be satisfied:

F2 1k

tanh (k) (6.95)

Both dunes and antidunes cause the water surface as well as the bed to undulate. In thecase of dunes, the undulation of the water surface is usually of much smaller amplitudethan that of the bed; the two are nearly 180o out of phase.

Dunes also can occur in the case of wind-blown sand. Barchan dunes are commonlyobserved in the desert. In addition, they can be found in the fluvial environment in the caseof sand (in supply insufficient to cover the bed completely) migrating over an immobilegravel bed.

6.7.1.2 Antidunes. Antidunes are distinguished from dunes by the fact that the undula-tions of the water surface are nearly in phase with those of the bed. They are associatedwith supercritical flow in the sense that

F2 1k tanh (k) (6.96)

Antidunes may migrate either upstream or downstream. Upstream-migrating antidunesare usually rather symmetrical in shape and lack a slip face. Downstream-migratingantidunes are rarer; they have a well-defined slip face and look rather like dunes. The dis-tinguishing feature is the water surface undulations, which are pronounced in the case ofantidunes.

The potential-flow criterion dividing upstream-migrating antidunes from downstream-migrating antidunes is

F2 k tan1h (k) (6.97)

Values lower than the value in Eq. 6.97 are associated with upstream-migrating antidunes.

6.7.1.3 Ripples. Ripples are dunelike features that occur only in the presence of a vis-cous sublayer. They look much like dunes because they migrate downstream and have apronounced slip face. They generally are much more three-dimensional in structure thanare dunes, however, and have little effect on the water surface.

A criterion for the existence of ripples is the existence of a viscous sublayer. Recallingthat the thickness of the viscous sublayer is given by δv 11.6v/u*, it follows that ripplesform when

UgH





u*

νD 11.6 (6.98)

6.7.1.4 Bars. Bars are bedforms in rivers that scale the channel width. They includealternate bars in straight streams, point bars in meandering streams, and pool bars in braid-ed streams. In straight streams, the minimum channel slope S necessary for alternate-barformation is given by

S (6.99)

(Jaeggi, 1984), where B is the channel width, Dg is the geometric mean size of the bed sed-iment, as given by Eq. (6.82a), and M is a parameter that varies from 0.34 for uniform-sized bed material to 0.7 for poorly sorted material.

Scour depth (Sd) caused by alternate bar formation can be estimated with

Sd 0.76∆AB , (6.100)

where ∆AB is the total height of the alternate bar.

6.7.1.5 Progression of bedforms. Various bedforms are associated with various flowregimes. In the case of a sand-bed stream with a characteristic size less than about 0.5 mm,a clear progression is evident as flow velocity increases. This is illustrated in Fig. 6.14.

The bed is assumed to be initially flat. At low imposed velocity U, the bed remains flatbecause no sediment is moved. As the velocity exceeds the critical value, ripples areformed first. At higher values, dunes form and coexist with ripples. For even higher veloc-ities, well-developed dunes form in the absences of ripples.

At some point, the velocity reaches a value near the critical value in the Froude sense:

B

6DB

g

0.15

exp1.07

DB

g

0.15 M

12.9DB

g

Ripples

Bars

Wavelength less thanapprox 1 ft; height lessthan approx 0.1 ft.

Lengths comparable tothe channel width.Height comparable tomean flow depth.

Roughly triangular inprofile, with gentle,slightly convexupstream slopes anddownstream slopesnearly equal to theangle of repose.Generally short-crestedand three-dimensional.

Profile similar toripples. Plan formvariable.

Move downstream withvelocity much less thanthat of the flow.Generally do not occurin sediments coarserthan about 0.6 mm.

Four types of bars aredistinguished: (1)point, (2) alternating,(3) transverse, and (4)tributary. Ripples mayoccur on upstreamslopes.

Table 6.4 Summary of Bedform Effects on Flow Configuration

Bed Form or Behavior and Configuration Dimensions Shape Occurrence

(1) (2) (3) (4)




6.38 Chapter Six

Dunes

Transition

Flat bed

Antidunes

Wavelength and heightgreater than ripples butless than bars.

Vary widely

—

Wave length 2πV2/g(approx)* Heightdepends on depth andvelocity of flow.

Similar to ripples.

Vary widely.

—

Nearly sinusoidal inprofile. Crest lengthcomparable towavelength.

Upstream slopes ofdunes may be coveredwith ripples. Dunesmigrate downstream inmanner similar to ripples.

A configurationconsisting of aheterogeneous array ofbed forms, primarilylow amplitude ripplesand dunes interspersedwith flat regions.

A bed surface devoidof bed forms. May notoccur for some rangesof depth and sand size.

In phase with andstrongly interact withgravity watersurfacewaves. May moveupstream, downstream,or remain stationary,depending on propertiesof flow and sediment.

Table 6.4 (Continued)

Bed Form or Behavior and Configuration Dimensions Shape Occurence

(1) (2) (3) (4)

*Reported by Kennedy (1969).Source: Vanoni (1975).

i.e., Eq. (6.94). Near this point, the dunes often are suddenly and dramatically washed out.This results in a flat bed known as an upper-regime (supercritical) flat bed. Furtherincreases in velocity lead to the formation of antidunes and, finally, to the chute and poolpattern. The last of these is characterized by a series of hydraulic jumps.

In the case of a bed coarser than 0.5 mm, the ripple regime is replaced by a zone char-acterized by a lower-regime (subcritical) flat bed. Above this lies the ranges for dunes, theupper-regime flat bed, and antidunes.

The effect of bedforms on flow resistance is summarized in Table 6.4. As noted earli-er for equilibrium flows in wide straight channels, the relation for bed resistance can beexpressed in the form

τb ρCfU2 (6.101)

where Cf denotes a bed-friction coefficient. If the bed were rigid and the flow wererough, Cf would vary only weakly with the flow, according to the logarithmic lawembodied in Eq. (6.12). As a result, the relation between τb and U is approximately par-abolic.





The effect of bedforms is to increase the bed shear stress to values often well abovethat associated with the skin friction of a rough bed alone. In Fig. 6.15, a plot of τb

versus U is given for the case of an erodible bed. At extremely low values of U, the parabolic law is followed. As ripples, then dunes are formed, the bed shear stress risesto a maximum value. At this maximum value, the value of Cf is seen to be as much as five times the value without dunes. It is clear that dunes play an important roleregarding bed resistance. The increased resistance results from form drag in the lee ofthe dune.

As the flow velocity increases further, dune wavelength gradually increases and duneheight diminishes, leading to a gradual reduction in resistance. At some point, the dunesare washed out, and the parabolic law is again satisfied. At even higher velocities, theform drag associated with antidunes appears; it is usually not as pronounced as that ofdunes.

6.7.2 Dimensionless Characterization of Bedform Regime

Based on the above arguments, it is possible to identify at least three parameters govern-ing bedforms at equilibrium flow. These are Shields stress τ*, shear Reynolds number Re u*D/ν, and Fr. A characteristic feature of sediment transport is the proliferation ofdimensionless parameters. This feature notwithstanding, Parker and Anderson (1977)showed that equilibrium relations of sediment transport for uniform material in a straightchannel can be expressed with just two dimensionless hydraulic parameters, along with aparticle Re (e.g., Rep or Re) and a measure of the denstiy difference (e.g., R).

FIGURE 6.15 Variations of bed shear stress τb and Darcy-Weisbach friction fac-tor f with mean velocity U in flow over a fine sand bed. (Raudkivi, 1990)




6.40 Chapter Six

FIGURE 6.16 Bedform predictor proposed by Simons andRichardson (1966).

In the case of bedforms, then, the following classification can be proposed:

bedform type function (π1, π2; Rep, R) (6.102)

Here, any independent pair of hydraulic variables π1, π2 applicable to the problem can bespecified because any one pair can be transformed into any other independent pair. Forexample, the pair τ* and Fr might be used or, alternatively, S and H/D.

One popular discriminator of bedform type is not expressed in dimensionless form atall. It is the diagram proposed by Simons and Richardson (1966), (Fig. 6.16). In the dia-gram, regimes for ripples and dunes, transition to the upper-regime plane bed, and upper-regime plane bed and antidunes are shown. The two hydraulic parameters are abbreviatedto a single one, stream power τbU, and the particle Re is replaced by grain size D. The dia-gram is applicable only for sand-bed streams of relatively small scale.

Liu’s discriminator (1957), shown in Fig. 6.17, uses one dimensionless hydraulic para-





FIGURE 6.17 Criteria for bedforms proposed by Liu (1957)

FIGURE 6.18 Bedform classification. (after Chabert and Chauvin,1963)




6.42 Chapter Six

FIGURE 6.19a Bed-form chart for Rg = 4.5–10 (D50 = 0.12 mm–0.200 mm)

FIGURE 6.19b Bed-form chart for Rg = 4.5–10 (D50 = 0.12mm–0.200 mm)

meter u*/vs (a surrogate for τ*) and the particle Rep. The diagram is of interest because itcovers sizes much coarser than those of Simons and Richardson. It is seen that the vari-ous regimes become compressed as grain size increases. In the case of extremely coarsematerial, the flow must be supercritical for any motion to occur. As a result, neither rip-ples or dunes are expected.

In fact, dunes can occur over a limited range in the case of coarse material. This isillustrated in Fig. 6.18. The diagram shows that Re must be less than approximately 10(δv

D) for ripples to form. Recalling that

Re u*

νD (τ*)1/2

RgνD D (6.103)

and using a critical value of τ* of approximately 0.03, it is seen from Eq. (6.101) and theconditions R 1.65, ν 0.01 cm2/s that the condition Re 10 corresponds to a value ofD of approximately 0.6 mm.

For coarser grain sizes, the dune regime is preceded by a fairly wide range consistingof a lower-regime flat bed. Many gravel-bed rivers never leave this lower-regime flat bed





FIGURE 6.19d Bed-form chart for Rg = 16–26 (D50 = 0.228 mm–0.45 mm)

FIGURE 6.19e Bed-form chart for Rg = 24–48 (D50 = 0.4mm–0.57 mm)

FIGURE 6.19c Bed-form chart for Rg = 4.5–10 (D50 = 0.15 mm–0.32 mm)




6.44 Chapter Six

region, even at bankfull flow. The diagram in Fig. 6.18 is not suited to the description ofupper-regime flow.

A complete set of diagrams for the case of sand is shown in Fig. 6.19a to f, (Vanoni,1974). The two hydraulic parameters are Fr and H/D; the particle Re used in the plot isequal to Rep/R, and constant R is set at 1.65. Note how the transition to upper regime

FIGURE 6.20 Bedform classification (after van Rijin, 1984)

FIGURE 6.19f Bedform chart. A, B, C, D, E, F (after Vanoni, 1974)





occurs at progressively lower values of Fr for relatively deeper flow (in the sense that H/Dbecomes large).

A bedform classification scheme that includes both the lower and the upper regimewas proposed by Van Rijn (1984). The scheme is based on a dimensionless particle diam-eter D* and the transport-stage parameter T defined, respectively, as

D* D50

Rνg2

1/3 R2/3ep (6.104)

and

T τ*

s

τ*c

τ*c

(6.105)

where τs* is the bed shear stress caused by skin or grain friction, and τc

* is the critical shearstress for motion from the Shields diagram.

Van Rijn (1984) suggested that ripples form when both D* 10 and T 3, as shownin Fig. 6.20. Dunes are present elsewhere when T 15, dunes wash out when 15 T 25, and upper flow regime starts when T 25.

In the lower regime, the geometry of bedforms refers to representative dune height ∆and wavelength ∆ as a function of the average flow depth H, median bed particle diame-ter D50, and other flow parameters such as the transport-stage parameter T, and the grainshear Reynolds number Re. The bedform height and steepness predictors proposed by vanRijn (1984) are

FIGURE 6.21a,b Bedform height and steepness (after van Rijn,1984)




6.46 Chapter Six

H∆

0.11DH

50

0.3(1 e 0.5T)(25 T) (6.106)

and

Λ∆

0.015DH

50

0.3(1 e 0.5T) (25 T) (6.107)

The bedform length obtained from dividing these two equations, Λ 7.3H, is closeto the theoretical value Λ 2πH, derived by Yalin (1964). The agreement with labora-tory data is good, as shown in Fig. 6.21a and b, but both curves tend to underestimate thebedform height and steepness of field data (Julien, 1995; Julien and Klaasen, 1995). Forinstance, lower-regime bedforms are observed in the Mississippi River at values of T wellbeyond 25. Large dunes on alluvial rivers often display small dunes moving along theirstoss face (Amsler and García, 1997; Klaasen et al., 1986), resulting in additional formdrag that is not accounted for in relations derived from laboratory observations. What isneeded is a predictor for bedforms in large alluvial rivers based on field observations,

6.7.3 Effect of Bedforms on River Stage

The presence or absence of bedforms on the bed of a river can lead to some curious effectson a river’s stage. According to a standard Manning-type relation for an nonerodible bed,the following should hold:

U 1n

H2/3S1/2 (6.108)

FIGURE 6.22 Flow velocity versus hydraulic radius forthe Rio Grande (after Nordin, 1964)




Here, the channel is assumed to be wide enough to allow the hydraulic radius to bereplaced with the depth H. According to Eq. (6.108), if the energy slope remains relative-ly constant, depth should increase monotonically with increasing velocity. This wouldindeed be the case for a rigid bed. In a sand-bed stream, however, resistance decreases asU increases over a wide range of conditions.

At equilibrium,

τb ρCfU2 ρgHS (6.109)

This decrease in resistance implies that depth does not increase as rapidly in U as it wouldfor a rigid-bed open channel. In fact, as transition to upper regime is approached, the bed-forms can be wiped out suddenly, resulting in a dramatic decrease in resistance. The resultcan be an actual decrease in depth as velocity increases (Fig. 6.22).

It is often found that the discharge at which the dunes are obliterated is a little belowbankfull in sand-bed streams. As a result, flooding is not as severe as it would be other-wise. The precise point of transition is generally different, depending on whether the dis-charge is increasing or decreasing. This can lead to double-valued stage-discharge rela-tions, (Fig. 6.22).

6.8 EFFECT OF BEDFORMS ON FLOW AND

SEDIMENT TRANSPORT

6.8.1 Form Drag and Skin Friction

As was seen in Sec. 6.7.3, bedforms can have a profound influence on the flow resistanceand thus on the sediment transport in an alluvial channel. To characterize the importanceof bedforms in this regard, it is of value to consider the forces that contribute to the dragforce on the bed.

Consider, for example, the case of normal flow in a wide rectangular channel. In thepresence of bedforms, Eq. (6.1) must be amended to

τb ρgHS (6.110)

where τb is an effective boundary shear stress, where the overbear denotes averaging overthe bedforms and can be defined as the streamwise drag force per unit area, where H nowrepresents the depth averaged over the bedforms.

In most cases of interest, the two major sources of the effective boundary shear stressτb are skin friction, which is associated with the shear stresses, and the form drag, whichis associated with the pressure. That is,

τb τbs τbf (6.111)

where τbs is the shear stress caused by skin friction and τbf is the shear stress caused byform drag.

The important thing to realize is that form drag results from a net pressure distributionover an entire bedform. At any given point along the surface of the bedform, the pressureforce acts normal to the body. For this reason, form drag is ineffective in either movingbedload sediment or entraining sediment into suspension. In the case of dunes in rivers,because the flow usually separates in the lee of the crest, the form drag is often substan-tial. The part of the effective shear stress that governs sediment transport is thus seen to





6.48 Chapter Six

be the skin friction.To render any of the bedload formulas presented in Sec. 6.6.4 valid in the presence of

bedforms, it is necessary to replace the Shields stress τ* by the Shields stress τ*s associat-

ed with skin friction only:

τ*s ρR

τb

gs

D (6.112)

The fact that the form drag needs to be excluded to compute sediment transport doesnot by any means imply that it is unimportant. It is often the dominant source of bound-ary resistance and thus plays a crucial role in determining the depth of flow. This will beconsidered in more detail below.

6.8.2 Shear Stress Partitions

6.8.2.1 Einstein partition. Einstein (1950) was among the first to recognize the neces-sity to distinguish between skin friction and form drag. He proposed the following simplescheme to partition the two. Equation (6.101) is amended to represent an effective bound-ary shear stress averaged over bedforms:

τb ρCfU2 (6.113)

where Cf now represents a resistance coefficient that includes both skin friction and formdrag. For a given flow velocity U, Einstein computed the skin friction as follows:

τbs ρCfsU2 (6.l14)

where Cfs is the frictional resistance coefficient that would result if bedforms were absent.For example, in the case of rough turbulent flow, Eq. (6.15) may be used:

Cfs

1

1n11

Hks

s

2(6.115)

(In fact, Einstein presented a slightly different formula, which allows for turbulent smoothand transitional flow as well.) The parameter Hs denotes the depth that would result in theabsence of bedforms (but with U held constant). This depth is per force less than Hbecause the resistance is less in the absence of bedforms.

The remaining problem is how to calculate Hs. Einstein restricted his arguments to thecase of normal flow. In this case, Eq. (6.15) holds: that is,

τb ρCfU2 ρgHS (6.116a)

and

τbs ρCfsU2 ρgHsS (6.116b)

Now, between Eqs. (6.113) and (6.116b), the following relation is obtained for Hs:

Hs UgS

2

1

1n(11 Hks

s)

2(6.117)

For given values of U, ks, and S (averaged over bedforms), Eq. (6.117) is easily solved




iteratively for Hs. Once Hs is known, it is not difficult to complete the partition. From Eq.(6.109), it follows that

τbf τb τbs. (6.118)

In analogy to Eqs. (6.111), (6.112), and (6.114), the following definitions are made:

τbf ρCffU2 ρgHfS (6.119)

from which it follows that

Cf Cfs Cff (6.120a)

and

H Hs Hf (6.120b)

Here, Cff denotes the resistance coefficient associated with form drag and Hs denotes theextra depth (compared to the case of skin friction alone) that results from form drag.

Up to this point, it is assumed the U, S, and ks are given. If, for example, H also isknown, τb can be calculated from Eq. (6.110). After Hs, Cfs, and τbs are computed fromEqs. (6.113) to (6.115), it is possible to compute τbf, Hf, and Cff from Eqs. (6.116) and(6.118).

6.8.2.2 Example of the Einstein partition. Consider a sand-bed stream at a given crosssection with a slope of 0.0004, a mean depth of 2.9 m, a value of median bed sedimentsize of 0.35 mm, and a discharge per unit width of 4.4 m2/s. Assume that the flow is atnear-normal conditions. Compute values of τbs, τbf, Cfs, Cff, Hs, and Hf.

Solution: U 4.4/2.9 1.52 m/s. An appropriate estimate of ks for a sand-bed stream is

ks 2.5D50 (6.121)

Solving Eq. (6.115) by successive approximation, it is found the Hs 1.047 m. Thefollowing values then hold:

τbs 4.11 newton’s/m2 (τ*s 0.725)

τbf 7.27 newtons/m2 (τ*f 1.283)

τb 11.38 newtons/m2 (τ* 2.008)

Cfs 0.00178

Cff 0.00315

Cf 0.00493 (Cf–1/2 14.5)

Hs 1.047 m

Hf 1.842 m

H 2.9 m

In the above relations,

τ*f ρR

τgbf

D (6.122)

denotes a form Shield stress. In the above case, only some 30% of the total Shields stress(skin form) contributes to moving sediment.





The Einstein method provides a way of partitioning the boundary shear stress if theflow is known. It does not provide a direct means of computing form drag. A method pro-posed by Nelson and Smith (1989) overcomes this difficulty.

6.8.2.3 Nelson-Smith partition. Nelson and Smith considered flow over a dune; theflow is taken to separate in the lee of the dune. On the basis of experimental observations,they use the following relation for form drag:

Df B 12 ρcD∆U2

r (6.123)

Here, Df denotes that portion of the streamwise drag force Dfs that is caused by form drag,B is the channel width, and Ur denotes a reference velocity to be defined below. They eval-uate the drag coefficient cD as

cD 0.21 (6.124)

It follows that

τbf 12 ρcD

U2r

BD·

f (6.125)

The reference velocity Ur is defined to be the mean velocity that would prevail betweenz ks and z ∆ if the bedforms were not there. From the logarithmic profile represent-ed by Eq. (6.7), this is found to be given by

U

τbr

s/ρ κ

1 [ln(30k

∆s

) 1] (6.126)

It is now assumed that a rough logarithmic law with roughness ks prevails from z ks toz D, and a different rough logarithmic law with roughness kc prevails from z D to z H. Here kc represents a composite roughness length, including the effects of both skinfriction and form drag. The two laws are thus

uτ(

bz

s

)

/ρ κ

1 ln30 k

z

s, ks z ∆ (6.127a)

and

κ1

ln30 kz

c, ∆ z H (6.127b)

Nelson and Smith (1989) matched the above two laws at the level z ∆. After somemanipulation, it is found that

τbs

τ

bs

τbf = l

lnn((3300

∆∆//kk

c

s

)2

(6.128)

The partition requires a prior knowledge of total boundary shear stress τb τbs τbf aswell as roughness height ks, dune height ∆, and dune wavelength Λ. Between Eqs. (6.123)and (6.124),

τbs/f τb τbs 12cD Λ

∆κ2 ln30 k

∆s

12τbs (6.129)

This equation can be solved for τbs, and thus τbf. The value of kc is then obtained from Eq.(6.128).

6.8.2.4 Example of the Nelson-Smith Partition. The example is chosen to be rather

u(z)(τbs τbf)/ρ

6.50 Chapter Six





similar to the previous one: H 2.9 m, S 0.0004, ks 2.5 D50, D50 0.35 mm, ∆ 0.4 m, and Λ 15 m. The technique, which requires no iteration, yields the followingresults:

τbs 4.45 newtons/m2 (τ*s 0.785)

τbf 6.93 newtons/m2 (τ*f 1.223)

τb 11.38 newtons/m2 (τ* 2.008)kc 0.0311 mCfs 0.00130Cff 0.00203Cf 0.00333 (C1/2

f 17.3)Hs 1.134 mHf 1.766 mH 2.9 m

In computing friction coefficients, the following relationship was used for the depth-aver-aged flow velocity:

(τbs

U

τbf)/ρ

1 1n

11

Hkc

(6.130)

The Nelson-Smith method does not require the assumption of quasi-normal flow.

6.8.3 Empirical Formulas for Stage-Discharge Relations

To use either the Einstein or Nelson-Smith partitions, it is necessary to know in advancethe total effective boundary shear stress τb. In general, this is not known. As a result, therelations in themselves cannot be used to predict the boundary shear stress (as well as thecontributions from skin friction and form drag), and thus depth H, for a flow of, say, giv-en slope S and discharge per unit width qw.

A number of empirical techniques have been proposed to accomplish this. Only threeare presented here; they are known to perform well for sand-bed streams with dune resistance.

6.8.3.1 Einstein-Barbarossa Method. The method of Einstein and Barbarossa (1952) isapplicable for the case of dune resistance in a sand-bed stream. It assumes an empiricalrelation of the following form:

Cff fnτ*

s35

(6.131)

Here,

τ*s35 ρR

τgb

Ds

35 (6.132)

The Einstein-Barbarossa plot is shown in Fig. 6.23. Note that it implies that Cff declinesfor increasing τ*

s35. That is, the relation applies in the range for which increased intensityof flow causes a decrease in form drag.




In the Einstein-Barbarossa method, Cfs is computed from a relation similar to Eq.(6.113). That relation is used here to illustrate the method, which uses the Einstein parti-tion for skin friction and form drag.

6.8.3.2 Application of the Einstein-Barbarossa Method. The Einstein-Barbarossamethod is now used to synthesize a depth-discharge relation: that is, a relation between Hand water discharge Q is obtained. It is assumed that the river slope S and the sizes D50

and D35 are known. The river is taken to be wide enough so that the hydraulic radius Rh ≅H; otherwise, Rh should be used in place of H. In addition, the cross-sectional shape isknown, allowing for specification of the following geometric relation:

B B(H) (6.133)

(It also is assumed that auxiliary relations for area A, wetted perimeter P and Rh as func-tions of H are known.)

A range of values of Hs is arbitrarily assumed, ranging from an extremely shallowdepth to nearly bankfull depth (recall that Hs H). For each value of Hs, the calculationproceeds as follows:

Hs → Cfs Eq. (6.115)

Cfs, Hs → U Eq. (6.116b)

Hs → τbs → τ*s35 Eq. (6.116b), (6.132)

τ*s35 → Cff Eq. (6.131); use the diagram

Cff, U → Hf Eq. (6.119)

6.52 Chapter Six

FIGURE 6.23 Flow resistance due to bedforms. [after Einstein et al. (1952).]





H Hs Hf Eq. (6.120b)

Q UH B(H) Eq. (6.133)

The result can be plotted in terms of H versus Q.The analysis can be continued for bedload transport rates. That is, the parameter τbs can

be computed from

τ*s ρR

τgb

Ds

50 (6.134)

and this parameter can be substituted into an appropriate bedload transport equation toobtain q. The volume bedload transport rate Qb is then computed as

Qb q·B (6.135)

6.8.3.3 Engelund-Hansen Method. The method of Engelund and Hansen (1967) alsoapplies specifically for sand-bed streams. It is generally more accurate than the method ofEinstein and Barbarossa, to which it is closely allied. The method assumes quasi-uniformmaterial; it is necessary to know only a single grain size D. Roughness height ks is com-puted from Eq. (6.121).

The method uses the Einstein partition. Skin friction is computed using Eq. (6.112).Form drag is computed from the following empirical relation:

τ*s f(τ*) (6.136)

where

τ* ρRτgb

D ; τ*s ρR

τb

gs

D (6.137a, b)

Equation (6.134) is shown graphically in Fig. 6.24. It has two branches, each correspond-ing to lower-regime and upper-regime flows. The two do not meet smoothly, implying thepossibility of a sudden transition. The point of transition is not specified, which suggeststhe possibility of double-valued rating curves. The lower-regime branch of Eq. (6.136) isgiven by

τ*s 0.06 0.4·(τ*)2 (6.138)

The upper branch satisfies the relation

τ∗s τ* (6.139)

over a range; this implies an upper-regime plane bed. For higher values of Shields stress,τ* again exceeds τ*

s implying antidune resistance.

6.8.3.4 Application of the Engelund-Hansen Method. The procedure parallels that ofEinstein-Barbarossa relatively closely. It is assumed that the values of S and D as well asthe cross-sectional geometry are known. Values of Hs are selected, ranging from a low val-ue to near bankfull. The calculation then proceeds as follows:

Hs → Cfs → U Eq. (6.115) and (6.116b)

Hs → τbs → τ*s Eq. (6,116b), (6.137b)




6.54 Chapter Six

FIG

UR

E 6

.24

Rel

atio

n be

twee

n gr

ain

shea

r st

ress

and

tota

l she

ar s

tres

s (a

fter

Eng

elun

d an

d H

anse

n,19

76)




τ*s → τ* Eq. (6.134); use Equation (6.136) or plot

τ* → τb → H Eq. (6.137b) and (6.116a)

Q = UH B(H) Eq. (6.133)

The value of τ*s can then be used to calculate bedload transport rates in a fashion that is

completely analogous to the procedure outlined for the Einstein-Barbarossa method.

6.8.3.5 Brownlie method. There are almost as many empirical resistance predictors forrivers as there are sediment transport relations. A fairly comprehensive summary of theolder methods can be found in ASCE Manual No. 54 (Vanoni, 1975). A recent empiricalmethod offered by Brownlie (1981a) has proved to be relatively accurate. It does notinvolve a decomposition of bed shear stress; instead it gives a direct predictor of depth-discharge relations.

The complete method can be found in Brownlie (1981a), where the relation is presented for the case of lower-regime dune resistance in a sand-bed stream. It takesthe form

DH

5

S0

0.3724(qS)0.6539 S0.09188 σ0.1050g (6.140)

where σg denotes the geometric standard deviation of the bed material, and q denotes adimensionless water discharge per unit width, given by

q Rg

q

Dw

50 D50

(6.141)

For known S, D50, and σg, qw, and thus Q qwB is computed directly as a function ofdepth H.

6.9 SUSPENDED LOAD

6.9.1 Mass Conservation of Suspended Sediment

Suspended sediment differs from bedload sediment in that it can be diffused throughoutthe vertical column of fluid via turbulence. Here, the local mean volume concentration ofsuspended sediment is denoted as c. As long as the suspended sediment under considera-tion is coarse enough not to undergo Brownian motion (i.e., silt or coarser), moleculareffects can be neglected. Suspended particles are transported solely by convective fluxes.

For an arbitrary volume of sediment-water mixture in the water column, the equationof mass balance of suspended sediment can be written in words as

∂∂t

[mass in volume] [net mass inflow rate] (6.142)

Insofar as the choice of volume V is entirely arbitrary, the following sediment conserva-tion equation, averaged over turbulence-induced fluctuations about the mean, can beobtained:





6.56 Chapter Six

∂∂ct

u ∂∂cs

v ∂∂nc (w vs)

∂∂cz

∂∂usc

∂∂vnc

∂w∂

zc

(6.143)

where u, v, and w are the mean flow velocities in the s, n, and z directions, respectively,and the terms uc, vc, and wc are sediment fluxes caused by turbulence, also knownas Reynolds fluxes. The simplest closure assumption for these terms is

uc Dd ∂∂cs

(6.144a)

vc Dd ∂∂nc (6.144b)

and

wc Dd ∂∂cz

(6.144c)

where the kinematic eddy diffusivity Dd is assumed to be a scalar quantity. To solve Eq.(6.143), boundary conditions are needed.

6.9.2 Boundary Conditions

Equation (6.143), when closed with a Fickian assumption, such as Eq. (6.144a, b, and c),represents an advection-diffusion equation for suspended sediment. The condition of van-ishing flux of suspended sediment across (normal to) the water surface defines the upperboundary condition.

If uniform steady flow over a flat (when averaged over bedforms) bed is considered,the surface boundary condition for the net vertical flux of sediment reduces to

Fszz H 0 (6.145)

where

Fsz vsc wc (6.146)

is the net vertical flux of sediment.The boundary condition at the bed differs from the one at the water surface because it

must account for entrainment of sediment into the flow from the bed and for depositionfrom the flow onto the bed. The mean flux of suspended sediment onto the bed is givenby D, where

D vscb (6.147)

denotes the volume rate of deposition of suspended sediment per unit time per unit bedarea. Here cb denotes a near-bed value of c.

The component of the Reynolds flux of suspended sediment near the bed that is direct-ed upward normal to the bed may be termed the rate of erosion, or more accurately,entrainment of bed sediment into suspension per unit bed area per unit time. The entrain-ment rate E is thus given by

E wc (6.148)

where w´ and c´ denote turbulent fluctuations around both the mean vertical fluid veloci-ty and the mean sediment concentration, respectively. The “overbar” denotes averagingover turbulence. The term near bed used to avoid possible singular behavior at the bed





(located at z 0).It is seen from the above equations that the net upward normal flux of suspended sed-

iment at (or rather just above) the bed is given by

Fsznear bed vs(Es cb) (6.149a)

where

Es ≡ vE

s (6.149b)

denotes a dimensionless rate of entrainment of bed sediment into suspension. The requiredbed boundary condition, then, is a specification of Es. Typically, a relation of the follow-ing form is assumed:

Es Es(τbs, other parameters) (6.150a)

where τbs denotes the boundary shear stress caused by skin friction.Furthermore, it is assumed that an equilibrium steady, uniform suspension has been

achieved. It follows that there should be neither net deposition on (Fsz 0) nor erosionfrom (Fsz 0) the bed. That is, Fsz 0, yielding

Es cb (6.150b)

This relation simply states that the entrainment rate equals the deposition rate; thus,there is no net normal flux of suspended sediment at the bed.

6.9.3 Equilibrium Suspension in a Wide Rectangular Channel

Consider normal flow in a wide, rectangular open channel. The bed is assumed to be

FIGURE 6.25 Definition diagram for sediment entrainment and deposition




6.58 Chapter Six

erodible and has no curvature when averaged over bedforms. The z-coordinate is quasi-vertical, implying low channel slope S. Similarly, the suspension is assumed to be in equi-librium. That is, c is a function of z alone (Fig. 6.25). The flow and suspension are uni-form in s and n and steady in time; thus, Eq. (6.143) reduces to

wc vsc 0 (6.151)

It is appropriate to close this equation with the assumption of an eddy diffusivity, as in Eq.(6.144c); thus, Eq. (6.151) becomes

Dd ddcz vsc 0 (6.152)

Equation (6.152) has a simple physical interpretation. The term vsc represents therate of sedimentation of suspended sediment under the influence of gravity; it is alwaysdirected downward. If all the sediment is not to settle out, there must be an upward fluxthat balances this term. The upward flux is provided by the effect of turbulence, acting toyield a Reynolds flux. According to Eq. (6.144c), this flux will be directed upward as longas dc/dz 0. It follows that the equilibrium suspended-sediment concentration decreasesfor increasing z: therefore, turbulence diffuses sediment from zones of high concentration(near the bed) to zones of low concentration (near the water surface).

6.9.4 Eddy Diffusivity

Further progress requires an assumption for the kinematic eddy diffusivity Dd. The simpleapproach taken here is that of Rouse (1957), which involves the use of the Prandtl analo-gy. The argument is as follows: Fluid mass, heat, momentum, and so on should all diffuseat the same kinematic rate because of turbulence and thus have the same kinematic eddydiffusivity because each is a property of the fluid particles, and the fluid particles are whatis being transported by Reynolds fluxes.

Although the Prandtl analogy is by no means exact, it has proved to be a reasonableapproximation for many turbulent flows. Its application to sediment is more of a problem.Inertial effects might cause the sediment particles to lag behind the fluid, resulting in alower eddy diffusivity for sediment than for the fluid. Furthermore, the mean fall veloci-ty of sediment grains should reduce their time of residence in any given eddy, again reduc-ing the diffusive effect. If the particles are not too large, however, it may be possible toequate the vertical diffusivity of the sediment with the vertical eddy viscosity (eddy dif-fusivity of momentum) of the fluid as the first approximation. This is done here.

The velocity profile is approximated as logarithmic throughout the depth. To accountfor the possible existence of bedforms, the turbulent rough law embodied in Eq. (6.127b)is used:

uu(z

*

)

1 ln

30 k

zc

(6.153)

Here kc is a composite roughness chosen to include the effect of bedforms, as outlined inSec. 6.8.2.3. Furthermore, according to Eq. (6.2), the bed shear stress is given by

ρu2* ≡ τb (6.154)

where b is chosen to be close to the bed: i.e.,





Hb

« 1 (6.155)

Now the kinematic eddy viscosity Dd is defined as

τ ρ Dd dduz

(6.156)

where the distribution of fluid shear stresses τ is given by

τ τb

1 H

z

(6.157)

From the above equations, it is quickly found that

Dd κu*z1 H

z

(6.158)

where κ 0.4 is Von Karman’s constant.

The above relation is the Rousean relation for the vertical kinematic eddy viscosity.The form predicted is parabolic in shape. Although strictly applying to the turbulent dif-fusion of fluid momentum, it is equated to the eddy diffusivity of suspended sedimentmass below. If Dd is averaged in the vertical, the following result is obtained:

Dd κ6 u*H 0.0667 u*H (6.159)

FIGURE 6.26 Vertical suspended sediment distribution (after Vanoni, 1961).




6.60 Chapter Six

This relation is useful to estimate the longitudinal dispersion of fine-grained sedimentin rivers and streams.

6.9.5 Rousean Distribution of Suspended Sediment

The nominal “near bed” elevation in applying the bottom boundary condition is taken tobe z b, where b is a distance taken to be extremely close to the bed: i.e., satisfying con-dition Eq. (6.155). In the Rousean analysis, this value cannot be taken as z 0 becauseEq. (6.153) is singular there.

Equation (6.158) is now substituted into Eq. (6.152), which is then integrated from thenominal bed level to distance z above the bed in z. The resulting form can be cast as

z

b

dcc Z

z

b

z(H

Hdz

z) ln

H

zz

Z

z

b(6.160)

where Z denotes the Rouse number, a dimensionless number given by

Z κvus

* (6.161)

Further reduction yields the following profile:

c cb

((HH

bz))//zb

Z

(6.162)

Some sample profiles of suspended sediment plotted in Rousean form are provided in Fig. 6.26.

Note that from Eq. (6.150b), cb is equal to the dimensionless sediment entrainment rateEs in the case of the present equilibrium suspension. This provides an empirical means toevaluate Es as a function of τbs and other parameters, as will be shown.

6.9.6 Vertically Averaged Concentrations: Suspended Load

Assuming that a value of near-bed elevation b is chosen approximately, Eq. (6.162) can beused to evaluate a depth-averaged volume suspended-sediment concentration C, defined by

C H1

H

b

c(z)dz (6.163)

Using Eq. (6.162), then

C cbI1 (Z,ζb) (6.164a)

where

Il z

ζb

((11

ζζ)

b)//ζζ

b

Z

dζ; ζb Hb

(6.164b, c)

In the above relation, ζ z/H; the integral is evaluated easily by means of numerical tech-niques. Einstein (1950) represented I1 in the form

Il (0.216)1ζbIl (6.165)





FIGURE 6.27a Function I1 in terms of ξb = b/H for values of Z:




6.62 Chapter Six

FIGURE 6.27b Function –I2 in terms of ξb = b/H for values of Z:





where I1 is given in tabular form in the attached Fig. 6.27a.The streamwise suspended load qs was seen in Eq. (6.61a) to be given by the relation

qs H

b

c(z)u(z)dz (6.166)

Reducing with the aid of Eqs. (6.153) and (6.162), we find that

qs 1

cbu*HIl·ln30

Hkc

I2 (6.167)

Here,

I2(Z, ζb) 1

ζb((11

ζζb

))//ζζb

Z

1n(ζ)dζ. (6.168)

The integral I2 is again evaluated easily numerically: Einstein provides the relation

I2 (0.216)1ζbI2 (6.169)

where I2 is given in tabular form in Fig. 6.27b. Brooks (1963) also proposed an interest-ing way to calculate suspended load discharge from velocity and concentration parameters.

It is apparent that further progress is predicated on a method for evaluating the “refer-ence concentration” cb, or equivalently (for the case of equilibrium suspensions) thesediment entrainment rate Es Such a relation is necessary to model transport of suspend-ed sediment (e.g., Celik and Rodi, 1988).

6.9.7 Relation for Sediment Entrainment

A number of relations are available in the literature for estimating the entrainment rate ofsediment into suspension Es (and thus the reference concentration cb for the case of equi-librium). Table 6.5 summarizes all the relations that are available. García and Parker(1991) performed a detailed comparison of eight such relations against data. The relationswere checked against a carefully selected set of data pertaining to equilibrium suspensionsof uniform sand. In this case, it is possible to measure cb directly at some near-bed eleva-tion z b, and to equate the result to Es according to Eq. (6.150b)

The data consisted of some 64 sets from 10 different sources, all pertaining to labora-tory suspensions of uniform sand with a submerged specific gravity R near 1.65.Information about the bedforms was typically not sufficient to allow for a partition ofboundary shear stress in accordance with Nelson and Smith (1989). As a result, the shearstress caused by skin friction alone τbs and the associated shear velocity caused by skinfriction u*s, given by

τbs ρu2*s (6.170)

were computed using Eq. (6.114) and the following relation for ks,

ks 2 D (6.171)

or a similar method.




6.64 Chapter Six

Ein

stei

n(1

950)

Eng

elun

d an

dFr

edso

e (1

976;

1982

)

Smith

and

McL

ean

(197

7)

Itak

ura

and

Kis

hi(1

980)

Van

Rijn

(19

84)

Cel

ik a

nd R

odi

(198

4)

Aki

yam

a an

dFu

kush

ima

(198

6)

Gar

cía

and

Park

er (

1991

)

Zys

erm

an a

ndFr

edso

e (1

994)

b

2Ds

b

2Ds

b

α oτ

* s

τ∗ cDs

k s

α o=

26.

3

b

0.05

H

b

∆ 2b if

∆ bkn

own

else

b

k s*b m

in

0.01

H

b

0.05

H

b

0.05

H

b

0.05

H

b

2Ds

c b 23

.2

q τ*

s* 0.5

c b (1

0.6 λ5 b

1 )3

c b

10.65

γ γo 0T T

c b

k 1k 2

u v* s τΩ *

1

c b

0.01

5D bs

DT1 *0. .5 3

c b

k 0C I

m

Es

0;

Z

Zc

Es

3

10

12

Z101

Z Zc ;Zc

Z

Z

m

Es

0.

3;Z

Z

m

Es

c b

0.33

1(θ'

0.

045)

1.75

1

0 0.3 .43 61

(θ

' –0.

045)

1.75

AZ

5 u

1

0A .3

Z5 u

λ b

0.5

;p1

+

4 0.

25

;β

1.0

T

τ* s τ* c

τ* c

;γo

2.

4·10

3

Ω

kτ* 3 k 4

+

1;

Ao

=k τ*3

k 4

;

k 1

0.00

8;k 2

0.

14;k

3

0.14

3 ; k

4

2.0

D*

D

s g vR 21/

3 ;∆b

is th

e m

ean

dune

hei

ght

Cm

0.

034 1

Hk s 0.

06

gRu2 * HU v sm

;I=

1 0.051

ηη

1

η b η bvs

/0.4

u*dη

;

η

z/H

;ηb

0.

05;k

o

1.13

Z

u v* sR

p0.5 ;

Zc

5;

Zm

13

.2

Zu

u v* ss

Rn p;

u *s

g C0.

'5 U

m;C

'18

log1 32 DR sb

;n

0.

6;

A

1.3

10-7

θ'

R(ug* Ds)

2 s

exp(

A

2 o)

∞ A

oex

p(–

ξ2)d

ξ

β 6π τ* s

0.

06

τ* s

0.06

βp 6π

0.02

7(R

1)

τ* s

TA

BL

E 6

.5E

xist

ing

form

ulas

to e

stim

ate

sedi

men

t ent

rain

men

t or

near

-bed

con

cent

ratio

n un

der

equi

libri

um c

ondi

tions

.

Aut

hor

Form

ula

Para

met

ers

Ref

eren

ce H

eigh

t





The data covered the following ranges:

Es: 0.0002 0.06

u*s/vs: 0.70 7.50

H/D: 240 2400

Rep 3.50 37.00

The range of values of Rep corresponds to a grain size ranging from 0.09 mm to 0.44mm. Except for the relatively small values of H/D, the values cover a range that includestypical field sand-bed streams.

Three of the relations for Es performed particularly well and are presented here. Thefirst is the relation of García and Parker (1991). The reference level is taken to be 5 per-cent of the depth: that is,

Hb

ζb 0.05. (6.172)

FIGURE 6.28 Sediment entrainment function (afterGarcía and Parker, 1991).

Es

Zu




6.66 Chapter Six

The good performance of this relation is not overly surprising because the relation was fit-ted to the data. The relation takes the form

Es (6.173a)

where

A 1.3 107 (6.173b)

and

Zu uv

*

s

s Rep

0.6 (6.173c)

Equation (6.173a) is compared against the data in Fig. 6.28. Predicted values of Es arecompared with observed values in Fig. 6.29.

A second relation that performed well is that of Van Rijn (1984), which takes the form

Es 0.015 Db (τ*

s /τ*c 1)1.5Rep

0.2 (6.174)

where τ*s denotes the Shields stress caused by skin friction, given by Eq. (6.112). For the

purposes of the present comparison, b was again set equal to 5 percent of the depth: i.e.,Eq. (6.172) was used. Van Rijn computed τbs from relations that are similar to Eqs. (6.115)and (6.116b). Van Rijn's relations are

Cfs 1

ln12 Hks

2(6.175a)

AZ5u

1 0A.3

Z5u

FIGURE 6.29 Comparison of predicted and observed near-bed concentration forGarcía-Parker function





FIGURE 6.31 Comparison of predicted and observednear-bed concentration for Smith-McLean function.

FIGURE 6.30 Comparison of predicted and observednear-bed concentration for van Rijn function

where, for uniform material,

ks 3 D (6.175b)

Note that in Eq. (6.175), the total depth H is used, in contrast to Eq. (6.115) where Hs is used.In performing the comparison, García and Parker (1991) estimated τ*

s from a fit to the




Shields curve due to Brownlie (1981a). This fit is given by Eq. (6.44). Predicted andobserved values of Es are presented in Fig. 6.30.

A third relation that performs well is that of Smith and McLean (1977) which can beexpressed as

Es 0.65 (6.176a)

where

γo 0.0024 (6.176b)

The value b at which E is to be evaluated is given by the following relation:

b 26.3(τ*s / τ*

c 1)D ks (6.176c)

where ks denotes the equivalent roughness height for a fixed bed.For the purpose of comparison, García and Parker used Eq. (6.171) to evaluate ks and

used Eq. (6.115) to evaluate τbs. Critical Shields stress was evaluated with Eq. (6.44).Predicted and observed values of Es are shown in Fig. 6.31.

6.9.8 Entrainment Relation for Sediment Mixtures

García and Parker (1991) provided a generalized treatment for the entrainment rate in thecase of mixtures. Let the grain-size range of bed material be divided into N subranges,each with mean size φj on the phi scale and geometric mean diameter Dj 2φj where j 1...N. Let Fj denote the volume fraction of material in the surface layer of the bed in thejth grain range. In analogy to Eq. (6.148), it is assumed that

Ej vsj Fj E(Zuj) (6.177a)

where Ej denotes the volume entrainment rate for the jth subrange and the functional rela-tion between Es and Zuj is given by Eq. (6.173a). The parameter Zuj is specified as

Zuj λm uv

*

sj

s Repj

0.6 DD

5

j

00.2

(6.177b)

In the above relations, vsj denotes the fall velocity of grain size Dj in quiescent water, D50

denotes the median size of the surface material in the bed

Repj (6.177c)

and the parameter λm is given by

m 1 0.288σφ (6.177d)

Here, σφ denotes the arithmetic standard deviation of the bed surface material on the phiscale, given by Eq. (6.30).

The García-Parker relation for mixtures reduces smoothly to the relation for uniformmaterial in the limit as σφ → 0. It was developed and tested with three sets of data fromtwo rivers: the Rio Grande and the Niobrara River. Recently, the García-Parker formula-tion also has been used to interpret observations of sediment entrainment into suspensionby bottom density currents (García and Parker, 1993).

RgDj Djν

γo(τ*s / τ*

c 1)1 γo(τ*

s / τ*c 1)

6.68 Chapter Six





6.9.9 Example of Computation of Sediment Load and Rating Curve. Consider theexample of a stream in Sec. 6.8.2. For this stream, S 0.0004 and D 0.35 mm (uni-form material). At bankfull flow, the stream width is 75 m. For flows below bankfull, thefollowing relation holds:

BB

bf

Q

Q

bf

0.1

where the subscript bf denotes bankfull. Assume that the stream is wide enough to equatethe hydraulic radius Rh with the cross-sectionally averaged depth H.

Compute the depth-discharge relations for flows up to bankfull (lower regime only)using the Engelund-Hansen method. Plot H versus Q. Use the results of the Engelund-Hansen method to compute values of τs

* as well.Use the values of τs

* to compute the bedload discharge Qb qb B using the Ashida-Michiue formulation. For each value of H and U, back-calculate the composite roughnesskc. Then compute the suspended load Qs qs B from the Einstein formulation and therelation for Es by García and Parker. Plot Qb, Qs, and QT Qb Qs as functions of waterdischarge Q.

Solution: In this example, the flow depth, bedload discharge, and suspended load dis-charge are computed as a function of flow discharge for a stream with the following prop-erties:

S = 0.0004

Ds = 0.35 mm = 3.5 10 -4 m

R = 1.65

B = 75 m at bankfull

H = 2.9 m at bankfullThe calculations are performed for flows up to bankfull. For flows below bankfull, the

following relation is used to calculate the stream width:

BB

bf Q

Qbf0.1

UQH

b

Bf

0.1(i)

where the subscript bf indicates bankfull values. Solving for the stream width B,

B BbfUQ

Hbf0.11/0.9

(ii)

The methods used to determine Q, Qb, Qs, and Qbf are described below. A computerprogram can be written, or a spreadsheet can be used, to perform the necessary calcula-tions. All computations and results are summarized in Table 6.6.

6.9.9.1 Depth-discharge calculations. The depth-discharge relation is computed usingthe Engelund-Hansen method. The calculations are performed by assuming a value for Hs

(the flow depth that would be expected in the absence of bedforms), then calculating theactual flow depth (H) and the flow discharge (Q). Hs is varied between 0.22 m and thebankfull value of 2.9 m. The first step in calculating the depth-discharge relation is tocompute the resistance coefficient caused by skin drag (Cfs) from Hs:

Cfs 1

1n11 Hks

s2

(iii)




Hs

Cfs

U*

s*

Flow

W

idth

Disc

harg

eq* b

q bQ

bk c

Z uE s

u *R

ouse

I 12I

2q s

Qs

Qt

(m/s

)D

epth

H(m

)B

(m

)(m

3 /s) Q

No.

Z0.

100.

0031

40.

353

0.06

90.

152

0.22

45.2

83.

520.

0129

63.

41E

-07

0.00

0015

50.

0196

022.

5199

71.

32E

-05

0.02

936

4.76

4548

652

0.04

80.

137.

36#-

093.

33E

-07

0.00

0015

8

0.20

0.00

261

0.54

80.

139

0.44

30.

6453

.54

18.7

80.

2236

25.

89E

-06

0.00

0315

40.

0883

143.

5637

87.

47E

-05

0.05

012.

7923

2451

10.

105

0.26

4.23

E-0

72.

26E

-05

0.00

0338

0

0.30

0.00

236

0.70

60.

208

0.60

80.

8857

.04

35.3

60.

6229

61.

64E

-05

0.00

0936

00.

0782

674.

3647

20.

0002

060.

0586

82.

3840

1690

20.

130.

322.

68E

-06

0.00

0153

0.00

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7

0.40

0.00

220

0.84

40.

277

0.73

71.

0659

.43

53.3

61.

1686

23.

08E

-05

0.00

1829

70.

0628

255.

0399

50.

0004

220.

0646

2.16

5604

409

0.14

0.38

8.26

E-0

60.

0004

910.

0023

207

0.50

0.00

209

0.96

90.

346

0.84

61.

2261

.28

72.5

01.

8380

84.

84E

-05

0.00

2967

40.

0499

015.

6348

30.

0007

370.

0692

32.

0207

3406

0.15

0.39

2.16

E-0

50.

0013

250.

0042

927

0.60

0.00

201

1.08

30.

416

0.94

31.

3662

.80

92.5

82.

6168

16.

89E

-05

0.00

4329

20.

0399

026.

1726

50.

0011

60.

0730

91.

9141

9657

40.

145

0.5

4.04

E-0

50.

0025

380.

0068

674

0.70

0.00

194

1.19

00.

485

1.03

11.

4964

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113.

483.

4944

39.

21E

-05

0.00

5899

90.

0322

456.

6672

20.

0017

030.

0764

11.

8308

9608

50.

175

0.40

59.

64E

-05

0.00

6181

0.01

2081

0

0.80

0.00

188

1.29

10.

554

1.11

11.

6065

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135.

094.

4630

40.

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180.

0076

678

0.02

6344

7.12

756

0.00

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0.07

935

1.76

3054

519

0.18

80.

410.

0001

750.

0114

240.

0190

916

0.90

0.00

184

1.38

70.

623

1.18

71.

7166

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157.

345.

5164

00.

0001

450.

0096

231

0.02

1746

7.55

992

0.00

3176

0.08

21.

7061

7081

50.

195

0.42

0.00

0283

0.01

8721

0.02

8343

8

1.00

0.00

180

1.47

80.

693

1.25

81.

8267

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180.

156.

6493

70.

0001

750.

0117

577

0.01

812

7.96

885

0.00

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0.08

441

1.65

7421

368

0.20

50.

450.

0004

350.

0292

310.

0409

882

1.10

0.00

176

1.56

60.

762

1.32

51.

9167

.94

203.

507.

8576

50.

0002

070.

0140

645

0.01

5228

8.35

780.

0052

10.

0866

31.

6149

2615

10.

208

0.47

0.00

0621

0.04

2174

0.05

6238

6

1.20

0.00

173

1.65

10.

831

1.38

82.

0068

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227.

329.

1375

80.

0002

410.

0165

376

0.01

2897

8.72

944

0.00

6448

0.08

869

1.57

7374

615

0.21

50.

480.

0008

870.

6090

50.

0774

422

1.30

0.00

170

1.73

20.

900

1.45

02.

0969

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251.

5910

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990.

0002

760.

0191

715

0.01

0998

9.08

589

0.00

7839

0.09

062

1.54

3819

540.

225

0.49

0.00

1253

0.86

947

0.10

6118

8

1.40

0.00

167

1.81

10.

970

1.50

82.

1870

.05

276.

2911

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080.

0003

130.

0219

616

0.00

9439

9.42

887

0.00

9385

0.92

431.

5135

5437

40.

230.

510.

0016

650.

1166

710.

1386

321

1.50

0.00

165

1.88

81.

039

1.56

42.

2670

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301.

3713

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370.

0003

520.

0249

034

0.00

8147

9.75

981

0.01

1087

0.09

414

1.48

6041

241

0.23

50.

525

0.00

2177

0.15

3872

0.17

8775

8

1.60

0.00

163

1.96

31.

108

1.61

92.

3471

.24

326.

8314

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650.

0003

930.

0279

932

0.00

7069

10.0

799

0.01

2944

0.95

771.

4608

5992

30.

245

0.53

0.00

2891

0.20

598

0.23

3972

9

1.70

0.00

161

2.03

61.

177

1.67

12.

4171

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352.

6416

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930.

0004

350.

0312

273

0.00

6163

10.3

901

0.01

4957

0.09

731

1.43

7677

192

0.24

0.53

50.

0034

840.

2500

920.

2813

193

1.80

0.00

159

2.10

71.

247

1.72

22.

4972

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378.

7818

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390.

0004

790.

0346

026

0.00

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10.6

913

0.01

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0.09

878

1.41

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513

0.25

0.54

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750.

3235

690.

3581

711

1.90

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2.17

61.

316

1.77

22.

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405.

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370.

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240.

0381

161

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10.9

843

0.01

9441

0.10

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1.39

6282

541

0.26

0.56

0.00

5651

0.41

1357

0.44

9473

1

2.00

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2.24

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385

1.82

02.

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432.

0121

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360.

0005

70.

0417

649

0.00

419

11.2

697

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0.10

155

1.37

7669

964

0.26

50.

570.

0068

960.

5051

550.

5469

200

2.10

0.00

154

2.31

11.

455

1.86

72.

7073

.70

459.

0723

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950.

0006

180.

0455

465

0.00

3711

11.5

480.

0245

160.

1028

51.

3602

3531

40.

260.

570.

0079

740.

5877

210.

6332

673

2.20

0.00

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2.37

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524

1.91

32.

7674

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4025

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670.

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5085

60.

270.

580.

0097

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7248

920.

7743

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2.30

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2.44

01.

593

1.95

82.

8374

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0027

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860.

0007

180.

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987

0.00

294

12.0

854

0.03

0147

0.10

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1.32

8407

943

0.27

0.59

0.01

1335

0.84

4959

0.89

8457

9

2.40

0.00

150

2.50

21.

662

2.00

12.

8974

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541.

8729

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830.

0007

70.

0576

648

0.00

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12.3

453

0.03

3158

0.10

648

1.31

3813

452

0.28

0.59

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1.02

7782

1.08

5447

2

2.41

60.

0015

02.

512

1.67

32.

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6.35

29.5

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0.05

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1103

496

TA

BL

E 6

.6C

ompu

tatio

n of

Tot

al S

edim

ent L

oad.

6.70Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)

Copyright © 2004 The McGraw-Hill Companies. All rights reserved.Any use is subject to the Terms of Use as given at the website.



where is the von Karman constant (0.4) and ks is given by

ks 2.5 Ds 2.5(3.5 104) 8.75 104m (iv)

The depth-averaged flow velocity (U) can be found from Cfs and Hs:

U gC

Hf

s

s

S (v)

The Shields stress caused by skin friction (τs*) is given by

τ*s ρg

τRbs

Ds R

HDsS

s (vi)

According to Engelund-Hansen, the total Shields stress for the lower regime can be foundfrom the following relation:

τ* τ∗

s

0.4

0.06 (vii)

The flow depth can be calculated from the Shields stress as follows:

H τ*R

SDs (viii)

Finally, the discharge can be calculated from the results of Eqs. (v) and (viii):

Q UHB (ix)

where B must be adjusted according to Eq. (ii) for flows less than bankfull. A plot of thedepth-discharge relation is shown in Fig. 6.32.

6.9.9.2 Bedload discharge calculations. The dimensionless bedload transport rate (q*)is found from the Ashida-Michiue formulation:

q* 17(τ*s τ*

c)[(τ∗s)0.5 (τ*

c)0.5] (x)

where τs* is calculated in Eq. (vi) and τc

* is taken to be 0.05. The bedload transport rateper unit width (qb) is given by

qb q* gRDsDs (xi)

Therefore, the bedload transport rate (in m3/s) is given by

Qb qbB (xii)

Again, B must be adjusted according to Eq. (ii) for flows less than bankfull.

6.9.9.3 Sediment load discharge calculations. The Einstein formulation is used tocompute the suspended load transport rate per unit width (qs):

qs κ1

cbu*HI1ln30 Hkc

I2 (xiii)




6.72 Chapter Six

FIG

UR

E 6

.32

Exa

mpl

e of

flo

w d

isch

arge

rat

ing

curv

e





where

u* gHS (xiv)

If the suspension is assumed to be at equilibrium, cb Es. The dimensionless rate ofentrainment (Es) is calculated with the relation of García and Parker (1991):

Es (xv)

where A is equal to 1.3 10-7 and

Zu uv

*

s

s Rep

0.6 (xvi)

u*s gHsS (xvii)

andRep

RgνDs Ds (xviii)

Notice that for the entrainment formulation, the shear velocity associated with skinfriction u*s must be used. The temperature is assumed to be about 20°C; therefore, thekinematic viscosity is about 106 m2/s. An iterative method, or Eq. (6.38), is used to calculate the terminal fall velocity of the sediment particles vs, which is found to be 5.596 102 m/s. The composite roughness (kc) is calculated according to the following relation:

kc 11 H exp

κuU*

(xix)

The parameters I1 and I2 are found by numerical integration of the following equations:

I1 1 b

ζb

Z

1

ζb1

ζζ

Z

dζ (xx)

and

I2 1

ζb

ζb

Z

1

ζb1

ζζ

Z

ln(ζ)dζ (xxi)

where ςb is taken to be 0.05 and

Z κvus

* (xxii)

The numerical integrations can be performed with Numerical Recipes subroutines(Press et al., 1986) or can be obtained from Figs. 6.27a and b. The suspended load trans-port rate per unit width calculated according to Eq. (xiii) is used to compute the suspend-ed load transport rate (in m3/s):

Qs qsB (xxiii)

AZ5u

1 0A.3

Z5u




For flows less than bankfull, B must be adjusted according to Eq. (ii).

6.9.9.4 Determination of bankfull flow discharge (Qbf). The flow discharge at bankfull(Qbf) is determined by assuming that up to bankfull flow, lower regime conditions exist.The bankfull flow depth for this stream is assumed to be 2.9 m. Then, for bankfull flow,the total shear stress τ* is

τ* R

H

D

Ss

1.625.9· 3

·.5

.000

1404 2.01 (xxiv)

From Engelund-Hansen,

τ*s 0.06 0.4(τ*)2 0.06 0.4 · (2.01)2 1.67 (xxv)

Hs τ*

sRS

Ds 2.42 m (xxvi)

Cfs κ1

ln11Hks

s2

01.4

ln11 8.752

.421042

1.5 103 (xxvii)

U gC

Hf

s

s

S 9.81

1 ·. 52 .

421·0

0

.30004

2.51 m/s (xxviii)

and

Qb UHB 2.51 2.9 75 546.35 m3/s (xxix)

A plot of Qb, Qs, and QT Qb Qs as functions of water discharge are shown in Fig.6.33. For flows up to 100 m3/s, the bedload discharge is larger than the suspended loaddischarge. As the flow discharge increases, the suspended load is much larger than thebedload all the way up to bankfull flow conditions.

Also notice that the composite roughness kc increases first with flow discharge for lowflows but, from then on, decreases monotonically as the bedforms begin to be washed outby the flow. For bankfull conditions, the bedforms have a small effect on flow resistancein this particular problem.

6.10 DIMENSIONLESS RELATIONS FOR TOTAL BED

MATERIAL LOAD IN SAND-BED STREAMS

6.10.1 Form of the Relations

In the analysis presented in previous sections, the guiding principle has been the develop-ment of mechanistically accurate models of the bedload and suspended load components

1.67 1.65 (3.5 104)0.0004

6.74 Chapter Six





FIG

UR

E 6

.33

Exa

mpl

e se

dim

ent d

isch

arge

rat

ing

curv

es f

or b

edlo

ad,s

uspe

nded

load

,and

tota

l loa

d




6.76 Chapter Six

of bed material load. The total bed material load is then computed as the sum of the two.That is, where q denotes the volume bedload transport rate per unit width and qS denotesthe volume suspended load transport rate per unit width (bed material only), the total vol-ume transport rate of bed material per unit width is given by

qt q qs (6.178)

Another simpler approach is to ignore the details of the physics of the problem andinstead use empirical techniques, such as regression analysis, to correlate dimensionlessparameters involving qt to dimensionless flow parameters inferred to be important for sed-iment transport. This can be implemented in the strict sense only for equilibrium or qua-si-equilibrium flows: i.e., for near-normal conditions. The resulting relations are no betterthan the choice of dimensionless parameters to be correlated. They also are less versatilethan physically based relations because their application to nonsteady, nonuniform flowfields is not obvious. On the other hand, they have the advantage of being relatively sim-ple to use and of having been calibrated to sets of both laboratory and field data oftendeemed to be trustworthy.

Here, four such relations are presented, those of Engelund and Hansen (1967),Brownlie (1981a), Yang (1973), and Ackers and White (1973). They apply only to sand-bed streams with relatively uniform bed sediment. The first two relations are the mostcomplete because each is presented as a pair of relations for total load and hydraulic resis-tance. The latter two are presented as relations for total load only. In most cases, it will benecessary for the user to specify a relation for hydraulic resistance as well to perform actu-al calculations; the latter relations for load give no guidelines for this.

The importance of using transport and hydraulic resistance relations as pairs cannot beoveremphasized. Consider, for example, the simplest generalization beyond the assump-tion of normal flow: i.e., the case of quasi-steady, gradually varied, one-dimensional flow.The governing equations for a wide rectangular channel can be written as

dds

2Vg2

H

S Sf (6.179a)

UH qw (6.179b)

where the friction slope Sf is given as

Sf ρτg

b

H Cf g

UH

2

(6.180)

A slightly more general form for nonrectangular channels is

dds

12

g

Q

A

2

2 ξb S Sf (6.181a)

UA Q (6.181b)

where A is the channel cross-sectional area and the friction slope Sf is given as

Sf ρgτRb

h Cf g

UR

2

h (6.182)

In the above equations, Rh denotes the hydraulic radius and ξb denotes the water surfaceelevation above the deepest point in the channel.

Note that in the case of normal flow, the momentum equations reduce to Sf S, or τb




ρgHS for the wide rectangular case and τb ρgRhS for the nonrectangular case.However, in the case of gradually varied flow, Sf ≠ S; in this case, the bed slope S cannotbe used as a basis for calculating sediment transport. The appropriate choice is Sf, so thatfrom Eq. (6.182), for example,

τb ρgRhSf (6.183)

For the case of gradually varying flow, then, it should be apparent that the friction slopenecessary to perform sediment transport calculations must be obtained from a predictor ofhydraulic resistance.

A few parameters are introduced here. Let Q denote the total water discharge and Qst

denote the total volume bed material sediment discharge. Furthermore, let Ba denote the“active” width of the river over which bed material is free to move. In general, Ba is usu-ally less than water-surface width B as a result of the common tendency for the banks tobe cohesive, vegetated, or both. Thus, it follows that

Q Bqw (6.184a)

and

Qst Baqt (6.184b)

One dimensionless form for dimensionless total bed material transport is qt*:

q*t (6.185)

where D is a grain size usually equated to D50. Another commonly used measure is con-centration by weight in parts per million, here called Cs, which can be given as

Cs 106 ρQρ

sQρst

sQst (6.186)

6.10.2 Engelund-Hansen Relations

6.10.2.1 Sediment transport. This relation is among the simplest to use for sedimenttransport and also among the most accurate. It was determined for a relatively small set oflaboratory data, but it also performs well as a field predictor. It takes the form

Cf qt* 0.05 (τ*)5/2 (6.187)

where Cf is the total resistance coefficient (skin friction plus form drag) and τ* denotes thetotal (skin friction plus form drag) Shields stress based on the size D50.

6.10.2.2 Hydraulic resistance. The hydraulic resistance relation of Engelund andHansen (1967) has already been introduced; it must be written in several parts. The keyrelation for skin friction is

C1/2fs

gU

RhsS 2.5 · 1n11

Rk

h

s

s (6.188a)

where ks (2 2.5) · D50. Here, Rhs denotes the hydraulic radius caused by skin friction,which often can be approximated by Hs. The relation for form drag can be written in the fol-lowing form:

qtRgD D





τ*s f(τ*) (6.188b)

where for lower regime,

τ*s 0.06 0.4 · (τ*)2 (6.188c)

and for upper regime,

τ* τ* 1

τ*s [0.298 0.702 · (τ*)1.8](1/1.8) τ* 1 (6.188d)

An approximate condition for the transition between lower and upper regime is

τ*s 0.55 (6.188e)

Computational procedure for normal flow. The water discharge Q, slope S, and grainsize D50 must be known. In addition, channel geometry must be known so that B, Ba, A,H, P, and Rh are all known functions of stage (water-surface elevation) ξ. The procedureis best outlined assuming that Rhs is known and that Q is to be calculated, rather than viceverse. For any given value of Rhs (or Hs), U can be computed from Eq. (6.188a). Notingthat τs

* RhsS/(RD50) and τ* RhS/(RD50), τ*, and thus Rh can be computed from Eq.(6.188b-e). The plot of Rh versus ξ is used to determine ξ, which is then used to determineB, Ba, H, A, P, and so on. Discharge Q is then given by Q UBH. In actual implementa-tion, this process is reversed (Q is given and Rhs and so forth are computed). This requiresan iterative technique; Newton-Raphson is not difficult to implement.

Once the calculation of hydraulic resistance is complete, it is possible to proceed to thecomputation of total bed material load Qst. The friction coefficient Cf is given by(gRhS)/U2. Putting the known values of Cf and τ* into (6.187), qt

*, and thus qt can be com-puted. It follows that Qst qtBa.

Computational procedure for gradually varied flow. To implement the method forgradually varied flow, it is necessary to recast the above formulation into an algorithm forfriction slope Sf, which replaces S everywhere in the formulation of Eqs. (6.188a-e). Theformulation is then solved in conjunction with Eqs. (6.179a and b) or Eqs. (6.181a and b)to determine the appropriate backwater curve. Once Cf and τb are known everywhere, thesediment transport rate can be calculated from Eq. (6.187).

6.10.3 Brownlie Relations

6.10.3.1 Sediment transport. The Brownlie relations are based on regressions of morethan 1000 data points pertaining to experimental and field data. For normal or quasi-nor-mal flow, the transport relation takes the form

Cs 7115cf (Fg Fgo)1.978 S 0.6601DR

5

h

00.3301

(6.189a)

where

Fg R

U

gD50 (6.189b)

6.78 Chapter Six





Fgo 4.596(τ*c)0.5293 S0.1045 σg

0.1606 (6.189c)

τ∗c 0.22Y 0.06 · 107.7Y (6.189d)

and

Y Rep0.6 (6.189e)

In Eq. (6.189a), cf 1 for laboratory flumes and 1.268 for field channels. The para-meters τ∗

c and Rep are the ones previously introduced in this chapter.

6.10.3.2 Hydraulic resistance. The Brownlie relations for hydraulic resistance weredetermined by regression from the same set of data used to determine the relation for sed-iment transport. The relation for lower regime flow is

DR

5

h

0 S 0.3724(qS)0.6539 S 0.09188 σg

0.1050 (6.190a)

The corresponding relation for upper regime flow is

DR

5

h

0 S 0.2836(qS)0.6248 S 0.08750 σg

0.08013 (6.190b)


q (6.190c)

The distinction between lower and upper regime is made as follows. For S 0.006, theflow is always assumed to be in upper regime. For S 0.006, the largest value of Fg atwhich lower regime can be maintained is taken to be

Fg 0.8Fg (6.190d)

and the smallest value of Fg for which upper regime can be maintained is taken to be

Fg 1.25F´g (6.190e)


F´g 1.74S1/3 (6.190f)

6.10.3.3 Computational procedure for normal flow. It is necessary to know Q, S, D50,σg, and cross-sectional geometry as a function of stage. The computation is explicit,although trial and error may be required to determine the flow regime. Hydraulic radiusis computed from Eq. (6.190a) or Eq. (6.190b), and the result can be substituted into Eq.(6.189a) to determine the concentration Cs in parts per million by weight. The transportrate Qst is then computed from Eq. (6.186).

6.10.3.4 Computational procedure for gradually varied flow The Brownlie relation isnot presented in a form which obviously allows for extension to gradually varied flow. Themost unambiguous procedure, however, is to replace S with Sf in the resistance relation,and couple it with a backwater calculation on order to determine Sf . The friction slope isthen substituted into Eq. (6.189a) in place of the bed slope in order to determine the sed-iment transport rate.

qwgD50 D50




6.80 Chapter Six

6.10.4 The Ackers–White Relation Like the Brownlie relation for sediment transport,this relation is based on a massive regression. Several years after is was presented, a cor-responding relation for hydraulic resistance was also presented. The relation for hydraulicresistance, however, does not appear to be among the best predictors. As a result, only theload equation is presented here. It takes the form

Cs 106 cs

DR

5

h

0 u

U

*n A

F

a

g

w

r 1m (6.191a)

where

Fgr ; U’* (6.191b,c)

The parameters n, and Aaw are determined as a functions of Dgr, where

Dgr Rep2/3 (6.191d)

in the following fashion. If Dgr 60, then

n 0; m 1.5 (6.191e and f)

and

Aaw 0.17; c 0.025 (6.191g and h)If 1 Dgr 60, then

n 1 0.56log(Dgr); m 9D.6

g

6r

1.34 (6.191i and j)

Aaw 0.

D23

gr 0.14 (6.191k)

log(c) 2.86log(Dgr) [log(Dgr)]2 3.53 (6.191)

Note that all logarithms here are base 10, and u* retains its previously introduced mean-ing as shear velocity.

6.10.5 Yang Relation

This relation also was determined by regression. Its form is

log(Cs) a1 a2 logUvs

S

Uv

c

s

S (6.192a)

where

a1 5.435 0.286 logvsD

50 0.457 log

uv

*

s (6.192b)

U

32 log 10 DR

5

h

0

U*n

U’*1n

RgD50





FIG

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6.82 Chapter Six

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6.84 Chapter Six

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6.86 Chapter Six

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6.88 Chapter Six

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a1 1.799 0.409 logvsD

50 0.314l og

uv

*

s (6.192c)

and Uc denotes a critical flow velocity given by

2.05 if u*D

50

70

Uvs

c

if 1.2

u*D

50 70. (6.192d)

Not that the logarithms are all base 10 and that vS retains its previous meaning as fallvelocity.

6.10.6 Comparison of the Relations Against Data

In the following eight diagrams (Fig. 6.34a-h) taken from Brownlie (1981b) all four rela-tions are compared against first laboratory, then field data. The plots are in terms of theratio of calculated versus observed concentration as a function of observed concentrationCs in parts per million by weight. In the case of a perfect fit, all the data would fall on theline corresponding to a ratio of unity. The middle dotted line on each diagram shows themedian value of this ratio; the upper and lower dotted lines correspond to the 84th per-centile and the 16th percentile. The closer the median value is to unity and the smaller thespread is between the two dotted lines, the better is the predictor.

The Engelund-Hansen relation is seen to be a good predictor of both laboratory andfield data despite its simplicity. The Brownlie relation gives the best fit of both the labo-ratory and field data shown. This is partly to be expected because the relation was deter-mined by regressing against the data shown in the figures. The Ackers-White relation pre-dicts the laboratory data essentially as well as the Brownlie relation does, but its predic-tions of field data are relatively low. The Yang equation does a good job with the labora-tory data but a rather poor job with the field data.

6.11 HYDRAULICS OF RESERVOIR SEDIMENTATION

6.11.1 Introduction

The construction of reservoirs allows for the controlled storage of water.To develop a suc-cessful reservoir, the characteristics of the sites sediment transport must be considered. Asa matter of course, water backed up behind a dam will experience a marked decrease insediment-carrying capacity. As a result, if site characteristics are correct, large quantitiesof sediment will be deposited within the reservoir basin. Over time, the reservoir will, ineffect, fill with sediment, greatly decreasing its storage capacity. In 1988, Morris and Fanpublished an excellent handbook on reservoir sedimentation.

When designing a reservoir, it is important to predict the progress of sedimentation. Inpractice, these predictions are often carried out using empirical and semiempirical meth-ods that have been developed through observation and measurements of operating reser-voirs. Although these methods do provide helpful design information, the drawback is that

2.5

logu*D

50 0.06




they are not firmly rooted in the physics of sediment transport. Instead, they provide a pre-diction based on a synthesis of past observations. As a result, an engineer conducting thesecalculations easily loses touch with the basic mechanisms governing reservoir sedimenta-tion. Sedimentation is treated as a bulk process, and the relative role of bed-load versussuspended-load transport is not always fully understood.

The following exercise presents a view of reservoir sedimentation based on theoreticalrelations. A gorgelike reservoir is considered to allow for a 1-D model (Hotchkiss andParker, 1991). The following conditions are given: the flow per unit width qw 1.427 m2/sis taken as constant. The stream has an initial slope S 0.0003. The sediments meandiameter and fall velocity are Ds 0.3 mm and vs 4.25 cm/s respectively. Suppose areservoir is placed at some point on the river so that the water surface is raised and heldat an elevation equal to 10 m above the elevation of the initial bed at the dam site.Obviously, the dam will generate a backwater effect, which in turn will reduce the flowsability to transport sediment through the reservoir. The quasi-steady state approximationwill be used to develop a model of reservoir sedimentation based on the governing equa-tions of conservation of momentum, bedload and suspended-load relations, and the Exnerequation. The model will be used to predict the level of reservoir sedimentation and deltaprogression for time intervals of 2, 5, 10, 20, and 30 years. First, the model will be runconsidering bed-load transport only; second, suspended load also will be included to helpidentify the relative roles these two forms of transport play.

The flow discharge per unit width qw used herein is equivalent to the “dominant” waterdischarge which, if continued constant for an entire year, would yield the mean annualsediment discharge.

Of course, it is impractical to assume that a model as simple as the one presented herecould replace the empirical methods of predicting reservoir sedimentation. After all, asteady flow, 1-D, constant reservoir-elevation model seriously limits the model’s applica-tion, and transport relations are not easily transposed from site to site. Still, the followingprovides an understanding of the physical mechanisms causing reservoir sedimentation.An ideal reservoir-sedimentation model would be based in sediment transport physicswhile respecting (and matching) the vast quantity of empirical observations available.

6.11.2 Theoretical Considerations

As in any sediment transport study, it is first necessary to identify the appropriate resis-tance and bedload transport relations that hold for the site under consideration. For thismodel, the following relations have been chosen:

q*b 11.2τ*1,51 τ

τ*

*c4.5

(6.193)

and

Cf1/2 8.1k

hs

1/6 u

U

* (6.194)

where τ* is the Shields Stress; τc* is the critical Shields stress, which is taken to have a val-

ue of 0.03; h stands for the flow depth; and ks is the roughness height. Cf is the resistancecoefficient, and qb

* is the Einstein dimensionless bedload transport defined below:

q*b

gRqb

D D (6.195)

where qb is the volumetric bedload transport per unit width having the dimensions of m2/s,

6.90 Chapter Six





where qb is the volumetric bedload transport per unit width having the dimensions of m2/s,R is the submerged specific gravity (taken as 1.65 for quartz), and D is the mean diame-ter of the sediment particles.

The following conservation relation can be used for suspended-load sediment routing:

dU

dhxC

qw ddCx vs(Es roC) (6.196)

where x is the coordinate in the streamwise direction, qw Uh, C is the average volu-metric suspended sediment concentration, and roC cb near-bed sediment concentra-tion. The shape factor ro is given by the approximate relationship (Parker et al., 1987):

ro 1 31.5uv

*

s1.46

(6.197)

Therefore, the suspended load transport (volume per unit width per unit time) through asection can be evaluated as the product of the average sediment concentration and the flowdischarge per unit width:

qs qwC (6.198)

All that remains is to evaluate Es, the sediment-entrainment coefficient. This is accom-plished with the García-Parker relation:

Es (6.199a)

where

A 1.3 107 (6.199b)

and

Zu uv

*

s Rep

0.6 (6.199c)

With the above equations and the assumption of a rectangular cross section (qw Uh),one can calculate the normal flow and equilibrium transport conditions for the river. Thesecalculations, shown next, will serve as the initial conditions for the sedimentation study.

6.11.3 Computation of Normal Flow Conditions

From the Manning-Strickler relation (Eq. 6.194),

qhw U 8.1k

hs

1/6(ghS)1/2

where ks 2.5 Ds and

h 0.6 0.987m

Now, qb can be computed:

τ* RhDS

s

0.19.8675m· 0

·.00.0000303

0.5982

1.427m2/s · (2.5 · 0.0003)1/6

8.19.8 · 0.0003qw (2.5Ds)1/6

8.1gS

AZ5u

1 0

A.3

Z5u




6.92 Chapter Six

Then

qb* 11.2 τ*1.51

ττ

*

*

c4.5

qb 11.2(0.5982)1.51 00.5.093824.5

4.108

qb* qb

*RgDs Ds 4.1081.65 · 9.8m/s2·0.0003 · 0.0003

qb 8.58 105 m2/s

Estimation of C, Es and ro:

u*s τ*s RgD 0.598 · 1.65 · 9.8m/s2·0.0003m 0.0538 m/s

Rep RgD D/υ 9.81m/s2·1.65 · 0.0003m 1 0.0

1000

36mm

2/s

Rep 20.98

Zu uv

*

s

s Rep

0.6 0.

40.52358m

/s1(02

02

.m89

/s)0.6

7.841

ro 1 31.6 uv

*

s1.46

1 31.54.205.053

180m

/2

sm/s1.46

ro 23.33.

Then

Es 3.8 · 103

For equilibrium conditions, entrainment and deposition rates are the same thus, with thehelp of Eq. (6.196),

C Ero

s

3.82

3.31303

1.63 105

and, finally, qs can be computed as

qs UCH qw C 1.427m2/s 1.63 105

qs 2.33 104m2/s

6.11.4 Governing Equations

1.3 107 (7.841)5

1 1.3

0.3107 (7.841)5





Next, it is necessary to identify the governing equations. Raising the water surface througha control structure results in the development of backwater effects. The backwater profilecan be calculated using the standard 1-D St. Venant equation expressed in terms aboveparameters, with U being the flow velocity in x, the streamwise direction. The symbol ηstands for the elevation of the bed above the datum:

∂∂Ut U

∂∂Ux g

∂(η∂

xh)

Cf Uh

2. (6.200)

The backwater change in the water depth will cause a change in the transport of sedi-ment. This phenomenon can be captured using the Exner equation with λp being the poros-ity of the bed sediment (taken at 0.3):

∂∂ηt (1

1λp)

∂∂x

(qb qs) (6.201)

Notice that a time differential appears in both of the above equations; this reflects thefact that both hydraulic and transport conditions change continuously in time. The twoequations are coupled through η, the bed elevation. Of course, a simultaneous solution ofboth equations, including the time derivative, is difficult.

To simplify the model and expedite a solution, the quasi-steady-state approximationcan be used. Not surprisingly, analysis has shown, that the time scale for sedimentologi-cal changes is much larger than that for changes in flow condition. Simply put, if the timechanges of hydraulic conditions are driven by changes in sediment transport, they willoccur slowly. Within an appropriate time step, the flow conditions can be considered to besteady. In this way, it is possible to drop the time differential in the St. Venant equation:

U ddUx

g d(η

d

xh)

Cf Uh

2 (6.202)

Equations (6.201) and (6.202), in conjunction with continuity (qw Uh), provide thetheoretical basis for the following analysis. In the quasi-steady-state analysis, the back-water curve resulting from a forced raise in water elevation is calculated first, (Eq. 6.202).The new water depths for the time step are used to calculate a new bed position (Eq.6.201), and these values are used in the next time step to determine a new backwater pro-file. The procedure repeats for each time step.

6.11.5 Discussion of Method

As discussed in a previous section, initial “normal” flow conditions can be calculatedthrough consideration of the resistance and transport relations. Far away from the dam,where backwater effects are negligible, normal flow and equilibrium transport conditionswill exist.

A numeric scheme and computational grid must be chosen to evaluate the quasi-steady-state governing equations as they relate to reservoir sedimentation. First, it is nec-essary to develop a spatial computational grid. The grid used in this numerical experimentbegins 40 km upstream of the front near the dam face. The length is divided into reachesof 200 m, resulting in 201 nodes to be evaluated. This length allows for initial backwatercomputation to very nearly reach the normal depth at the upstream end.

Using this grid, it is possible to develop a numerical scheme for solving the governingequations. To begin the simulation, a backwater calculation starting at the downstream endof the grid must be conducted. Combining the momentum equation (Eq. 6.202) and water




6.94 Chapter Six

FIGURE 6.35 Water surface elevation before and after reservoir construction

continuity (qw Uh) yields:

ddHx Sf (6.203)

where

H 2qg

2

hw

2 h η (6.204)

and the friction slope Sf is given by

Sf Cg

f

hq3

w2

(6.205)

If the value of h (and thus H) is known at node i 1, its value at node i (upstream) canbe calculated using the following finite difference scheme:

Hi Hi1 12(Sf,i 1 Sf,i)∆x (6.206)

The above expression can be expanded and written as a function of hi:

D(hi) 2qg

w

h

2

i2 hi ηi Hi 1

12Sf,i 1x

12 xCf g

qhw2

i3 0 (6.207)

Now, a Newton-Raphson method can be used to evaluate hi. In this method, an arbi-trary guess at hi can be refined by hi using the expression

∆hi DD

´((hh

i

i

)) (6.208)





FIGURE 6.36 Development of delta for bedload only.

FIGURE 6.37 Development of delta for both bedload and suspended load




6.96 Chapter Six

FIGURE 6.38a Delta location and height after 30 years; (A) bedload only.

where

D′(hi) ddDhi 1 Fr,i

2 32 x

Sh

f

i

,i (6.209)

The computation begins at the downstream end of the problem, where the initial bedelevation must be specified according to the normal flow conditions that existed beforeraising the water surface by 10 m. The first jump is ∆x 200 m, and each subsequentjump is from node, to node with ∆x 200 m. Once the computation has progressed to thefinal node, the hydraulic conditions for the initial bed condition are known. Variables suchas U, qb , and qs can be calculated easily for each node once the water depth is known.

Knowing the hydraulic conditions, the next necessary step is to evaluate the corre-sponding change in bed elevation. This is accomplished with the Exner equation writtenin the backward finite difference form

ηi,j1 ηi,j ∆x(1∆

tλp)

[(qb,i1 qs,i1) (qb,i qs,i)] (6.210)

where j is the current time step, j 1 is the next time step, and i 1 is the node imme-diately upstream of the node being calculated. The calculation proceeds in the down-stream direction. For the first node, the same technique is used, and the initial “normal”bedload and suspended-load transport “feed rate” are used for the upstream values. It iscrucial to choose a time step that upholds the assumptions inherent to the quasi-steady-state approximation. Here, for bedload transport only, a time step of 0.01 year (3.65days) is used, and for runs with both suspended and bedload transport, a time step of0.001 year (365 days) is used. These time steps are small enough to maintain theoreti-cal integrity and numeric stability. The calculated elevations are fed into the next timestep for the adjustment of hydraulic conditions. The above procedure is continued foreach time step.

To complete the numerous computations necessary for this procedure, a computer pro-





FIGURE 6.39 Turbidity current flowing into a laboratory reservoir. (Bell, 1942)

FIGURE 6.38b Delta location and height after 30 years.

gram must be written to facilitate the numerical computations.

6.11.6 Results

The initial conditions for the water surface profile and bed elevation are shown in Fig.6.35. Changes in river profile with time under the consideration of bedload transport onlyare shown in Fig. 6.36. The initial condition and the conditions after 2, 5, 10, 20, and 30years are plotted. Figure 6.37 presents the same data, taking into account both bed andsuspended sediment transport. Not surprisingly, the delta formation is accelerated consid-erably when total load (bed and suspended) is considered.

Both the heights and lengths of deltas are greater for total load calculations for all timesteps. Of course, varying delta formations result in a variation in backwater effects. Whenboth bedload and suspended-load are considered, the backwater effects are more dramatic.

The delta formed after 30 years are shown in Fig. 6.38A and B. After 30 years, the totalload condition produces a delta reaching a length of approximately 36 km. Considering only




6.98 Chapter Six

bedload results in a prediction of a delta only 28.2 km long. Holding the downstream eleva-tion constant results in a considerable backwater effect driven by sedimentation. Althoughthis elevation assumption is not wholly realistic, the results serve to illustrate the threat offlooding associated with reservoir sedimentation. As the reservoir “silts” in, it will be unableto hold the same amount of water without producing a commensurate increase in water stage.

If one is interested in estimating the amount of time necessary to fill a reservoir, it ispossible to simply divide the reservoirs “filling” volume per unit width by the normal sed-iment inflow at the upstream end. To determine the “filling” volume, it is necessary to con-sider the initial bed condition and the full bed conditions. The filling volume per unit widthis estimated at 360,000 m3/m. Assuming bedload only and dividing the filling volume bythe normal bedload inflow results in an approximate filling time of approximately 130years. If total load is considered, an approximate filling time of 35 years is determined. Thisagrees well with the results of the model; Fig. 6.38b shows that the total-load model pre-dicts that the reservoir will be approximately “full” around 40 years. Neither the above cal-culation nor the developed computer model considers the effect of sediment compaction,which may play an important role in increasing the time required to fill a reservoir.

In general, the above results clearly indicate that suspended load plays a major role inreservoir sedimentation. Not considering suspended-load results in a considerable under-estimation of the progress and effects of reservoir sedimentation. If the suspended load ofthe incoming flows is high, plunging may occur and turbidity currents will develop.Turbidity flows can transport fine-grained sediment for long distances, hence having aprofound effect on reservoir sedimentation and water quality.

6.12 HYDRAULICS OF TURBIDITY CURRENTS

6.12.1 Introduction

Turbidity currents are currents of water laden with sediment that move downslope in oth-erwise still bodies of water. Consider the situation illustration in Fig. 6.39. After plunging,a turbidity current moves along the bed of a laboratory reservoir (Bell, 1942). It is seenthat when the flow goes from the sloping portion onto the flat portion, there is a two-foldincrease in current thickness, indicating a change in flow regime through a hydraulicjump. There are a number of field situations where a similar slope-induced hydraulic jumpcan take place (García, 1993, García, and Parker, 1989).An important engineering aspect of turbidity currents concerns the impact these flows

have on the water quality and sedimentation in lakes and reservoirs. Turbidity flows wereobserved in lakes and man-made reservoirs long before their occurrence in the oceanbecame apparent. This situation usually occurs during flood periods, when rivers carry alarge amount of sediment in suspension. In China, where the suspended load in mostrivers is extremely large, the venting of turbidity currents through bottom outlets to reducethe siltation of reservoirs has become common practice. Even though the bed slopes oflakes and reservoirs are orders of magnitude smaller than those in the ocean, turbidity cur-rents are still capable of traveling long distances without losing their identities: e.g., morethan 100 km in Lake Mead. An excellent account of numerical methods to model turbid-ity currents in reservoirs can be found in Sloff (1997).

The ability of turbidity currents to transport sediment also has been put to use for thedisposal of mining tailings (Normark and Dickson, 1976) and ash from power station boil-ers. Environmental concern has reduced waste disposal into lakes, but in the ocean, the





FIG

UR

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.40

Tur

bidi

ty c

urre

nt f

low

ing

dow

nslo

pe th

roug

h a

quie

scen

t bod

y of

wat

er. (

afte

r G

arcí

a,19

94)




6.100 Chapter Six

dumping of mining tailing continues (Hay, 1987a, 1987b).

6.12.2 Governing Equations

A detailed derivation of the governing equations for two-dimensional turbidity currentscan be found in Parker et al. (1986). Here, the equations of motion are presented in layer-averaged form. The situation described in Fig. 6.40 is considered. A steady, continuousturbidity current is flowing downslope through a quiescent body of water, which isassumed to be infinitely deep and unstratified except for the turbidity current itself. Thecross section is taken to be rectangular, with a width many times longer than the under-flow thickness; therefore, variation in the lateral direction can be neglected. The bed hasa constant small slope S and is covered with uniform sediment of geometric mean diam-eter Dsg and fall velocity vs; the x coordinate is directed downslope tangential to the bed,and the z coordinate is directed upward normal to the bed. The submerged specific grav-ity of the sediment is denoted by R (ρs/ρ 1), where ρs is the density of the sedimentand ρ is the density of the clear water. Local mean downstream-flow velocity and volu-metric sediment concentration are denoted as u and c, respectively. The suspension isdilute, hence c « 1 and Rc « 1 are assumed to hold everywhere. The parameters u and care assumed to maintain similar profiles as the current develops in the downslope direc-tion. The layer-averaged current velocity U and volumetric concentration C and the layerthickness h are defined via a set of moments (Parker et al., 1986):

Uh ∞

0

udz (6.211a)

U2h ∞

0

u2dz (6.211b)

UCh ∞

0

ucdz (6.211c)

The equation of fluid mass balance integrates in the upward normal direction to yield

ddUxh

ewU (6.212)

where ew is the coefficient of entrainment of water from the quiescent water above the cur-rent. The equation of sediment conservation takes the layer-averaged form

dU

dCx

H vs(Es cb) (6.213)

where cb is the near-bed concentration of suspended sediment evaluated at z 0.05h andEs is a dimensionless coefficient of bed sediment entrainment into suspension. The inte-gral momentum balance equation takes the form

dU

dx2h gRChS

12gR d

dx

(Ch2) u2* (6.214)

where u* denotes the bed-shear velocity. The equations of sediment mass, fluid mass, andflow momentum balance must be closed appropriately with algebraic laws for ew, u*, Es,and cb. The water entrainment coefficient ew is known to be a function of the bulkRichardson number (Ri), which can be defined as

Ri gR

UC2

h (6.215)





FIGURE 6.41 Water entrainment coefficient as a function of Richardson number(After Parker et al., 1987)




6.102 Chapter Six

FIG

UR

E 6

.42

Plot

of

bed

fric

tion

coef

fici

ent c

Dve

rsus

Rey

nold

s nu

mbe

r. (a

fter

Par

ker

et a

l.,19

87)





FIGURE 6.43 Plot of shape factor ro versus µ u*/Vs. (afterParker et al., 1987)

FIGURE 6.44 Plot of the sediment entrainment coefficient Esfor both open-channel suspensions and density currents. (afterGarcía and Parker, 1993)




6.104 Chapter Six

FIGURE 6.45 Plot of plunging flow depth versus (q2/g)1/3, includ-ing field and laboratory data García (1996)

and is equal to one over the square of the densimetric Fr U/(gRCh)1/2, often used instratified flow studies. A useful equation for the water entrainment coefficient plotted inFig. 6.41 is the following (Parker et al., 1987):

ew (1 07.1087R5i2.4)0.5 (6.216)

It is customary to take the bed shear stress to be proportional to the square of the flowvelocity so that u*

2 CDU2, where CD is a bed friction coefficient. Values of CD for tur-bidity currents have been found to vary between 0.002 and 0.05, as shown in Fig. 6.42(Parker et al., 1987). The near-bed concentration cb can be related to the layer-averagedconcentration C by a shape factor ro cb/C, which is approximately equal to 2 for sedi-ment-laden underflows, as shown in Fig. 6.43 (Parker et al., 1987). The sediment entrain-ment coefficient Es is known to be a function of bed shear stress and sediment character-istics (García and Parker, 1991). The formulation of García and Parker (Eq. 6.173a) isplotted in Fig. 6.44, where data on sediment entrainment by sediment-laden density cur-rents also are included (García and Parker, 1993).

6.12.3 Plunging Flow

The necessary conditions for plunging to occur in a reservoir may vary as a function ofthe physical parameters that produce flow stratification. These parameters are sometimesknown in advance from measurements in the field. Akiyama and Stefan (1984) general-ized several expressions that were derived from laboratory experiments, field measure-





ments, or theoretical analysis as a function of the parameters involved in plunging:

hp F1r2

pg

qR

2w

C1/3(6.217)

where hp flow depth at plunging, qw flow discharge per unit width, and Frp densi-metric Fr at plunging defined by

Frp (gRCUhp)1/2 (6.218)

The value of Frp has been found to range from 0.2 to 0.8 (Morris and Fan, 1998). If thereis not enough suspended sediment, plunging will not occur and a turbidity current will notdevelop. Figure 6.45 shows field and laboratory data for the flow depth at plunging as afunction of the inflow parameters (García, 1996).

6.12.4 Internal Hydraulic Jump

The bulk Ri, given by Eq. (6.215), is an important parameter governing the behavior ofstratified slender flows, such as turbidity currents (Turner, 1973). This parameter has a crit-ical value Ric near unity so that the range Ri Ric corresponds to a high-velocity super-critical turbid flow regime, and the range Ri Ric corresponds to a low-velocity subcriti-cal turbid flow regime. The change from supercritical flow to subcritical flow is accom-plished via an internal hydraulic jump, as illustrated in Fig. 6.39. Therein, a turbidity cur-rent undergoes a hydraulic jump induced by a change in bed slope in the proximity of a lab-oratory reservoir. Conservation of momentum gives the following relation (García, 1993):

hh

2

1

12

1 8Ri11 1

(6.219)

which is analogous to Belanger’s equation for open-channel flow hydraulic jumps. For aknown prejump Ri1, Eq. (6.219) gives the ratio of the sequent current thickness h2 to theinitial current thickness h1. The subcritical flow, forced by some type of control acting far-ther downstream, will influence the location of the jump and thus the length of the water-entraining supercritical flow upstream of the jump. In laboratory experiments, the down-stream boundary conditions are usually imposed by the experimenter (e.g., weir, sluicegate, outfall) because of the finite length of experimental facilities. In the ocean or lakes,where a current may travel several hundred kilometers without losing its identity, the con-trol of the flow will operate through deposition of sediment and bed friction.

6.12.5 Application: Turbidity Current in Lake Superior

As an example, the case of turbidity currents produced by the discharge of taconite tail-ings by the Reserve Mining Company into Lake Superior at Silver Bay, Minnesota, is con-sidered. Over a period of 20 years, the man-made turbidity currents formed a delta with asteep front followed by a depositional fan. Normark and Dickson (1976) used field obser-vations to infer that the transition from the delta slope to the fan slope took place througha hydraulic jump, whereas Akiyama and Stefan (1985) used numerical modeling to showa clear tendency by the flow to become supercritical shortly after reaching the fan region.However, the lack of knowledge about the role played by the hydraulic jump has made




6.106 Chapter Six

FIG

UR

E 6

.46

Sim

ulat

ion

of tu

rbid

ity c

urre

nt u

nder

goin

g a

hydr

aulic

jum

p in

Lak

e Su

peri

or,

Min

neso

ta. (

afte

r G

arcí

a,19

93)





flow computations along the subcritical region practically impossible. Such computationscan now be simplified through the knowledge gained in laboratory experiments (García,1994). This is illustrated by the following numerical experiment.

The lake bed topography at Silver Bay in Lake Superior is modeled in a one-dimen-sional configuration, as illustrated in Fig. 6.46. The delta slop angle is 17º, and the fanslope is 1.5º. The delta-fan slope transition takes place between 600 m to 900 m from theshore. The equations of motion (6.212), (6.213), and (6.214) are solved using a simplestandard step method (García and Parker, 1986). The water entrainment ew and sedimententrainment Es coefficients are estimated with relationships proposed by Parker et al.(1987) and García and Parker (1991; 1993), respectively. A constant bed friction coeffi-cient CD 0.02 and a shape factor ro 2 are used. Initial flow conditions at the tailingsdischarge point similar to those used by Akiyama and Fukushima (1986) are used: i.e., Uo

0.6 m/s, ho 1 m, Φo 0.1 m2/s, and Rio 0.5. The tailings have a mean particle sizeDsg 40 mm and a submerged specific gravity R 2.1. Particle fall velocity is estimat-ed to be vs 0.14 cm/s. Lateral spreading of the flows is ignored. The computationsmarch downslope starting at the head of the delta, and at approximately 0.6 km from thetailings’ discharge point, the flow starts to slow down because of the slope transition. Ifthe current depth at the end of the fan region could be known, the jump location could bedetermined with a simple “backwater” computation. Because this information is not avail-able, the hydraulic jump is assumed to take place at 0.9 km from the inlet. According tothe laboratory observations, water entrainment from above, as well as bed sedimententrainment into suspension, can be neglected after the jump. Under these assumptions,Eq. (6.213) can be integrated with the help of Eq. (6.212), and an expression for the spa-tial variation of the volumetric layer-averaged sediment concentration C is obtained,

C Cje vs

qro

w

x´ (6.220)

where C is the value of Cj at the hydraulic jump and x′ is distance measured from thejump’s location. Since the flow discharge per unit width qw is constant in the subcriticalflow region (ew 0), Eq. (6.220) can be used to compute the volumetric sediment trans-port rate per unit width CUh, at any location after the jump. The variation in current thick-ness between the jump’s location and a point located 2.2 km from the inlet is shown inFig. 6.46. The profile is obtained by first computing the value of C at 2.2 km with the helpof Eq. (6.220), then by doing a “backwater” computation in an iterative manner until thecomputed current thickness at 0.9 km coincides with the current thickness obtained withthe supercritical flow computation and the hydraulic jump Eq. (6.219). The flow dischargeper unit width computed at the jump’s location is qw 48 m2/s. For such flow discharge,Eq. (6.220) predicts that the turbidity current, after experiencing a hydraulic jump, willtravel approximately 80 km before dying out as a result of deposition of sediment.

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