CHAPTER 2

Sediment Transport and Morphodynamics Marcelo H. García

ASCE Manual 54 , Engineering, prepared , fl uid comes into play, carrying under the leadership of Professor Vito A. Vanoni, has pro- the well up into the column. In both cases, the vided guidance to theoreticians and practitioners’ world driving force for transport is the action of gravity wide on the primary topic of sediment problems involved on the fl uid phase; this force is transmitted to the particles in the development, use, and conservation of water and via . Whether the mode of transport is or sus- land resources. First published in 1975, Manual 54 gives pension, the volume concentration of solids anywhere in the an understanding of the nature and scope of sedimentation water column tends to be rather dilute in . As a result, it problems, of the methods for their investigation, and of prac- is generally possible to treat the two phases separately. tical approaches to their solution. It is essentially a textbook In the geophysical domain, the fi eld is much broader on sedimentation engineering, as its title accurately refl ects. than rivers alone. The same phenomena of and sus- Manual 5 4 was the fi rst and most comprehensive text of its pended load transport occur in a variety of other geophysical kind and has been circulated throughout the world for the contexts. is accomplished in the near- past 30 years as the most complete reference on sedimenta- shore of and by wave action. Turbidity currents tion engineering in the world. It has recently been published act to carry suspended sediment into lakes, reservoirs, and again as the Classic Edition (Vanoni 2006). In the spirit the deep . , debris fl ows and fl ows pro- of its predecessor, this chapter of Manual of Practice 110, vide mass transport mechanisms for the delivery of sediment Sedimentation Engineering, aims at presenting the state of from highlands to lowlands. the art concerning the hydraulics of sediment transport in fl u- The solid phase can vary greatly in size, ranging from vial systems based on the knowledge gained in the last three particles to , , , cobbles, and . decades. A concerted effort is made to relate the mechanics Rock types can include quartz, , , granite, of sediment transport in rivers and by turbidity currents to basalt, and other less common types such as magnetite. The the morphodynamics of and reservoir sedimentation, fl uid phase can, in principle, be almost anything that con- including the formation of fl uvial deltas. stitutes a fl uid. In the geophysical sense, however, the two fl uids of major importance are water and air. The phenomenon of sediment transport can sometimes 2.1 SEDIMENT TRANSPORT MECHANICS be disguised as rather esoteric phenomena. When water is AND RELATED PHENOMENA supercooled, large quantities of particulate frazil ice can form. As the water moves under a frozen ice cover, one has The fi eld of sediment transport might just as well be called the phenomenon of sediment transport in rivers stood on its “transport of granular particles by fl uids.” As such, it embod- head. The frazil ice particles fl oat rather than sink, and thus ies a type of two-phase fl ow, in which one phase is fl uid tend to accumulate on the bottom side of the ice cover rather and the other phase is solid. The prototype for the fi eld is than on the riverbed. Turbulence tends to suspend the par- the . Here, the fl uid phase is river water, and the solid ticles downward rather than upward. phase is sediment grains, e.g., quartz sand. The most com- In the case of a powder , the fl uid phase mon modes of sediment transport in rivers are those of bed is air and the solid phase consists of snow particles. The load and suspended load. In bed load, particles roll, slide, or dominant mode of transport is . These fl ows are saltate over each other, never rising too far above the bed. In close analogues of turbidity currents, insofar as the driving

21

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 2211 11/21/2008/21/2008 5:42:335:42:33 PMPM 22 sediment transport and morphodynamics force for the fl ow is the action of gravity on the solid phase , and alternate bars. The fi rst three of these can have rather than the fl uid phase. That is, if all the particles drop a profound effect on the resistance to fl ow offered by the out of suspension, the fl ow ceases. In the case of sediment riverbed. Thus, they act to control river depth. Riverbanks transport in rivers, it is accurate to say that the fl uid phase themselves can also be considered to be a self-formed mor- drags the solid phase along. In the cases of turbidity currents phological feature, thus specifying the entire container. and powder snow , the solid phase drags the fl uid The container itself can deform in plan. Alternate bars phase along. cause rivers to erode their banks in a rhythmic pattern, thus sand provide an example for which the fl uid allowing the onset of meandering. Fully developed river phase is air, but the dominant mode of transport is saltation meandering implies an intricate balance between sediment rather than suspension. Because air is so much lighter than and . If a is suffi ciently wide, it will water, quartz sand particles saltate in long, high trajectories, braid rather than , dividing into several intertwining relatively unaffected by the direct action of turbulent fl uctua- channels. Braided rivers are an important component of the tions. The dunes themselves are created by the effect of the Earth’s surface. The deposits of ancient braided rivers may fl uid phase acting on the solid phase. They, in turn, affect the contain signifi cant reserves of water and hydrocarbon. fl uid phase by changing the resistance. Rivers create morphological structures on much larger In the limiting case of vanishing solids, the fi eld reduces scales as well. These include , alluvial fans, and del- to pure fl uid mechanics. As a result, sediment transport must tas. Turbidity currents act to create similar structures in the be considered to be a subfi eld of fl uid mechanics. In the lim- oceanic environment. In the coastal environment, the iting case of vanishing fl uid, the problem reduces to that of profi le itself is created by the interaction of water and sedi- the fl ow of a granular substance in a vacuum. The driving ment. On a larger scale, offshore bars, spits, and capes con- force now typically, but not always, becomes gravity. This stitute rhythmic features created by wave--sediment problem, as well, can be treated with the techniques of fl uid interaction. The often created by debris fl ows mechanics, as long as one is willing to move far afi eld of provide another example of a morphological structure cre- traditional Newtonian fl uid mechanics. Martian rock ava- ated by a sediment-bearing fl ow. lanches constitute a geophysical realization of grain fl ows This chapter is an introduction to the mechanics of sedi- in a near vacuum, and it is likely that the fl uid phase plays ment transport and river morphodynamics. Rivers evolve only a subsidiary role in many terrestrial rock avalanches. over time in accordance with the interaction between the fl ow Another example of grain fl ow is a slab avalanche of snow. If and sediment-transport fi elds over an erodible bed (which they attain suffi cient speed, slab avalanches tend to devolve changes the bed) and the changing morphology of the bed into more dilute powder snow avalanches in which the fl uid (which changes the fl ow and sediment-transport fi elds). This phase plays a greater role. co-evolution is termed morphodynamics. Sediment transport Among the more interesting intermediate cases are debris by turbidity currents and the mechanics of lake and reser- fl ows, mud fl ows, and hyperconcentrated fl ows. In all of voir sedimentation are also considered in this chapter. The these cases, the solid and fl uid phases are present in similar approach is intended to be as mechanistic and deductive as quantities. A debris fl ow typically carries a heterogeneous possible so that readers will be able to gain a fi rm founda- mixture of grain sizes ranging from boulders to clay. Mud tion in the mechanics of sediment transport. This should be fl ows and hyperconcentrated fl ows are generally restricted benefi cial both for understanding the rest of the material pre- to fi ner grain sizes. In most cases, it proves useful to think of sented in the manual as well as for sedimentation engineer- such fl ows as consisting of a single phase, the mechanics of ing and teaching purposes. which is highly non-Newtonian. The study of the movement of grains under the infl u- 2.1.1 The Sediment Cycle in the Environment ence of fl uid drag and gravity constitutes a fascinating fi eld in its own right. The subject becomes even more interest- The sediment cycle starts with the process of erosion, ing when one considers the link between sediment transport whereby particles or fragments are weathered from rock and morphology. In the laboratory, the phenomenon can material. Action by water, , and as well as be studied in the context of a variety of containers, such as plant and animal activities, contributes to the erosion of the fl umes and wave tanks, specifi ed by the experimentalist. In earth’s surface. Fluvial sediment is the term used to describe the fi eld, however, the fl uid-sediment mixture constructs its the case where water is the key agent for erosion. Natural, own container: the river. This new degree of freedom opens or geological, erosion takes place slowly, over centuries or up a variety of intriguing possibilities for river and coastal millennia. Erosion that occurs as a result of human activity morphodynamics (Parker and Garcia 2006). may take place much faster. It is important to understand the Consider a river. Depending on the existence or lack role of each when studying sediment transport. thereof of a viscous sublayer and the relative importance of The dynamics of sediment in the environment and its mor- bed load and suspended load, a variety of rhythmic struc- phological consequences are schematized in Fig. 2-1. Any tures can form on the riverbed. These include ripples, dunes, material that can be dislodged is ready to be transported. The

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Fig. 2-1. Sedimentation processes and associated morphological changes in a Watershed (adapted from Dietrich and Gallinatti 1985).

transportation process is initiated on the land surface when erosion, rapid deposition, and uncertainty in fl ow path make raindrops result in sheet erosion. , , , and the prediction of fl ood evolution and extent rather diffi cult rivers then act as conduits for sediment movement. The (NRC 1996). greater the , or rate of fl ow, the higher the capac- ity for sediment transport. Mass sediment transport can 2.1.2 Scope of this Chapter also occur through landslides, debris fl ows, and mudfl ows. Hyperconcentrated fl ows have also a tremendous capacity This chapter presents fundamental aspects of the erosion, to transport vast amounts of sediment as observed after the entrainment into suspension, transport, and deposition of release of large amounts of sediment following the eruption sediment in fl uvial systems. The emphasis is on providing of Mt. St. Helens in Washington State, USA (Chapter 19). an introduction to the fl uid mechanics of sediment trans- The fi nal process in the sediment transport cycle is depo- port in rivers and the morphodynamics of lake and reservoir sition. When there is not enough energy to transport the sedimentation by turbidity currents, with the objective of sediment, it comes to rest. Sinks, or depositional areas, can establishing the background needed for sedimentation engi- be visible as newly deposited material on a fl oodplain, bars neering and management. Emphasis is placed on the trans- and islands in a , and deltas. Considerable deposi- port of noncohesive sediment, where the material involved tion occurs that may not be apparent, as on lake and river is in granular form and ranges in size from fi ne silt to coarse beds. Alluvial fans are depositional environments typically sand. The transport of gravel and sediment mixtures is encountered at the base of a front. Flooding pro- treated in Chapter 3, whereas the transport of fi ne-grained, cesses occurring on alluvial fans are considerably different cohesive sediment is considered in Chapter 4. Fluvial pro- from those occurring along single-thread rivers with well- cesses are addressed in Chapter 6 while engineering aspects defi ned fl oodplains (French 1987; Bridge 2003). Active of are covered in Chapter 16. Sediment

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 2233 11/21/2008/21/2008 5:42:345:42:34 PMPM 24 sediment transport and morphodynamics transport in ice-covered rivers is the subject of Chapter 13. consideration. With the help of the boundary , it

Hyperconcentrated fl ows, including mud fl ows and debris is possible to defi ne the shear velocity u * as fl ows as well as sediment hazards related to fl ows in alluvial τρ/ fans, are treated in Chapter 19. This chapter is intended to u* b (2-2) provide the foundation for the rest of the manual. The shear velocity, and thus the boundary shear stress, provides a direct measure of the fl ow intensity and its abil- 2.2 FLUID MECHANICS AND HYDRAULICS ity to entrain and transport sediment particles. The size of FOR SEDIMENT TRANSPORT the sediment particles on the bottom determines the sur- face roughness, which in turn affects the fl ow velocity dis- In this section, basic fl uid mechanics and hydraulics con- tribution and its sediment transport capacity. Because fl ow cepts needed for the analysis of sedimentation processes are resistance and sediment transport rates are interrelated, it is presented. important to be able to determine the role played by the bot- tom roughness. 2.2.1 Flow Velocity Distribution: Law of the Wall In the case of steady, uniform fl ow the shear stress varies linearly in the vertical direction as shown in Fig. 2-2 and is Consider a steady, turbulent, uniform, open-channel fl ow given by the following expression: having a mean depth H and a mean fl ow velocity U (Fig. 2-2). The channel has a mean width B that is much greater ⎛ z ⎞ than the mean fl ow depth H , and its bottom has a mean slope ττ1 (2-3) b ⎝⎜ ⎠⎟ S and a surface roughness that can be characterized by the H effective height k (Brownlie 1981). For very wide channels s It is well established, both experimentally and from (i.e. B / H 1), the hydraulic radius of the channel, R (cross- h dimensional arguments (Schlichting 1979; Nezu and sectional area over ), can be approximated Rodi 1986) that the fl ow velocity distribution is well by the mean fl ow depth H . When the bottom of the channel represented by: is covered with sediment having a mean size or diameter D,

the roughness height k s will be proportional to this diameter. Due to the weight of the water, the fl ow exerts on the bottom u 1 ⎛ z ⎞ ln⎜ ⎟ (2-4) a tangential force per unit bed area known as the bed shear u κ ⎝ z ⎠ τ * 0 stress b , which in the case of steady, uniform fl ow can be expressed as: Here τ ρ b gHS (2-1) u time-averaged fl ow velocity at a distance z above the bed; where z0 bed roughness length (i.e., distance above the bed ρ where the fl ow velocity goes to zero); and water density and κ g gravitational acceleration. is known as von Karman’s constant and has a value of approximately 0.41 (Nezu and Rodi 1986; Long et al. This equation is simply the one-dimensional momen- 1993). The above law is known as the “law of the wall.” It tum conservation equation for the channel reach under strictly applies only in a relatively thin layer (z /H 0.2) near the bed (Nezu and Nakagawa 1993). It is commonly used as a reasonable approximation throughout most of the fl ow in many streams and rivers. If the bottom boundary is suffi ciently smooth, a condi- tion rarely satisfi ed in rivers, turbulence will be drastically suppressed in an extremely thin layer near the bed, known as the viscous sublayer. In this region, a linear velocity profi le holds (O’Connor 1995):

u u z * (2-5) u* v

Fig. 2-2. Defi nition diagram for open-channel fl ow over a where v is the kinematic of water. This law merges δ sediment bed. with the logarithmic law near z v , where

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 2244 11/21/2008/21/2008 5:42:355:42:35 PMPM fl uid mechanics and hydraulics for sediment transport 25 represents both ranges, as well as the transitional range v (2-6) δ v 11.6 between them, can be written as (Yalin 1992) u* denotes the height of the viscous sublayer. In the loga- u 1 ⎛ z ⎞ ri thmic region, the constant of integration introduced above ln (2-9a) κ ⎝⎜ ⎠⎟ Bs has been evaluated from data to yield u*sk

with B as a function of Re u k / v which can be esti- u 1 ⎛ u z⎞ s * * s ln * 5.5 (2-7) mated with the following empirical fi t κ ⎝⎜ ⎠⎟ u* v ⎡ ()⎤ 242 . 8.5⎡ 2.5 ln ( ) 3⎤ 0.121 ⎣ ln Re* ⎦ / Bs ⎣ Re* ⎦ e (2-9b) Comparing Eqs. (2-7) and (2-4), it follows that z o v 9u * for a hydraulically smooth fl ow. Understanding the physics of the fl ow in the viscous sub- A plot of this function can be seen in Fig. 2-3. layer is of relevance in benthic boundary layer fl ows (e.g., An alternative way of writing Eq. (2-9a) is Boudreau and Jorgensen 2001). For example, sediment oxy- gen demand is affected by viscous effects as well as near-bed u 1 ⎛ z ⎞ turbulence levels, as is shown in Chapter 22 of this manual. ln A (2-9c) κ ⎝⎜ s ⎠⎟ Also, the existence of a viscous sublayer seems to be a nec- u*sk essary condition for the development of ripples in unidirec-

tional fl ows (e.g., Raudkivi 1997; Coleman and Melville It follows then that A s and Bs are related by 1994, 1996). κ Bs Most boundaries in alluvial rivers are hydraulically rough. Aes (2-9d) /δ Let k s denote an effective roughness height. If ks v 1, then no viscous sublayer will exist, because the roughness Another useful fi t to the vertical velocity distribution in elements will protrude through such layer. In this case the open-channel fl ows, which also covers the entire range from corresponding logarithmic velocity profi le is given by hydraulically smooth to hydraulically rough as well as the transition, was proposed by Swamee (1993), u 1 ⎛ z ⎞ 1 ⎛ z ⎞ ln ⎜ ⎟ 8.5 ln ⎜30 ⎟ (2-8) / 03. κκ⎝ ⎠ ⎝ ⎠ ⎧ 10/ 3 ⎡ ⎤ 10 3 ⎫ u* k ssk u ⎪⎛ v ⎞ ⎛ 9()uz/v ⎞ ⎪ ⎨ ⎢κ 1 ln 1 * ⎥ ⎬ ⎝⎜ ⎠⎟ ⎝⎜ / ⎠⎟ uuz**⎪ ⎣⎢ 103.(uk* s v ) ⎦⎥ ⎪ / ⎩ ⎭ It follows that z o ks 30 for a hydraulically rough fl ow. As noted above, the logarithmic velocity distribution often holds (2-10) as a fi rst approximation throughout the fl ow depth in a river. It is by no means exact since wake effects near the free sur- Typically, muddy bottoms as well as beds covered with face can be important (Coleman 1981; Lyn 1986). Sediment- silt and fi ne sand are hydraulically smooth, whereas the pres- induced stratifi cation as well as the presence of bed forms can ence of coarse and gravel leads, in general, to hydrau- also infl uence the fl ow velocity distribution. For many years, lically rough conditions. the effect of suspended sediment was understood to be in a change of von Karman’s constant k (Einstein and Chien 1955; 2.2.2 Flow Velocity Distribution: Velocity-Defect Vanoni 1975). However, there is now conclusive evidence that and Log-Wake Laws von Karman’s constant is not affected by the presence of sus- pended sediment as previously believed, and its clear-water The fl ow velocity distribution given by the law of the wall, value ( k ≈ 0.41) can be considered to be a universal one (Smith Eq. (2-4), requires some knowledge of the bed roughness and McLean 1977; Coleman 1981, 1986; Lyn 1986; Soulsby characteristics. An alternative formulation can be obtained if and Wainright 1987; Wright and Parker 2004a). the fl ow depth H is introduced as the relevant length scale.

As is to be shown, it is not uncommon under fi eld con- Assuming that the maximum fl ow velocity u max takes place ditions to fi nd that the fl ow regime is neither hydraulically at the water surface, z H , Eq. (2-4) can be manipulated to / δ smooth nor hydraulically rough. The conditions k s v 1 obtain the so-called velocity-defect law, also known as the /δ for hydraulically rough fl ow and ks v 1 for hydraulically outer form of the law of the wall (Schlichting 1979) smooth fl ow can be rewritten to indicate that the roughness / , given by u * ks v, should be much larger than 11.6 for turbulent rough fl ow, and much smaller than uu 1 z max ln (2-11) 11.6 for turbulent smooth fl ow. A composite form that κ u* H

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Fig. 2-3. Plot of Bs function in log-law velocity distribution.

A number of researchers have argued that the loga- In this relation W0 is known as the Coles wake parameter, rithmic behavior of the velocity distribution, either in the expressing the strength of the wake function. Through trigo- inner form given by Eq. (2-4) or in the outer form given by nometric substitution, Eq. (2-11) can also be written in log- Eq. (2-11), can be justifi ed only for a restricted region near wake form (Coleman 1981; Coleman and Alonso 1983) . the bed ( z/H 0.2), and that, for z/H 0.2, a correction of the logarithmic function is necessary (Coleman and Alonso 1983; Sarma et al. 1983). uu 1 ⎛ z ⎞ 2W ⎛ π z ⎞ max ln⎜ ⎟ 0 cos2 ⎜ ⎟ (2-13) Nezu and Nakagawa (1993) added a wake function to the u κκ⎝ H ⎠ ⎝ 2H ⎠ standard log law given by Eq. (2-7), calling it the “log-wake * law,” as follows, A procedure to estimate the Coles wake parameter from fl ow velocity measurements, originally proposed by Coleman u 1 ⎛ uz⎞ ⎛ z ⎞ ln* 55 . w⎜ ⎟ (2-12a) (1981), can be found in Julien (1995 p. 103). Nezu and Rodi κν⎝⎜ ⎠⎟ ⎝ ⎠ u* H (1986), in experiments on fl at-bed, smooth-bed, turbulent

fl ows, found W 0 to vary from 0 to 0.253, with a mean value of W ≈ 0.2. This result was confi rmed independently by Lyn / 0 where w ( z H ) is the wake function fi rst proposed by Coles (1991). Coleman (1981) and Parker and Coleman (1986) (1956) for turbulent boundary-layer fl ows, which takes the demonstrated that for the case of sediment-laden fl ows over form fl at beds, W 0 increases with increasing sediment concentra- tion, ranging from 0.191 to 0.861. Lyn (1993) found that for π ⎛ z ⎞ 2W0 2 ⎛ z ⎞ w⎜ ⎟ sin ⎜ ⎟ (2-12b) fl ow over artifi cial bed forms, W 0 ranged from –0.05 to 0.1, ⎝ H ⎠ κ ⎝ 2 H ⎠ and suggested that negative values of W 0 are the result of

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 2266 11/21/2008/21/2008 5:42:365:42:36 PMPM fl uid mechanics and hydraulics for sediment transport 27 strong, favorable pressure gradients. Lyn also found good to estimate grain-induced resistance in gravel-bed streams results in replicating measured velocity profi les over bed (e.g. Bray 1979; Parker 1980). forms with the log-wake law. It can be shown that the logarithmic form of Eqs. (2-8) and Most knowledge of fl ow velocity distribution in turbulent, (2-16) can be approximated by power laws of the Manning- free-surface fl ows stems from laboratory studies (e.g., Nezu Strickler form, as follows: and Rodi 1986; Nelson et al. 1993; Song et al. 1994; Bennett 16/ and Best 1995; 1994; Best et al. 1997; Lemmin and Rolland u 1 ⎛ z ⎞ ⎛ z ⎞ 1997; Muste and Patel 1997; Graf and Cellino 2002). In the ln ⎜30 ⎟ ≅ 934. ⎜ ⎟ (2-17a) u κ ⎝ k ⎠ ⎝ k ⎠ past few years, however, new acoustic technology for fl ow * s s measurement has made possible the observation of velocity

profi les in streams and rivers as well (Kostaschuk et al. 2004; 16/ Dinehart and Burau 2005). With the help of observations U 1 ⎛ H ⎞ ⎛ H ⎞ ln11 ≅ 8.1 (2-17b) made in the , Holmes (2003) has found that κ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ u* kssk the velocity-defect law, Eq. (2-13), works well for fi eld con- ditions and the Coles wake parameter takes values ranging To facilitate their comparison, a plot of Eqs. (2-17a) and from 0.035 to 0.36. In all cases, -like bed forms were (2-17b) is shown in Fig. 2-4. It is similar to the one presented present, suggesting that such features might be responsible by Brownlie (1983). The relative error between the log law for the deviations from the logarithmic velocity distribu- and the power law is less than 4.2% in Eq. (2-17a) and less tion, observed away from the bottom. More fi eld observa- than 3% in the case of Eq. (2-17b). Keulegan (1938) was the tions need to be made to quantify the effect of bed forms on fi rst to point out the equivalence between the log-law and the velocity distribution in alluvial rivers as well as the role the power-law, given by Eq. (2-17b), in the context of open- played by stratifi cation induced by suspended . A channel fl ows. Chen (1991) provides a rigorous discussion recent review of mean fl ow, turbulence and bed form dynam- of logarithmic and power-law velocity distributions, includ- ics in alluvial rivers can be found in Best (2005). ing a comparison of the associated fl ow resistance relations for both hydraulically smooth fl ows and fully rough fl ows. 2.2.3 Relations for Channel Flow Resistance Now, between Eqs. (2-2) and (2-16), a resistance relation can be found for the bed shear stress: Most river fl ows are commonly considered to be hydrauli- cally rough. Neglecting wake effects, Eq. (2-8) can be used to τρ 2 b C f U (2-18) obtain an approximate expression for depth-averaged veloc- ity U that is reasonably accurate for most fl ows. Integrating the mean fl ow velocity distribution given by Eq. (2-8) and where the friction coeffi cient C f is given by dividing by the mean fl ow depth yields 2 ⎡ 1 ⎛ H ⎞ ⎤ H C ⎢ ln 11 ⎥ 1 f κ ⎝⎜ ⎠⎟ (2-19) U ∫ u dz (2-14) ⎣ k s ⎦ H 0

Now by slightly changing the lower limit of integration to avoid the fact that the logarithmic law is singular at z 0, the following result is obtained:

H U 11⎡ ⎛ z ⎞ ⎤ ⎢ ln 8.5⎥ dz (2-15) ∫ κ ⎝⎜ ⎠⎟ u* H ⎣ k s ⎦ k s

or after the integration is performed

U 1 ⎛ H ⎞ 1 ⎛ H ⎞ ln 6 ln 11 (2-16) κκ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ u* k ssk

This relation is known as Keulegan’s resistance law for Fig. 2-4. Comparison of logarithmic laws versus power laws for rough fl ow (Keulegan 1938) and it has been extensively used velocity distribution and fl ow resistance.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 2277 11/21/2008/21/2008 5:42:385:42:38 PMPM 28 sediment transport and morphodynamics If Eq. (2-17b) is used instead of Eq. (2-16), the friction hydraulically rough, fully turbulent fl ows. Extensive tables of coeffi cient takes the form: Manning’s n values for different channel characteristics are given in Chow (1959) and Yen (1991). 2 It is also important to notice that according to Eq. (2-23b), ⎡ 16/ ⎤ ⎛ H ⎞ ⎢81. ⎥ (2-20) Manning’s n is not a dimensionless parameter. Yen (1992, C f ⎝⎜ ⎠⎟ ⎣⎢ k s ⎦⎥ 2002) and Dooge (1992), as well as Mostafa and McDermid (1971), have proposed dimensionally homogeneous forms of Manning’s equation. Such dimensionless equation can be It is important to emphasize that Eq. (2-18) provides a readily obtained from Eqs. (2-23a) and (2-23b) as follows: local point estimate of bed shear stress, while Eq. (2-1) gives a reach-averaged value of the bed shear stress (Yen 2002). 1/6 It is useful to show the relationship between the fric- ⎛ H ⎞ UM gHS tion coeffi cient C and the roughness parameters in open- ⎝⎜ ⎠⎟ (2-24a) f ks channel fl ow relations commonly used in practice. Between Eqs. (2-1) and (2-18), a form of Chezy’s law can be derived (Chow 1959): Where the dimensionless constant M 8.1 in this case and is valid for very wide channels. Different values for M can be found in the literature depending on the Strickler (1923) U C 12/ 12/ z H S (2-21) coeffi cient used in Eq. (2-23b). Yen (1993) reports values of M between 6.71 and 12.82, while Julien (2002) fi ts a value of where the Chezy coeffi cient Cz is given by the relation M 5 to fi eld observations. With the help of Eq. (2-23b), it is possible to defi ne a dimensionless Strickler number

12/ ⎛ g ⎞ ng 1 C Z ⎜ ⎟ (2-22) St . (2-24b) ⎝ C f ⎠ 1/6 012 ks 81.

A specifi c evaluation of Chezy’s coeffi cient can be It follows that the constant M in Eq. (2-24a) is the inverse obtained by substituting Eq. (2-19) into Eq. (2-22). It is seen of the Strickler number St . An alternative way to express the that the coeffi cient is not constant, but varies as the logarithm Strickler number is with Keulegan’s equation and power- of the relative roughness H/k . A logarithmic dependence s law equivalent. Assuming that k s D, the identity given in is typically a weak one, partially justifying the common Eq. (2-17b) can also be used to estimate the Strickler number assumption that Chezy’s coeffi cient in Eq. (2-21) is roughly a constant. By substituting Eq. (2-20) into Eq. (2-21) and 1/6 (HD/ ) Eq. (2-22), Manning’s equation in metric units is obtained St (2-24c) (111/κ)ln( HD / )

1 / / U H 23 S12 n (2-23a) This relation gives values of St close to 0.12, as obtained from Eq. (2-24b) in the range of relative fl ow depth H / D from 10 to 1,000 (Niño 2002). For values H/D lower than about Here Manning’s n is given by 10, a sharp increase of St has been reported (e.g., Limerinos / 1970), due to form resistance added to the grain (skin) fric- k16 n s tion, associated with fl ow separation in the wake of large bed 12/ (2-23b) 8.1 g elements relative to the fl ow depth. For instance, Ayala and Oyarce (1993) calibrated the following relation from fi eld This relation is often called the Manning-Strickler form of data obtained in the Mapocho River in the Chilean side of Manning’s n (Brownlie 1983). It is deceptively simple but it the Andean , for values of H / D lower than 10 and also contains important information. Even for large increases taking D D90 , in roughness height k s , Manning’s n does not change much. The opposite behavior is seen if large values of Manning’s n 040. ⎛ H ⎞ are considered, and the corresponding value of k is estimated St 030. (2-24d) s ⎝⎜ ⎠⎟ with the help of Eq. (2-23b). Often the back-calculated values D90

of k s turn out to be larger than the mean fl ow depth H, sug- gesting that the value of Manning’s n being used is not a real- This implies that M in the relation to estimate the mean istic one. From the analysis above, it should also be apparent fl ow velocity (Eq. 2-24a) would no longer be constant but that Manning’s equation can only be applied to uniform, would change as a function of fl ow depth for H / D 10.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 2288 11/21/2008/21/2008 5:42:395:42:39 PMPM fl uid mechanics and hydraulics for sediment transport 29 There is no accepted standard equation for predicting dimensionless Darcy-Weisbach friction coeffi cient, fl ow velocities in channels with large relative roughnes, i.e. which, for a pipe with diameter D is known to be a func- / where channel bed material is large relative to water depth. tion of the fl ow Reynolds number R e UD v and the / This is typical of mountain streams (Jarrett 1984; Aguirre-Pe relative roughness D ks . and Fuentes 1990). Smart et al. (2002) conducted an analysis of existing fl ow resistance equations which points to the dif- Brownlie (1981) re-examined Nikuradse’s data and pro- fi culties associated with the defi nition of depth and hydraulic posed the friction factor diagram shown in Fig 2-5. The dia- radius when the bed roughness is large relative to the fl ow gram provides the values of the friction factor f, introduced in / depth. They found that the log law, or the equivalent power Eq. (2-26), as a function of the Reynolds number Re UD v / law, is only applicable when the roughness is of suffi ciently and the relative roughness D k s . This diagram is equiva- small scale, and recommended the use of a square-root lent to the well known Moody diagram shown in Fig. C-2 power law to estimate fl ow velocity in the presence of large- of Appendix C, and can be used for sidewall corrections in scale roughness. laboratory experiments (Vanoni and Brooks 1957) as well In the case of sand-bed streams, fl ow resistance is infl u- as for separating total resistance into grain resistance and enced by both grain or skin friction as well as form drag form resistance in alluvial streams with dunes (Brownlie induced by the development of bed forms such as ripples, 1981; Fedele and García 2001). For open-channel fl ow cal- dunes and bars, so any estimate of Manning’s n , or any culations, the pipe diameter D should be replaced by 4R h , other roughness coeffi cient, has to account for the possibil- in which R h is the hydraulic radius. Again, for a very wide ity of different fl ow regimes (i.e., lower and upper regimes). channel, the hydraulic radius can be replaced by the mean Bruschin (1985), Camacho and Yen (1991), Wu and Wang fl ow depth. The sidewall correction procedure is explained (1999), and Hager and Del Giudice (2001) have proposed in detail both in Brownlie (1981) as well as in ASCE Manual equations to estimate Manning’s n for the case of sand-bed 54 (Vanoni 2006) and therefore is not repeated here. rivers. A modifi ed Manning-Strickler formula for fl ow in In the late 1930s, Zegzhda conducted a set of experiments alluvial channels with sand beds has also been advanced by in straight rectangular fl umes of varying roughness, using an Yu and Lim (2003). Flow resistance predictors for sand-bed experimental method (gluing sand to the walls) similar to streams are discussed later in the chapter. the one used by Nikuradse for fl ow in pipes (see Novak and Cabelka 1981, p. 124). Because this work was not published in English, it is not as well known as Nikuradse’s work on 2.2.4 Fixed-Bed (Skin or Grain) Roughness pipes. However, this experimental study was conducted for / It is clear that to use these relations for channel fl ow resis- a set of relative-roughness (R h ks ) values more representative tance, a criterion for evaluating the equivalent roughness of the conditions observed in the fi eld for the case of sand- bed streams with plane beds. In fact, the relation obtained by height k s is necessary. Friction factors for turbulent fl ow in pipes and in fi xed-bed channels have their roots in the classic Zegzhda for fully-rough hydraulic conditions is very similar sand-roughened pipe experiments conducted by Nikuradse to the expression advanced independenly at about the same (1933). He conducted a set of pioneer experiments and pro- time by Keulegan (Eq. 2-16). posed the following criterion. Suppose a rough surface is A fi t to the experimental results of Nikuradse that can be used to estimate the roughness length parameter z in subjected to a fl ow. Then the equivalent roughness height k s 0 of the surface would be equal to the diameter of sand grains Eq. (2-4) as a function of ks was proposed by Christofferson that, when glued uniformly to a completely smooth wall, and Jonsson (1985) and then subjected to the same external conditions, yields the same velocity profi le. Nikuradse used sand glued to the kuk⎡ ⎛ ⎞ ⎤ ν inside of pipes to conduct this evaluation. z ss⎢1 exp * ⎥ (2-27) 0 ⎝⎜ ν ⎠⎟ To analyze the work of Nikuradse, it is convenient to 30 ⎣ 27⎦ 9u* introduce another relation that can be used to estimate mean fl ow velocity in open-channel fl ows, known as the Darcy- Smith (1977) seems to have been the fi rst to plot Weisbach equation Nikuradse’s data in a way useful to estimate the roughness length. Similar empirical relations have been proposed by 88 Fuentes and Carrasquel (1981) and Dade et al. (2001). For U gRh S u* (2-26) / / f f u* ks v 3, the fl ow is hydraulically smooth and z 0 0.11v u* ; / whereas for u * ks v 100 the fl ow is hydraulically rough and In this equation z0 0.033k s . In many interesting sediment transport situa- g gravitational acceleration; tions the fl ow is hydraulically transitional and an equation u* shear velocity; such as Eq. (2-27) has to be used to estimate the roughness Rh is the hydraulic radius (approximately equal to length in Eq. (2-4) associated with grain-induced roughness the fl ow depth H for very wide channels); and f is the (Kamphuis 1974). Typically, and fl at fi ne sands are

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Fig. 2-5. Revised Nikuradse friction factor diagram for fl ow in pipes of diameter D or open-channel / fl ows with hydraulic radius Rh D 4 (after Brownlie, 1981).

hydraulically smooth or transitional, and coarse sands and size larger than D50 is more meaningful to estimate fl ow are hydraulically rough (Soulsby, 1997). It is com- resistance because of the dominant effect of large sediment α mon practice to treat all fl ows over sands as being hydro- particles. The range of s values and the diverse representa- dynamically rough since this simplifi es the analysis. This tive sediment size used for D x indicate that further research simplifying approximation makes less than 10% error in the on this concept is necessary.

estimation of the shear velocity u *, for all values u* above In a study of fl ow resistance associated with rip-rapped the threshold of motion (see Section 2.4.2) of grains larger surfaces, Maynord (1991) reviewed a number of formula- than 60 μm . tions commonly used to estimate the Darcy-Weisbach fric- Although it is clear that the sediment size distribution tion coeffi cient and found that a power-law equation can be in most rivers is not as uniform as the material used in his used for most riprap (i.e., fi xed-bed) problems in very wide experiments, Nikuradse’s concept of grain-induced rough- open-channel fl ows, as follows: ness for pipe fl ows has been extended to estimate friction 1/2 1/6 factors in streams and rivers as well (Yen 1992). Nikuradse’s ⎛ 8 ⎞ ⎛ H ⎞ ⎜ ⎟ 689. ⎜ ⎟ (2-29) equivalent sand-grain roughness, k s , is commonly taken to be ⎝ ⎠ ⎝ ⎠ f D50 proportional to a representative sediment size D x , k ssα D x (2-28) Notice the similarity with the power-law equations for fl ow resistance presented earlier. Maynord (1991) also found α Suggested values of s which have appeared in the lit- a logarithmic expression for fl ow resistance, based on his own erature are listed in 2-1, originally compiled by Yen experiments as well as on data from other sources, given by (1992; 2002) and updated for this manual. Different sediment sizes have been suggested for D in Eq. (2-28). Statistically, 1/2 x ⎛ 8 ⎞ ⎛ H ⎞ D (the for which 50% of the bed material is 392..log 686 (2-30) 50 ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ fi ner) is most readily available. Physically, a representative f D50

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 3300 11/21/2008/21/2008 5:42:415:42:41 PMPM fl uid mechanics and hydraulics for sediment transport 31 Table 2-1 Ratio of Nikuradse Equivalent Roughness Size and Sediment Size for Rivers

Investigator Measure of sediment size, D α k /D x s s x

Ackers andWhite (1973) D35 1.23

Aguirre-Pe and Fuentes (1990) D84 1.6

Strickler (1923) D50 3.3

Katul et al (2002) D84 3.5

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

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

/ which applies in the range 2.2 H D50 23. Similar empir- Marchand et al. (1984). At about the same time, Aguirre-Pe ical relations have been advanced by Hey (1979), Thompson and Fuentes (1990) proposed a theory for fl ow resistance in and Campbell (1979), Grifi ths (1981), Pyle and Novak steep, rough streams that takes into account the existence of (1981), and Bathurst (1985). the highly turbulent wake zone near a very rough bed. Their From these equations, it follows that for wide, open model predictions compare favorably against fi eld observa- channel fl ows the Darcy-Weisbach friction coeffi cient and tions by several authors. As shown by Smart (1999; 2002), Manning’s roughness coeffi cient are related by most of the uncertainty when dealing with coarse gravel and cobbles in shallow channels is in the determination of the mean bed location so that the origin of the fl ow veloc- 1/2 /6 ⎛ 8 ⎞ KH1 gHS ity profi le can be ascertained. In relation to the diffi culties ⎜ ⎟ n (2-31) ⎝ f ⎠ ng1/2 U associated with defi ning the mean bed elevation, Nikora et al. (2001) show the importance of spatial averaging when dealing with shallow fl ows over gravel bed streams.

in which Kn is a constant equal to 1 in metric units and In the absence of a logarithmic velocity distribution, Katul equal to 1.486 in English units (Yen 2002). The velocity dis- et al. (2002) developed a velocity distribution equation tribution in high-gradient streams with relatively low values based on mixing-length theory capable of reproducing fl ow / of relative submergence H D50 is no longer logarithmic near resistance characteristics observed in shallow streams with the bed due to the wake effect produced by large roughness large relative roughness. More recently, Buffi ngton et al. elements. Wiberg and Smith (1991) have developed a model (2004) studied the effects of channel type and associated for the velocity fi eld in steep streams with coarse gravel beds hydraulic roughness on salmonid spawning-gravel availabil- that is capable of reproducing the fi eld observations made by ity in mountain catchments .

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 3311 11/21/2008/21/2008 5:42:425:42:42 PMPM 32 sediment transport and morphodynamics 2.2.5 Movable -Bed Roughness Since both estimators depend on the fl ow intensity as given by the bed shear stress, Eqs. (2-32a) and (2-32b) pro- In fl ows over geometrically smooth, fi xed boundaries, the vide an estimate of a variable roughness appropriate for apparent roughness of the bed k can be computed using s movable beds without the presence of bed forms. Nikuradse’s approach, as shown above. However, once the Wiberg and Rubin (1989) evaluated several expressions for transport of bed material has been instigated, the character- characterizing bed roughness produced by a layer of saltating istic grain diameter and the viscous sublayer thickness no sediment grains; they proposed with the help of a formulation longer provide the relevant length scales. The characteris- for the vertical eddy diffusivity coeffi cient (Gelfenbaum and tic length scale in this situation is the thickness of the layer Smith 1986; Long et al. 1993) a formulation which makes where the sediment particles are being transported by the use of a vertical fl ow velocity distribution given by the fol- fl ow, usually referred to as the bed-load layer height (Wiberg lowing expression and Rubin 1989). As the grains start to roll and saltate along the bed, they take momentum away from the mean fl ow via drag, resulting in an increase in fl ow resistance that trans- z u 1()zH/ lates into an increase in bed roughness. uz() * dz κ ∫ ⎡ 23⎤ τ zzHzHzHexp ⎣().()//33 22 .() /⎦ Once the bed shear stress b exceeds the critical shear stress z0 τ for motion c , the roughness length can be estimated with an expression inspired by the work of Owen (1964) for (2-33) wind-induced sediment transport, and fi rst proposed by Smith where (1977) for sediment transport by water currents, z distance from the bed; ()ττ H fl ow depth; and zz α b c κ 000 ()ρ ρ N (2-32a) 0.41 von Karman’s constant. s g Seven upper plane-bed experiments of Guy et al. (1966) were used to obtain best fi t values for the shear velocity u where * and bed roughness length z 0 with the help of Eq. (2-33). The α 0 26.3; analysis of Wiberg and Rubin (1989) shows that the bed z0 P 0.033 k s and k s Nikuradse roughness length; and roughness associated with sediment transport can reach val- ρ s bed sediment density. ues about an order of magnitude larger than the Nikuradse grain roughness in plane-bed fl ows, but this roughness will This approach is particularly suitable for sand-bed riv- in general be signifi cantly smaller than the roughness asso- ers and has been widely used in coastal sedimentation (e.g., ciated with ripples and dunes when they are present on the Smith and McLean 1977). bed surface. The roughness parameter also can be estimated with a At high bed shear stresses and sediment transport inten- scheme proposed by Dietrich and Whiting (1990), sities in sand-bed streams, dunes are washed out and the bed becomes plane. In this regime, sediment is transported near αδ the bed in a layer with a thickness that is much larger than zz01bN 0 (2-32b) the grain size. Collisions between grains are intense in this layer, resulting in a grain fl ow or granular fl uid fl ow. This where regime is known as sheet fl ow and measurements taken by researchers (Wilson 1987, 1989; Nnadi and Wilson 1992) α empirical constant equal to 0.077; 1 have shown that fl ow resistance increases drastically with z 0.033k and k Nikuradse roughness length; and 0 P s s fl ow intensity in this regime. Sumer et al (1996) found δ bedload-layer height, b that fl ow resistance induced by the sheet-fl ow layer can which is computed as be expressed in terms of the ratio of Nikuradse’s equiva- / lent sand roughness to the grain diameter (k s D ). This ratio ⎡τ ⎤ was found to behave differently whether of not the grains 12. D() 1 cosφ b ⎢ τ ⎥ became suspended near the bed. In the absence of sus- δ ⎣ c ⎦ b ⎡τ ⎤ (2-32c) pension mode, k / D depends only on the Shields param- b s 102. ⎢ τ ⎥ eter (τ * ) defi ned by Eq. (2-56). In the suspension mode, ⎣ c ⎦ / τ* k s D depends not only on but also on a dimensionless sediment fall velocity parameter R defi ned by Eq. (2-46b). where f There is also evidence that sediment transport in the sheet- angle of friction, and fl ow layer is infl uenced by the turbulent bursting process D mean diameter of the bed material. (e.g., Sumer et al. 2003).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 3322 11/21/2008/21/2008 5:42:435:42:43 PMPM fl uid mechanics and hydraulics for sediment transport 33 2.2.6 Equivalent Roughness of Bed Forms of fl ow (Nikora et al. 2001) over dunes are available, this method can be used to estimate a spatially-averaged com- As the fl ow intensity increases, bed forms such as ripples and posite roughness k due to the combined effect of both grain dunes can develop (e.g., Raudkivi 1997). In this situation, the c friction and form drag due to bed forms in large sand-bed bed roughness also will be infl uenced by form drag due to rivers. Boundary layer studies have shown that an alternative the presence of bed forms. The fundamental problem is that to Eq. (2-9a) for describing the vertical fl ow velocity distri- the bed form characteristics and, hence, the bed roughness bution in fl ows where the geometry and size of the rough- depend on the main fl ow characteristics (e.g., mean velocity, ness elements is such that skin friction and form drag are depth) and sediment characteristics (e.g., grain size, density). present, is given by the following equation Thus, the hydraulic roughness in the presence of bed forms is a dynamic parameter that depends strongly on fl ow condi- u 1 ⎛ zu ⎞ Δu ⎡uk ⎤ tions as well as on the bed sediment properties. The equiva- ln * A * c (2-35a) κν⎝⎜ ⎠⎟ ⎢ ν ⎥ lent roughness of alluvial beds in the presence of ripples and u**u ⎣ ⎦ dunes was addressed with the Nikuradse hydraulic roughness approach by Brownlie (1981) and van Rijn (1982, 1984c). In In Eq. (2-35a), κ 0.41 and A 5.5 are universal con- Δ / van Rijn’s approach, the height due to grain-induced rough- stants previously introduced, and u u* is a roughness func- ness (Eq. 2-28) was added to an estimate of the equivalent tion which is equal to zero for smooth walls (square brackets / roughness height produced by ripples and dunes obtained indicate functional relationship). When plotting u u * versus / from fi eld and laboratory observations, to obtain a measure of ln ( u* z v), this equation represents a family of parallel lines, the total (grain plus form resistance) effective roughness, each being displaced downwards from the smooth-wall Δ / velocity profi le by an amount u u * (Schlichting 1979). The roughness function for alluvial streams with dunes kDαγΔ. e25Δ/λ Δ / ss90 sf11() 1 (2-34a) is shown in Fig. 2-6. It shows u u* as a function of the / parameter k c u * v for laboratory and fi eld streams with fully- developed dunes (Fedele and García 2001). It is observed that where / for values of the roughness Reynolds number k c u * T v larger α than 100-200, most of the data collapse along a straight line, s 3 (see Eq. 2-28); along the fully-rough hydraulic regime, which is well repre- D90 grain size for which 90% of the bed material is fi ner; sented by the following fi t, γ sf dune shape factor 1. Δ λ Δu ⎛ w k ⎞ and bed form height and length, respectively; and 243.."np , c 324 Δ / λ ⎝⎜ ν ⎠⎟ (2-35b) steepness. u* The effective roughness height was then used to estimate An application of the alluvial roughness function is its the Chezy friction coeffi cient (Eq. 2-22), potential use to assess the effect of temperature changes on fl ow structure and bed morphology. It is observed in Fig. 2-6 1/2 ⎛ g ⎞ ⎛12 R ⎞ that even though the fl ows are under fully-rough hydraulic C⎜ ⎟ 18log⎜ hb ⎟ (2-34b) conditions, temperature variations will affect the viscosity of Z ⎝ ⎠ ⎝ k ⎠ C f s the water and this in turn will cause variations in the rough- ness Reynolds number and the fl ow structure. In this equation, R hb hydraulic radius of the river bed (i.e., substracting streambank effects on fl ow resistance) according to Vanoni-Brooks (1957) (see Vanoni 2006, p. 91). Notice that the Chezy coeffi cient is not dimensionless. A dimensionless expression of the Chezy coeffi cient applicable to -full sand bed and gravel bed streams can be found in Chapter 3. Application of Eq. (2-34a) to fi eld conditions resulted in considerable overestimation of the hydraulic roughness (van Rijn 1996). Further analysis showed that the lee-side slopes of natural sand dunes in rivers were less steep than those of γ dunes in the laboratory and a shape factor sf 0.7 was rec- ommended for application to natural river dunes. A different approach based on boundary-layer theory and measured velocity profi les was proposed by Fedele and Fig. 2-6. Roughness function for alluvial streams with dunes García (2001). When spatially-averaged velocity profi les (after Fedele and García 2001).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 3333 11/21/2008/21/2008 5:42:435:42:43 PMPM 34 sediment transport and morphodynamics Fedele and García (2001) also found that the composite 2.3 SEDIMENT PROPERTIES

roughness kc could be approximated with In this section, rock types, as well as fundamental charac- 1/3 k 3/2 ⎛ H ⎞ teristics of sediment particles such as size, size distribution, c 145. τ* R ⎜ ⎟ (2-35c) D ep ⎝ D⎠ density, and fall velocity are considered. The role of sedi- ment size on stream morphology is analyzed also, with the / 3 which is valid for (H D) 10 and Rep 30, which are com- goal of understanding the behavior of sand-bed and gravel- monly found conditions for large alluvial rivers with sand bed streams. dunes. Here, 2.3.1 Rock Types τ* dimensionless bed shear stress (i.e., ) for uniform fl ow (HS)/(RD) ; The solid phase in sediment transport can be any granular H fl ow depth; substance. In terms of engineering applications, however, the S channel slope; granular substance in question typically consists of fragments ρ /ρ ultimately derived from rocks—hence the name “sediment” R s 1 submerged specifi c gravity of sedi- ment; transport. The properties of these rock-derived fragments, D sediment size; taken singly or in groups of many particles, all play a role in determining the transportability of the grains under fl uid R gRDD/ν particle Reynolds number; and ep action. The important properties of groups of particles include H/D relative fl ow depth. and size distribution. The most common rock type one is likely to encounter in the river or coastal environment A simple method to estimate the composite roughness k c is quartz. Quartz is a highly resistant rock and can travel long has been proposed by Wright and Parker (2004b) and can be distances or remain in place for long periods of time without found in Section 2.8.3.3 below. losing its integrity. Another highly resistant rock type that is The total friction coeffi cient for fl ow in an alluvial chan- often found together with quartz is feldspar. Other common nel in the presence of dunes can be estimated with the help of rock types include limestone, basalt, granite, and more esoteric Keulegan’s Eq. (2-16), Eq. (2-19), and Eq. (2-35c), types such as magnetite. Limestone is not a resistant rock; it tends to abrade to silt rather easily. Silt-sized limestone parti- ⎡ 3/2 ⎤ /2 U 1 ()HD/ C 1 ln⎢ ⎥ 51 . (2-35d) cles are susceptible to solution unless the water is suffi ciently f κ τ*3/2 u* ⎣⎢ Rep ⎦⎥ buffered. As a result, limestone is not typically found to be a major component of sediments at locations distant from its The data used to by Fedele and García (2001) to obtain source. On the other hand, it can often be the dominant rock this fi t are shown in Fig. 2-7. This expression provides only type in mountain environments. a crude approximation for the friction factor, but clearly Basaltic rocks tend to be heavier than most rocks com- indicates that the roughness in alluvial streams with dunes posing the crust of the earth. They are typically brought to is a dynamic parameter that depends nonlinearly on the fl ow the surface by volcanic activity. Basaltic gravels are rela- intensity given by the Shields stress parameter (τ * ), the relative tively common in rivers that derive their sediment supply / from areas subjected to volcanism in recent geologic history. fl ow depth ( H D), and the particle Reynolds number (R ep). Basaltic sands are much less common. Regions of weathered granite often provide copious supplies of sediment. The par- ticles produced by are often in the size but often quickly break down to sand sizes. Sediments in the fl uvial or coastal environment in the size range of silt, or coarser, are generally produced by mechanical means, including fracture or . The clay minerals, on the other hand, are produced by chemical action. As a result, they are fundamentally different from other sediments in many ways. Their ability to absorb water means that the porosity of clay deposits can vary greatly over time. Clays also display cohesiveness, which renders them more resistant to erosion.

2.3.2 Specifi c Gravity

Fig. 2-7. Total friction coeffi cient for alluvial fl ows with sand Sediment specifi c gravity is defi ned as the ratio between the ρ ρ dunes (after Fedele and García 2001). sediment density s and the density of water . Some typical

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 3344 11/21/2008/21/2008 5:42:455:42:45 PMPM sediment properties 35 specifi c gravities for various natural and artifi cial sediments where are listed in Table 2-2. g acceleration of gravity. Quartz, for example, is a mineral with a specifi c gravity 2.3.3 Model Laboratory Sediments ρ /ρ s near 2.65. If a grain of the same volume were modeled In the laboratory, it is often of value to employ light weight in the laboratory using crushed with a specifi c gravity model sediment (Shen 1990). To see the utility of this, it is use- of 1.3, it would follow from Eq. (2-36a) that the coal grain ful to consider a movable-bed scale model of an actual river. would be only 1.3/2.65 or 0.49 times the weight of the quartz Consider a reach of the Minnesota River, Minnesota, with a grain. Rephrasing, the coal grain is 2.04 times lighter than bank-full width of 90 m, a bank-full depth of 4 m, a streamwise the quartz grain, and thus, in some sense, twice as mobile.

slope of 0.0002, and a median sediment size D 50 of 0.5 mm. In fact, the benefi t of using lightweight material is much The reach is scaled down by a factor of 100 to fi t into a typical greater than this, because the effective weight determining

laboratory model basin, resulting in a bank-full width of 90 cm the mobility of a grain is the submerged weight W s , i.e., the and a bank-full depth of 4 cm. In an undistorted model, slope actual weight minus the buoyancy force associated with the remains constant at 0.0002. hydrostatic pressure distribution about the particle. That is, If the sediment employed in the model were to be the ρρ ρ same as in the fi eld, it would most likely not move at all in WgVRgVSS() P P (2-36b) the scale model. Carrying the analogy to its logical conclu- sion, it would be as if the sediment in the fi eld Minnesota where River had a median size of 0.5 mm 100 0.5 m, i.e., ⎛ ρ ⎞ boulders. It should be clear that, in this case, the fi eld sedi- S R ⎜ 1⎟ (2-36c) ment cannot be employed directly in the model. The obvious ⎝ ρ ⎠ alternative is to scale down sediment size by the same factor denotes the submerged specifi c gravity of the sediment. as all other lengths, i.e., by a factor of 100. This would yield a Comparing coal and quartz again in terms of submerged size of 5 μm, which is so close to the clay range that it can weight, it is seen that be expected to display some kind of pseudocohesiveness. In addition, viscous effects are expected to be greatly exagger- () ( ) ated due to the small size. The net result is model sediment WS R 030. coal coal 018. (2-36d) that is much less mobile than it ought to be and, in addi- () (R) . WS quartz 165 tion, behaves in ways radically different from the prototype quartz sediment. It follows that under water, the coal grain is 1/0.18 5.5 There are several ways out of this dilemma. One of them times lighter than a quartz grain of the same size. Lightweight involves using artifi cial sediment with a low specifi c gravity. model sediments are thus a very effective way of increasing ρ ρ Let denote the density of water, and s denote the specifi c mobility in laboratory experiments (Zwamborn 1981; ASCE gravity of the material in question. The weight W of a par- 2000, p. 105). More material on physical modeling of sedi-

ticle of volume Vp is given by mentation processes can be found in Appendix C. WgV ρ (2-36a) SP 2.3.4 Size The notation D will be used to denote sediment size, the typ- ical units of which are millimeters (mm—sand and coarser Table 2-2 Specifi c Gravity of Rock Types and material) or micrometers (μ m –clay and silt). Another stan- Artifi cial Materials dard way of classifying grain sizes is the sedimentological Φ scale, according to which Specifiρ c/ρ gravity s Rock type or material Φ D 2 (2-37a) Quartz 2.60–2.70 Taking the logarithm of both sides, it is seen that Limestone 2.60–2.80 Basalt 2.70–2.90 ln()D Magnetite 3.20–3.50 Φ log ()D (2-37b) 2 ln()2 Bakelite 1.30–1.45 Coal 1.30–1.50 Note that the size Φ 0 corresponds to D 1 mm. The Ground walnut shells 1.30–1.40 utility of the Φ scale will become apparent upon a consid- PVC 1.14–1.25 eration of grain size distributions. The minus sign has been

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 3355 11/21/2008/21/2008 5:42:465:42:46 PMPM 36 sediment transport and morphodynamics inserted into Eq. (2-37b) simply as a matter of convenience In practical terms, there are several ways to determine to sedimentologists, who are more accustomed to working grain size. The most popular way for grains ranging from with material fi ner than 1 mm rather than coarser material. Φ 4 to Φ 4 (0.0625 to 16 mm) is with sieves. Each The reader should always recall that larger Φ implies fi ner sieve has a square mesh, the gap size of which corresponds material. to the diameter of the largest sphere that would fi t through. The Φ scale provides a very simple way of classifying The grain size D thus measured exactly corresponds to diam- grain sizes into the following size ranges in descending eter only in the case of a sphere. In general, the sieve size D order: boulders, cobbles, gravel, sand, silt, and clay. This is corresponds to the smallest sieve gap size through which a illustrated in Table 2-3. given grain can be fi tted. It should be noted that the defi nition of clay according to For coarser grain sizes, it is customary to approximate size (D 2 μm) does not always correspond to the defi nition the grain as an ellipsoid. Three lengths can be defi ned. The of clay according to mineral. That is, some clay mineral par- length along the major (longest) axis is denoted as a , that ticles can be coarser than this limit, and some silt particles along the intermediate axis is denoted as b , and that along produced by grinding can be fi ner than this. In general, how- the minor (smallest) axis is denoted as c . These lengths ever, the effect of viscosity makes it quite diffi cult to grind are typically measured with a caliper. The value b is then up particles in water to sizes fi ner than 2 μm . equated to grain size D .

Table 2-3 Sediment Scale Approximate sieve mesh Size range openings per inch Class Name Millimeters F MicronsInches Tyler U.S. standard Very large boulders 4096 ~ 2048 160 ~ 80 Large boulders 2048 ~ 1024 80 ~ 40 Medium boulders 1024 ~ 512 40 ~ 20 Small boulders 512 ~ 256 9 ~ 8 20 ~ 10 Large cobbles 256 ~ 128 8 ~ 7 10 ~ 5 Small cobbles 128 ~ 64 7 ~ 6 5 ~ 2.5 Very coarse gravel 64 ~ 32 6 ~ 5 2.5 ~ 1.3 Coarse gravel 32 ~ 16 5 ~ 4 1.3 ~ 0.6 2 ~ 1/2 Medium gravel 16 ~ 8 4 ~ 3 0.6 ~ 0.3 5 5 Fine gravel 8 ~ 4 3 ~ 2 0.3 ~ 0.16 9 10 Very fi ne gravel 4 ~ 2 2 ~ 1 0.16 ~ 0.08 16 18 Very coarse sand 2.000 ~ 1.000 1 ~ 0 2000 ~ 1000 32 35 Coarse sand 1.000 ~ 0.500 0 ~ 1 1000 ~ 500 60 60 Medium sand 0.500 ~ 0.250 1 ~ 2 500 ~ 250 115 120 Fine sand 0.250 ~ 0.125 2 ~ 3 250 ~ 125 250 230 Very fi ne sand 0.125 ~ 0.062 3 ~ 4 125 ~ 62 Coarse silt 0.062 ~ 0.031 4 ~ 5 62 ~ 31 Medium silt 0.031 ~ 0.016 5 ~ 6 31 ~ 16 Fine silt 0.016 ~ 0.008 6 ~ 7 16 ~ 8 Very fi ne silt 0.008 ~ 0.004 7 ~ 8 8 ~ 4 Coarse clay 0.004 ~ 0.002 8 ~ 9 4 ~ 2 Medium clay 0.002 ~ 0.001 2 ~ 1 Fine clay 0.001 ~ 0.0005 1 ~ 0.5 Very fi ne clay 0.0005 ~ 0.00024 0.5 ~ 0.24

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For grains in the silt and clay sizes, many methods (percent fi ner) versus log10 ( D )—that is, a semilogarithmic (hydrometer, sedigraph, etc.) are based on the concept of plot is employed. The plot, then, would look like the one in

equivalent fall diameter. That is, the terminal fall velocity vs Fig. 2-8(a) . of a grain in water at a standard temperature is measured. The The same size distribution plotted in sedimentological Φ equivalent fall diameter D is the diameter of the sphere hav- form would involve plotting pf 100 versus on a linear ing exactly the same fall velocity under the same conditions. plot, like shown in Fig. 2-8(b). Sediment fall velocity is discussed in more detail below. Note that Φ on a linear axis is completely equivalent A variety of other more recent methods for sizing fi ne par- to D on a logarithmic axis because Φ is related linearly to

ticles rely on blockage of light beams. The area blocked can log10 ( D ): be used to determine the diameter of the equivalent circle, i.e., the projection of the equivalent sphere. It can be seen Φ 1 that all of these methods can be expected to operate con- log10 ()D (2-38) sistently as long as grain shape does not deviate too greatly log10 ()2 from that of a sphere. In general, this turns out to be the case. There are some important exceptions, however. At the fi ne The utility of a logarithmic scale for grain size now end of the spectrum, mica particles tend to be platelike; the becomes apparent. Consider a sediment sample in which same is true of shale grains at the coarser end. Comparison one-third of the material lies in the range 0.1–1.0 mm, one- with a sphere is not necessarily a particularly useful way to third lies in the range 1.0–10 mm, and one-third lies in the characterize grain size for such materials. More recently, range 10–100 mm. In Fig. 2-8(c) p f 100 is plotted versus techniques employing light-scattering are becoming more D on a linear scale, and in Fig. 2-8(d) p f 100 is plotted popular for both particle-size analysis and settling velocity versus D on a logarithmic scale—p f 100 is plotted against measurements (e.g., Pedocchi and García 2006). More mate- log10 ( D). Plot (c) is virtually unreadable, as the fi nest two rial can be found in Chapter 5. ranges are crowded off the scale. Plot (d) provides a use- ful and consistent characterization of the distribution. It can be concluded that for the purposes of statistics, the relevant 2.3.5 Size Distribution grain size should be on a logarithmic scale, e.g., Φ rather Any sediment sample normally contains a range of sizes. than D itself. Φ Φ An appropriate way to characterize these samples is in terms The size distribution p f ( ) and size density p ( ) by of a grain size distribution. Consider a large bulk sample of weight (Fig. 2-8(e)) can be used to extract useful statistics Φ sediment of given weight. Let pf (D )—or pf ( ) —denote the concerning the sediment in question. Let x denote some per- Φ fraction by weight of material in the sample of material fi ner centage, say 50%; the grain size x denotes the size such that than size D(Φ). The customary engineering representation x % of the weight of the sample is composed of fi ner grains. Φ of the grain size distribution consists of a plot of p f 100 That is, x is defi ned such that

(a) (b)

Fig. 2-8. Sediment grain size distribution in (a) semilog scale, (b) sedimentological scale Φ, (c) linear scale, (d) log scale, (e) size distribution and size density, and (f) discretization of grain size distribution.

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(d) (c)

(e) (f) Fig. 2-8. Sediment grain size distribution in (a) semilog scale, (b) sedimentological scale Φ, (c) linear scale, (d) log scale, (e) size distribution and size density, and (f) discretization of grain size distribution. (Continued)

particularly useful for characterizing bed roughness, as dis- ( ) x (2-39a) p f Φ x cussed previously. 100 The density p ( Φ ) can be used to extract statistical Φ moments. Of these, the most useful are the mean size m and It follows that the corresponding grain size in terms of the standard deviation σ. These are given by the relations

equivalent diameter is given by D x , where ∫ ΦΦΦ Φ m p ( ) d (2-40a) Φ x D x 2 (2-39b) 2 ∫ Φ 2 ΦΦ σ ( Φ m) p ( ) d (2-40b) The most commonly used grain sizes of this type are

the median size D 50 and the size D90 such that 90% of the The corresponding geometric mean diameter D g and geo- σ sample by weight consists of fi ner grains. The latter size is metric standard deviation g are given as

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Φm D g 2 (2-41a) 1 ΦΦΦ ( ) (2-44b) m 2 84 16 σ σ 2 (2-41b) g Rearranging the relations with the aid of Eqs. (2-40a), (2-40b) and (2-43) and (2-44a), it is found that Note that for a perfectly uniform material, σ 0 and σ g 1. As a practical matter, a sediment mixture with a σ 12/ value of g of less than 1.3 is often termed well-sorted and ⎛ ⎞ can be treated as a uniform material. When the geometric σ D84 g ⎝⎜ ⎠⎟ (2-45a) standard deviation exceeds 1.6, the material can be said to D16 be poorly-sorted. In point of fact, one never has the continuous function Φ 12/ p ( ) with which to compute the moments of Eqs. (2-40a) D g (DD84 16) (2-45b) and (2-40b). One must rather rely on a discretization. To this end, the size range covered by a given sediment sample is discretized in terms of n intervals bounded by n It must be emphasized that the relations are exact only Φ Φ Φ Φ Φ 1 grain sizes , 2, . . ., n 1 in ascending order of , for a Gaussian distribution in . This is often not the case as illustrated in Fig. 2-8(f). The following defi nitions are in nature. As a result, it is strongly recommended that D g made from i 1 to n : and σ g be computed from the full size distribution via Eqs. (2-43a), (2-43b), (2-41a), and (2-41b) rather than the approx- 1 imate form embodied in the above relations. Φ (ΦΦ ) (2-42a) i 2 i i 1 2.3.6 Porosity ( ) ( ) ppif ΦΦi p f i 1 (2-42b) The porosity λ p quantifi es the fraction of a given volume of sediment that is composed of void space. That is, Relations (2-40a) and (2-40b) now discretize to

n volume of voids λ p Φ m ∑Φi pi (2-43a) volume of total space i = 1

n If a given mass of sediment of known density is depos- σ2 () 2 (2-43b) ited, the volume of the deposit must be computed assuming ∑ Φi Φ m pi i 1 that at least part of it will consist of voids. In the case of well-sorted sand, the porosity can often take values between In some cases, especially when the material in question is 0.3 and 0.4. Gravels tend to be more poorly-sorted. In this sand, the size distribution can be approximated as Gaussian case, fi ner particles can occupy the spaces between coarser on the Φ scale (i.e., log-normal in D). For a perfectly particles, reducing the void ratio to as low as 0.2. So-called Gaussian distribution, the mean and median sizes coincide: open work gravels are essentially devoid of sand and fi ner material in their interstices; these may have simi- lar to that of sand. Freshly deposited clays are notorious for 1 ( ) having high porosities. As time passes, clay deposits tend ΦΦm 50 ΦΦ 84 16 (2-43c) 2 to consolidate under its own weight so that porosity slowly decreases. Wu and Wang (2006) proposed an empirical Furthermore, it can be demonstrated from a standard relation to estimate the initial porosity of sediments, which table that in the case of the Gauss distribution the size Φ dis- have bed deposited within a year or less, as a function of Φ the median diameter D 50 of the sediment mixture. In situ placed one standard deviation larger that m is accurately Φ measurements of porosity indicate that biological activity given by 84 ; by symmetry, the corresponding size one Φ Φ can have an important effect on the porosity of sediments standard deviation smaller than 84 is 16 . The following relations thus hold: (Wheatcroft 2002). The issue of porosity becomes of practical importance as regards, for example, salmon spawning grounds in gravel- 1 σ (ΦΦ) (2-44a) bed rivers (Alonso and Mendoza 1992; Huang and García 2 84 16 2000). The percentage of sand and silt contained in the

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 3399 11/21/2008/21/2008 5:42:505:42:50 PMPM 40 sediment transport and morphodynamics sediment is often referred to the percentage of “fi nes” in the gravel-bed rivers is shown in Fig. 2-9 (Shirazi and Seim gravel deposit. When this fraction rises above 20–26 % by 1981). It is clear that as the material becomes coarser, the weight, the deposit is often rendered unsuitable for spawn- substrate porosity can be expected to increase accordingly, ing. Salmon bury their eggs within the gravel, and high fi nes augmenting the embryo survival rates. Chief causes of ele- content implies low porosity and thus reduced permeability. vated fi nes in gravel-bed rivers include road building and The fl ow of groundwater necessary to carry oxygen to the clear-cutting of timber in the watershed. eggs and remove metabolic waste products is impeded. In addition, newly hatched fry may encounter diffi culty in 2.3.7 Shape fi nding pore space through which to emerge to the sur- face. All of the above factors dictate lowered survival rates. There are a number of ways in which to classify grain An empirical relationship between percent embryo sur- shape (Vanoni 2006). One of these, the Zingg classifi ca- vival and the geometric mean diameter of the substrate in tion scheme, is illustrated here. According to the defi nitions introduced earlier, a simple way to characterize the shape of an irregular clast (stone) is in terms of the lengths a , b , and c of the major, intermediate, and minor axes, respec- tively. If these three are all equal, the grain can be said to be close to a sphere in shape. If a and b are equal but c is much smaller, the grain is rodlike. Finally, if c is much smaller than b, which is, in turn, much smaller than a , the resulting shape should be bladelike. This is illustrated in terms of the Zingg diagram in Fig. 2-10. In studies of the fall velocity of geometric shapes and sand grains by McNown, Albertson and others, the shape of the particles has been expressed by the Corey shape factor SF, which makes use of the characteristic lengths, defi ned above, and is given by (Vanoni 2006, p. 14)

c SF ab Fig. 2-9. Relationship between percent embryo survival and the geometric mean diameter of the substrate (after Shirazi and Seim, It follows that a spherical particle will have a SF 1. For 1981). natural sands SF 0.7. The shape factor has been used in

Fig. 2-10. Defi nition of Zingg diagram.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 4400 11/21/2008/21/2008 5:42:515:42:51 PMPM sediment properties 41 studies of particle fall velocity (Dietrich 1982; Jimenez and the ranges for silt, sand, and gravel are plotted for a kine- Madsen 2003; Wu and Wang 2006). More material on sedi- matic viscosity v 0.01 cm 2 /s (clear water at 20°C) and a ment particle shape and its effect on particle fall velocity can submerged specifi c gravity R 1.65 (quartz). An equivalent be found in Vanoni (2006, p. 14). diagram to estimate fall velocity of particles was proposed earlier by Parker (1978). Notice that for fi ne , R is smaller than one and the 2.3.8 Fall Velocity p drag coeffi cient given by Eq. (2-46d) reduces to A fundamental property of sediment particles is their fall or settling velocity. The fall velocity of sediment grains in water is determined by their diameter and density and by the 24 C (2-46f) viscosity of the water. Falling under the action of gravity, a D R particle will reach a constant, once the drag p equals the submerged weight of the particle. The relation for terminal fall velocity for a spherical particle in quiescent Substitution of (2-46f) into (2-46a) yields the well-known v fl uid s can be presented as Stokes law for settling velocity of fi ne particles,

12/ ⎡ ⎤ gRD2 ⎢ 4 1 ⎥ R vs (2-46g) f ⎢ 3 ⎥ (2-46a) 18ν2 ⎣ CRDp () ⎦

A useful empirical relation to estimate the kinematic vis- where cosity of clear water is:

ν s R 6 f (2-46b) 17910. gRD ν/ ()ms2 (2-46h) 1 0.. 03368TT 0 00021 2

ν D R s p ν (2-46c) where T temperature of the water in degrees centigrade (° C ). and the functional relation C D f (R p ) denotes the drag A number of relations for terminal fall velocity for the coeffi cient for spheres (García 1999). Here g is the accel- case of nonspherical (natural) particles can be found in the ρ ρ /ρ eration of gravity, R ( s ) is the submerged specifi c literature. Dietrich (1982) analyzed fall velocity data for gravity of the sediment, and ν is the kinematic viscosity natural particles and used dimensional analysis to obtain the of water. This relation is not very useful because it is not useful fi t ν explicit in s ; one must compute fall velocity by trial and error. One can use the following equation for the drag coeffi cient C D ⎡ ⎤ 2 R fepepexp{ bb12 ln ()R b 3 ⎣ln ()R ⎦ (2-47a) 3 4 24 ⎡ ⎤ ⎡ ⎤ 1/2 b4 ⎣ln ()Rep⎦ b5 ⎣ln ()R ep ⎦ } CD ()1 0.152 RRpp 0.0151 (2-46d) Rp

where and the defi nition b1 2.891394, b2 0.95296, b 3 0.056835, gRD D R (2-46e) b 0.002892, b 0.000245 (2-47b) ep ν 4 5

In an attempt to obtain a more practical relation, Jimenez

to obtain an explicit relation for fall velocity in the form of Rf and Madsen (2003) fi tted the formula of Dietrich (1982) to versus R ep. Such a diagram is presented in Fig. 2-11, where the expression

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 4411 11/21/2008/21/2008 5:42:585:42:58 PMPM 42 sediment transport and morphodynamics

Fig. 2-11. Diagram of Rf versus Rep calculated from the drag coeffi cient for spheres.

1 ν v ⎛ B ⎞ ⎡ 231/2 ⎤ W s A vs ⎣()10.. 36 1 049D* 10 . 36⎦ (2-49a) * ⎝⎜ ⎠⎟ (2-48a) D gRDN S*

where in which / ⎡ gR⎤13 D D D * ⎢ 2 ⎥ (2-49b) S N gRD (2-48b) ⎣ ν ⎦ * 4ν N

Here Here, D nominal particle diameter. The coeffi cients N g acceleration of gravity; A and B in Eq. (2-48a) are functions of Corey shape fac- ν kinematic viscosity of water; and particle and are expressed graphi- D mean sieve diameter of grains; and cally by Jimenez and Madsen (2003). In many practical R ( ρ ρ ) /ρ is the submerged specifi c gravity of the applications, the sediment is naturally worn quartz sands s grains. characterized by their sieve diameter D s. For this typical / application, D N D s 0.9, A 0.954 and B 5.12, are Equations very similar to (2-49a) have been proposed recommended. With these values incorporated in to it, independently by Zanke (1977) and van Rijn (1984). Eq. (2-48a) was found to provide reliable predictions At high concentrations the fl ows around adjacent settling of fall velocity for natural quartz sediment with sieving grains interact resulting in a larger drag than for the same diameters ranging from 0.063 mm up to 2 mm (Jimenez grain in isolation. This phenomenon is known as hindered

and Madsen 2003). settling and results in the hindered settling velocity v sC for Another simple relation to estimate the fall velocity of high sediment concentrations to be smaller that the fall

natural sand particles has been proposed by Soulsby (1997) velocity v s at low sediment concentrations (less than 0.05). for use in the marine environment, Applying reasoning similar to the one that led to Eq. (2-49a),

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 4422 11/21/2008/21/2008 5:42:585:42:58 PMPM sediment properties 43 Soulsby (1997) proposed the following relation for the hin- 2.3.9 Relation between Size Distribution and

dered fall velocity v sc of grains in a dense suspension having Stream Morphology a volumetric sediment concentration C : The study of sediment properties, and in particular size distribution, is most relevant in the context of stream mor- ν ⎡ 2 47. 3 12/ ⎤ phology. The material that follows is intended to point out v ()10.. 36 1 049( 1CD) 10 . 36 (2-49c) sC D ⎣ * ⎦ some of the more interesting issues, and in particular, mor- phological differences between sand-bed and gravel-bed streams. More discussion on the subject can be found in which is valid for all values of D* and C . When C 0, Chapters 3 and 6. Eq. (2-49c) reduces to Eq. (2-49a). In Fig. 2-12, several size distributions from the sand- The subject of sediment fall velocity is far from being bed Kankakee River, Illinois, are shown (Bhowmik et al. resolved. However, the empirical fi ts presented here 1980). The characteristic S -shape suggests that these distri- should suffi ce for engineering purposes. Other useful rela- butions might be approximated by a Gaussian curve. The tions to estimate sediment fall velocity can be found in median size D50 falls near 0.3–0.4 mm. The distributions Swamee and Ojha (1991), Cheng (1997), Ahrens (2000), are very tight with a near-absence of either gravel or silt. and Ahrens (2003). Recently, Wu and Wang (2006) com- For practical purposes, the material can be approximated pared different formulations and developed another as uniform. empirical fi t to estimate fall velocity which accounts for In Fig. 2-13, several size distributions pertaining to the the effect of particle shape through the Corey shape factor gravel-bed Oak Creek, Oregon, are shown (Milhous 1973). In (SF c/ ab). gravel-bed streams, the surface layer (“” or “pavement”) Material on particle settling for the case of fi ne-grained tends to be coarser than the substrate (identifi ed as “subpave- cohesive sediment is presented in Chapter 4. ment” in the fi gure). Whether the surface or substrate is

Fig. 2-12. Particle size distributions of bed materials in Kankakee River, Illinois (after Bhowmik et al. 1980).

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Fig. 2-13. Size distribution of bed material samples in Oak Creek, Oregon (after Milhous 1973).

considered, it is apparent that the distribution ranges over gravel-bed reaches. All of the sand-bed distributions are a much wider range of grain sizes than in the case of Fig. S -shaped, and all have a lower spread than the gravel-bed 2-12. More specifi cally, in the distributions of the sand-bed distributions. The fi gure indicates that the standard deviation Kankakee River, Φ varies from about 0 to about 3, whereas of the grain size distribution can be expected to increase sys- in Oak Creek, Φ varies from about -8 to about 3. In addi- tematically with increasing sediment size (White et al. 1973). tion, the distribution of Fig. 2-13 is upward concave almost The three gravel-bed size distributions differ systematically everywhere, and thus deviates strongly from the Gaussian from the sand-bed distributions in a fashion that accurately distribution. refl ects Oak Creek (Fig. 2-13). The standard deviation is, in These two examples provide a window toward generaliza- all cases, markedly larger than for any of the sand-bed dis- tion. A river may loosely be classifi ed as sand-bed or gravel- tributions, and the distributions are upward concave except

bed according to whether the median size D50 of the surface perhaps near the coarsest sizes. material or substrate is less or greater than 2 mm. The size distributions of sand-bed streams tend to be relatively nar- row and also tend to be S-shaped. The size distributions of 2.4 THRESHOLD CONDITION FOR gravel-bed streams tend to be much broader and to display SEDIMENT MOVEMENT an upward-concave shape. There are, of course, many excep- tions to this behavior, but it is suffi ciently general to warrant In this section the threshold conditions for initiation of emphasis. motion are analyzed. A mechanistic model for initiation of More evidence for this behavior is provided in Fig. 2.14. motion is presented. The Shields diagram and other methods Here, the grain size distributions for a variety of stream for assessing initiation of motion are introduced. The analy- reaches have been normalized using the median sediment sis is limited to noncohesive granular sediments such as silt,

size D 50 . Four sand-bed reaches are included with three sand and gravel.

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Fig. 2-14. Dimensionless grain-size distribution for different rivers (after White et al. 1973).

2.4.1 Submerged ρρ φ FgVgn()cos s p (2-50c) If granular particles are allowed to pile up while submerged φ μ in a fl uid, there is a specifi c slope angle beyond which FFrgn (2-50d) spontaneous failure of the slope occurs. This angle is termed the angle of repose, or alternatively, the friction angle. To The condition for incipient motion is given by study this in more detail, consider a typical grain resting on FF the surface of such a slope as shown in Fig. 2-15. gt r (2-50e) The coeffi cient of Coulomb friction is defi ned to be μ , where That is, the downslope impelling force of gravity should just balance with the Coulomb resistive force. From the above four relations, it is found that tangential resistive force μφ μ (2-50a) tan (2-50f) downward normal force The angle of repose is an empirical quantity. Tests with well-sorted material indicate that φ is near 30° for sand, The forces acting on the particle along the slope are the gradually increasing to 40° for gravel. Poorly sorted, angular submerged force of gravity (gravitational force minus buoy- material tends to interlock, giving greater resistance to fail- φ ancy force), which has a downslope component F gt and a ure, and as a result, a higher friction angle . Such material normal component Fgn , and a tangential resistive force Fr is thus often chosen for riprap (see Appendix B). due to Coulomb friction. These are given by Friction angle measurements obtained in gravel-bed streams, including implications for critical shear stress estimations, can be found in Kirchner et al. (1990) and ρρ φ FgVgt()sin s p (2-50b) Buffi ngton et al. (1992).

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Fig. 2-15. Defi nition diagram for angle of repose.

2.4.2 Critical Stress for Flow over a Granular Bed of motion presented by Ikeda (1982), which is based on the work of Iwagaki (1956) and Coleman (1967). A similar When a granular bed is subjected to a turbulent fl ow, it is analysis was presented by Wiberg and Smith (1987) for the found that virtually no motion of the grains is observed at case of nonuniform sediment size. Consider the granular some fl ows, but that the bed is noticeably mobilized at other bed of Fig. 2-16. The fl ow forces on a dangerously placed fl ows (Cheng and Chiew 1998; Papanicolau et al. 2002; Niño spherical sediment particle protruding upward from the et al. 2003). Literature reviews on incipient motion can be mean bed are considered in order to analyze the threshold found in Miller et al. (1977); Lavelle and Mofjeld (1987); of motion. and Buffi ngton and Montgomery (1997). Certain assumptions enter into the Ikeda-Coleman-Iwagaki Factors that affect the mobility of grains subjected to a analysis. The fl ow is taken to follow the logarithmic law near fl ow are summarized as follows, the boundary (Eq. 2-4). The origin of the z -coordinate for evalu- ating the logarithmic law is taken to be the base of the dan- Grain placement gerously exposed particle. Turbulent forces on the particle are Randomness neglected. Drag and lift forces act through the particle cen- Turbulence ter (in general, they do not, giving rise to torque as well as

forces). The value of the drag coeffi cient c D can be approxi- Lift mated by the free-stream value (Coleman 1967). The coordi- Fluid Mean & turbulent nate z is taken to be vertically upward, corresponding to very Drag low streamwise slopes S . The roughness height k s is equated Forces on grains to the particle diameter D .

Gravity

In the presence of turbulent fl ow, random fl uctuations typically prevent the clear defi nition of a critical or thresh- old condition for motion: the probability of grain movement is never precisely zero (Paintal 1971; Graf and Paziz 1977; Lopez and García 2001; Zanke 2003). It is, nevertheless, possible to defi ne a condition below which movement can be neglected for many practical purposes. The following analysis is a slightly generalized version of the derivation of the full Shields curve for the threshold Fig. 2-16. Forces acting on a “dangerously” placed particle.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 4466 11/21/2008/21/2008 5:43:045:43:04 PMPM threshold condition for sediment movement 47 It is seen from the above assumptions that the particle Now the forces acting on the particle can be considered. / center is located at z D 2. It is necessary to use some The streamwise fl uid drag force D f , upward normal (verti- information about turbulent boundary layers to defi ne the cal in this case) fl uid lift force L f , and downward vertical effective fl uid velocity uf acting on the particle in order to submerged gravitational force F g acting on the particle of the facilitate computation of the fl uid forces. A viscous sub- previous fi gure are thus /δ layer exists (see Eq. 2-6) when D v is less than about 0.5, or ( u D /v ) 5. In this case, the effective fl uid velocity u acting 2 * f 1 ⎛ D⎞ on the particle is estimated with Eq. (2-5) as D ρπ⎜ ⎟ cu2 (2-52a) fDf22⎝ ⎠

u 2 f 1 uD* ⎛ ⎞ (2-51a) ρπ1 D 2 L fLf⎜ ⎟ cu (2-52b) u* 2 v 22⎝ ⎠

On the other hand, if D / δ 2, no viscous sublayer exists, v 4 ⎛ D⎞ 3 and the logarithmic law applies near (but not at) the bed. So FRg ρπ g ⎝⎜ ⎠⎟ (2-52c) the fl ow velocity acting on the particle can be estimated by 32

From Eq. (2-50d), it is seen that the Coulomb resistive u f ⎛ z ⎞ 2.5ln⎜ 30⎟ 6.77 force Fr is given by ⎝ ⎠ 1 (2-51b) u* D zD 2 μ FFLrgf() (2-52d) It follows then that Eqs. (2-51a) and (2-51b) can be writ- ten in the more general form The critical condition for incipient motion of the particle is that the impelling drag force is just balanced by the resist- ing Coulomb frictional force: u f ⎛ uD⎞ F ⎜ * ⎟ (2-51c) u ⎝ ν ⎠ * DFfr (2-53)

where That is, if D f Fr , the particle will not move, and if D f F , it will move. Between Eqs. (2-52a), (2-52b), (2-52c), uD uD r 1 ** F for 13. 5 (2-51d) (2-52d), and (2-53), the following relation is obtained for 2 νν / critical fl uid velocity uf at z D 2:

and 2 u 4 μ f (2-54) RgD3 c μ c uD DL F 677.. for * 135 (2-51e) ν This relation is now converted to a relation in terms of boundary shear stress. It may be recalled that by defi nition However, it would be more convenient to have a con- ρu 2 τ , where τ denotes the boundary shear stress. In this tinuous function F , so that the transition between hydrauli- * b b case, the shear stress in question is the critical one for the cally smooth and fully rough condition, is smooth. The fi t onset of motion and is denoted by τ . Between Eqs. (2-51c) proposed by Swamee (1993) can be used to evaluate F in bc and (2-54), the Ikeda-Coleman-Iwagaki relation is obtained Eq. (2-51c) by setting z D / 2 and k D in Eq. (2-10), s for the critical shear stress:

⎧ 10/3 ⎫ 03. μ ⎡ ⎛ 9 uD∗ ⎞ ⎤ * 4 1 ⎪ 10/3 ⎪ τ (2-55a) ⎪⎛ ν ⎞ ⎢ ⎜ ⎟ ⎥ ⎪ c () μ 2 / 2 1 2 ν 3 ccDLFuDv( ) F ⎨ ⎢κ ln⎜1 ⎟ ⎥ ⎬ *c ⎝⎜ ⎠⎟ uD ⎪ uD* ⎢ ⎜ ∗ ⎟ ⎥ ⎪ ⎢ ⎝ 103. ⎠ ⎥ ⎩⎪ ⎣ ν ⎦ ⎭⎪ The equation is valid for nearly horizontal beds but the effect of channel slope can be readily incorporated. For a (2-51f) channel with a downstream slope angle α , the downslope

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 4477 11/21/2008/21/2008 5:43:065:43:06 PMPM 48 sediment transport and morphodynamics effect of gravity has to be included in the force balance presented earlier, resulting in the following expression for the critical shear stress

(μαcos sin α) τ* 4 1 (2-55b) c μ 2 / 3 ()ccDLFuDv( *c )

Notice that for α 0, Eq. (2-55b) reduces to Eq. (2-55a). A predecessor of all these equations was advanced by Egiazaroff (1965). It can be found in Vanoni (2006, p. 58) and is used for sediment mixtures in Chapter 3. Similar rela- tions were also obtained by Fredsøe and Deigaard (1992, p. 203) and can also be found, albeit without derivation, in Fig. 2-17. Comparison of Ikeda-Coleman-Iwagaki model for ini- Chien and Wan (1999, p. 319). tiation of motion with Shields data. τ * In the above relation, c is a dimensionless measure of boundary shear stress known as the Shields parameter and given by the defi nition (Buffi ngton 1999). The theoretical model developed here is for idealized spherical particles for which the friction angle τ can be expected to be lower than for natural sediments. For τ* bc (2-56) c the case of φ 40° ( μ 0.84), the curve predicted with ρgRD Eq. (2-55a), follows the trend of Shields data but predicts values of critical shear stress that are smaller by a factor of where about 1.6. It is interesting that several researchers have found τ ρ 2 that Shields crtical shear stress values are indeed higher than bc u * c critical bed shear stress for initiation of motion; those observed. More discussion on the internal friction angle is given below when the model of Wiberg and Smith (1987) u* c critical shear velocity; ρ water density; is presented. ρ ρ /ρ Although there are a number of assumptions made in its R ( s ) is the submerged specifi c gravity of the sediment; derivation, the mechanistic Ikeda-Coleman-Iwagaki model g acceleration of gravity; and (Eq. 2.55a) makes it possible to visualize the sources of D sediment particle diameter. uncertainty (i.e., angle of repose, drag and lift coeffi cients, particle location, etc.) and helps to understand why it is so dif- The most relevant fact about the mechanistic approach fi cult to characterize the threshold condition with a determin- to this problem of initiation of motion relates to the pos- istic model (e.g., Bettess 1984; Lavelle and Mojfeld 1987; sibility of obtaining an explicit formulation of the rela- Komar 1996; Papanicolau et al. 2001; Shvidchenko and tion explored by Shields with dimensional analysis and Pender 2001; Niño et al. 2001; Dancey et al. 2002). Recently experiments. Eq. (2-55a) can be evaluated with the aid of the role played by turbulence on initiation of motion has Eq. (2-51f ) and certain realistic assumptions about the inter- been examined by Zanke (2003), who found that turbulence- φ nal angle of friction , and the drag c D and lift c L coeffi cients. induced fl uctuations in the lift force make particles “lighter” As an example, two internal friction angles are considered, and easier to move. φ 40° (μ 0.84) and φ 60° (μ 1.73), and the fol- lowing assumptions are made: k 2D , and c 0.85 c . s L D 2.4.3 Shields Diagram It is furthermore assumed that c D is given as a function of / uf D v according to the standard drag curve for spheres (i.e., Shields (1936) conducted his set of pioneering experi- Eq. 2-46d). A plot of Eq. (2-55a) is shown in Fig. 2-17, ments to elucidate the conditions for which sediment grains together with the data of Shields (1936). Considering all the would be at the verge of moving. While doing this, Shields assumptions made for developing the threoretical model, the introduced the fundamental concepts of similarity and agreement is quite reasonable. The best agreement between dimensional analysis and made a set of observations that the Ikeda-Coleman-Iwagaki model and Shields observations have become legendary in the fi eld of sediment transport is found for φ 60° ( μ 1.73). Such friction angle is rather (Kennedy 1995). Shields deduced from dimensional analy- τ * high but is not possible to know the exact value of this param- sis and fl uid mechanics considerations that c should be a / eter for the sediment used by Shields in his experiments, and function of shear Reynolds number u * c D v , as implied by Eq. whether or not incipient transport conditions were present (2-55a). The Shields diagram is expressed by dimensionless

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 4488 11/21/2008/21/2008 5:43:075:43:07 PMPM threshold condition for sediment movement 49 τ / combinations of critical shear stress bc , sediment and less critical shear stress for values of Re * u * D v in excess γ γ water specifi c weights s and , respectively, sediment size of 500, while Gessler (1971) suggests using a value of τρ/ τ * 0.046 for such condition. D, critical shear velocity u*cbc, and kinematic c τ* viscosity of water ν. These quantities can be expressed in any The value of c to be used in design depends on the par- consistent set of units. The Shields dimensionless parameters ticular case at hand. If the situation is such that grains that are related by a simple expression, are moved can be replaced by others moving from upstream, some motion can be tolerated, and the values from the Shields curve may be used. On the other hand, if grains τ ⎛ u D⎞ τ* bc* c c F* ⎜ ⎟ (2-57) removed cannot be replaced as on a stream bank, the Shields ρνgR D ⎝ ⎠ τ* value of c are too large and should be reduced. As already mentioned it is well known from observations by Neill and The Shields diagram shown in Fig. 2-18 was originally Yalin (1969) and Gessler (1970) that Shields original values prepared by Vanoni (1964). This diagram is the predecessor for initiation of motion of coarse material are too high and of the one that fi nally appeared when Manual 54 was fi rst should be divided by a factor of 2 for engineering purposes. published in 1975 (Vanoni 2006, p. 57). A modern account As fi rst noticed by Vanoni (1964), the Shields diagram of the Shields diagram and its history can be found in is not practical in the form of Fig. 2-18, because in order τ* to fi nd the critical shear stress for incipient motion τ , one Kennedy (1995). Critical Shields values c are commonly bc τρ/ used to denote conditions under which bed sediment par- must know the critical shear velocity u*cbc . The ticles are stable but on the verge of being entrained. The τ* relation can be cast in explicit form by plotting c versus Rep , curve in the Shields diagram was originally introduced by noting the internal relation Rouse (1939), whereas the auxiliary scale was proposed by Vanoni (1964) to facilitate the determination of the criti- gR D D 12/ cal shear stress τ once the submerged specifi c gravity, the u** D u * bc ()τ Rep (2-58) particle diameter D and the kinematic viscosity of water ννgR D ν are specifi ed. It is known that the values obtained from the Shields diagram (Fig. 2-18) for initiation of motion are indeed larger than those observed by other research- ρ ρ where R s is the submerged specifi c gravity of ers, in particular for coarse material. For example, Neill ρ (1968) gives τ * 0.03 instead of 0.06 for the dimension- the sediment. c

Fig. 2.18. Shields diagram for initiation of motion (source Vanoni, 1964).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 4499 11/21/2008/21/2008 5:43:075:43:07 PMPM 50 sediment transport and morphodynamics Brownlie (1981) used this relation to convert the original conducted a series of experiments and observed that for fi ne- τ* Shields diagram into one with c versus R ep . Similar dia- grained, noncohesive sediments the critical shear stresses * grams using D (see Eq. 2-49b) instead of Rep have been can be estimated with the following relation advanced among others by Bonnefi lle (1963), Smith (1977), van Rijn (1984a), García and Maza (1997), and Soulsby and τ* 0.261 Whitehouse (1997). A useful fi t to the Shields data was pro- cep0. 135R (2-59c) posed by Brownlie (1981, p.161): which is valid for the range 0.056 Rep 3.16. Equations (2-59a) and (2-59c) merge for R 4.22. * 0.6 0.6 ep τc 0.22 Rep 0.06 exp () 17.77 R ep (2-59a) Lavelle and Mofjeld (1987) used the pioneering bed- stability observations made by Grass (1970) to question the existance of a deterministic value of critical stress for incipi- With this relation, the value of τ * can be readily computed c ent motion as foreseen by Shieds and to promote a probabi- when the properties of the water and the sediment are given. listic approach to address threshold conditions for initiation As already mentioned, to be on the safe side the values given of motion and entrainment into suspension. Along the same by Eq. (2-59a) should be divided by 2 for engineering pur- line of thought, Lopez and García (2001) have proposed a poses, resulting in the following expression risk-based approach showing that the Shields diagram can be interpreted in a probabilistic way. At the same time, there is also evidence that the Shields diagram is quite useful for * 1 ⎡ 0.6 0.6 ⎤ τc ⎣0.22 Rep 0.06 exp () 17.77 R ep ⎦ (2-59b) 2 fi eld application. For instance, Fisher et al. (1983) experi- mentally investigated incipient motion of organic detritus and inorganic sediment particles on sand and gravel beds This equation is plotted in the modifi ed Shields diagram and found that their observations followed the character- shown in Fig. 2-19, where the size ranges for silt, sand and istics of the Shields diagram. Recently, Marsh et al (2004) gravel are also shown. tested the Shields approach together with three other meth- For fi ne-grained sediments (silt and fi ner), the Shields ods available in the literature and showed that it is still one diagram does not provide realistic results. Mantz (1977) of the best methods available for sand-bed streams. More

Fig. 2-19. Modifi ed Shields diagram (after Parker 2005).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5500 11/21/2008/21/2008 5:43:095:43:09 PMPM threshold condition for sediment movement 51 recently, Sarmiento and Falcon (2006) introduced the novel Eq. B.5 in Appendix B). In particular, this relation is very idea of using spatially-averaged (over many particles) shear similar to the one proposed by Neill (1968) for initiation of stresses to defi ne incipient conditions for particle motion at motion of coarse material low transport rates. Buffi ngton (1999) thoroughly reanalyzed Shields’ work, 1/6 U ⎛ H ⎞ pointing out some inconsistencies in the way Shields obser- (vi) 1. 414⎜ ⎟ D vations had been interpreted and used by others. This moti- RgDRR ⎝ RR ⎠ vated a discussion that analyzed the universality of the Shields diagram in the context of sand- and gravel-bed rivers Suppose that the riprap is to be designed to be able to (García 2000), resulting in the river sedimentation diagram withstand a 10-year fl ood, at which the mean fl ow velocity presented below (Fig. 2-29). U is estimated to be 3 m/s and the fl ow depth H is estimated 2.4.3.1 Application to Riprap Sizing and Flow Com- to be 2.5 m. Using a submerged specifi c gravity R 1.65, petence It is worthwhile to show how knowledge about Eq. (v) gives a riprap size D RR 15.2 cm (6 inches), and velocity distribution and initiation of motion can be used for Neill’s relation yields DRR 9.3 cm (3.65 inches). A safety a practical problem. Consider the design of a riprap cap to factor should be built into the design and if the material is protect contaminated river-bed sediment against erosion. A poorly-sorted, the riprap size should be selected so that D RR geotextile or a fi lter layer can be used to cover the contami- D 90 . The reader is referred to Appendix B for a full treatment nated river-bed portion and then this layer can be protected of this topic.

with riprap material having a size D RR . The riprap size has to A similar analysis can be used to estimate the fl ow be determined to ensure the stability of the cap design. competence to move coarse river-bed material of a given As introduced earlier, the Manning-Strickler relation for size. In this case, the question would be what mean fl ow fl ow resistance is velocity and depth are needed to move coarse material of a certain size? This is a typical problem when analyzing 1/6 U ⎛ H ⎞ salmonid spawning gravel streams (e.g., Buffi ngton et al. (i) 8.1 ⎝⎜ ⎠⎟ 2004). u* ks

Here 2.4.4 Yalin and Karahan Diagram In a study of temperature effects on initiation of motion, kD α (ii) ssRR Taylor and Vanoni (1972) reported that small but fi nite Typical values for the coeffi cient α can be found in amounts of fi ne-grained sediment were transported in fl ows s with values of τ * well below those given by the Shields curve. Table 2-1. c The critical condition for motion of the coarse material They found that as the size of sediment grains decreases, making up the riprap can be written as the dimensionless critical shear stress increases more slowly than one would infer by extrapolating the Shields curve. Similar observations were made by Mantz (1977; 1980) but u2 τ* *c (iii) c the most conclusive evidence for such behavior was pro- RgDRR vided by Yalin and Karahan (1979) through carefully con- ducted experiments. τ* Where c should be between 0.02 and 0.03 depend- Yalin and Karahan (1979) compiled a substantial number ing upon how broadly the bed is covered with riprap (see of data while conducting their own set of experiments with Fig. 2-19). Combining the above relations yields sand sizes ranging from 0.10 to 2.86 mm for both laminar τ* and turbulent fl ow conditions. In this diagram, Y cr c and 1/6 1/2 ⎛ ⎞ X Re . They used glycerine in some of the experiments to U τα* 1/6 H cr * (iv) 81. ()cs⎜ ⎟ increase the thickness of the viscous sublayer, thus making RgD ⎝ D ⎠ RR RR it possible to observe initiation of motion under laminar and turbulent fl ow conditions. As shown in Fig. 2-20, Yalin and For example, if τ * 0.03 and α 2.5, this relation c s Karahan were able to elucidate the nature of transport incep- reduces to: tion conditions for a wide range of grain Reynolds number 1 Re u D /v . For Re 70, hydraulically rough conditions, ⎛ ⎞ * *c * U H τ* (v) 1. 414⎜ ⎟ c takes a value of about 0.045. For values of Re* 10, the * RgD ⎝ DRR ⎠ τ RR relation between c and Re * depends on the fl ow regime, i.e., whether the fl ow is laminar or turbulent. This equation is very similar to the many empirical Like the original Shields diagram, the Yalin-Karahan dia- τ* equations that have been determined for riprap design (see gram can only be used in an iterative way since c appears

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Fig. 2-20. Diagram for Initiation of Motion, Yalin and Karahan (1979).

both in the abscissas and in the ordinates. To obtain an For laminar fl ow conditions, explicit set of curves, a transformation similar to Eq. (2-58) can be introduced, as follows 921. 0. 1439 ⎡5. 6243⎤ τ 0.exp 0084 ; *c 0. 352 ⎢ ⎥ 13/ 13/ D*c ⎣ D*c ⎦ ⎛ Re2 ⎞ ⎛ ()γγ g ⎞ (2-61d) DD⎜ *c ⎟ ⎜ s ⎟ (2-60) 0. 2164 D 11. 252 *c ⎝ τ* ⎠ ⎝ γ ν2 ⎠ *c c

γ γ Here and s specifi c weight of water and sediment, respectively. García and Maza (1997) have proposed the fol- These relations can be used to estimate critical shear stress lowing useful fi t to the Yalin-Karahan data: for a wide range of sediment sizes and fl ow conditions. Dey For turbulent fl ow conditions, (1999) has also proposed a rather simple model for threshold conditions that captures the behavior displayed by the Yalin τ 0. 377 **cc0... 137DD; 0 1074* c 2 084 and Karahan (1979) laboratory observations. (2-61a) 2.4.5 Wiberg and Smith Diagram for Heterogeneous Sediments 2. 453 0. 178 ⎡ 31. 954 ⎤ τ* 0.exp 0437 ⎢ ⎥ Most of the work on initiation of motion has been done c 0. 7303 D 10 D*c ⎣ *c ⎦ (2-61b) for uniform size sediment. One exception is the model advanced by Wiberg and Smith (1987). They derived an 2. 084 D*c 47. 75 expression for the critical shear stress of noncohesive sedi- ment using a balance of forces on individual particles very similar to the one shown previously. For a given grain size and density, the resulting equation depends on the near-bed τ* (2-61c) cc0.. 045; D* 47 75 drag force, lift force to drag force ratio, and particle angle

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5522 11/21/2008/21/2008 5:43:115:43:11 PMPM threshold condition for sediment movement 53 of repose. They were able to reproduce the observations of Fig. 2-21. The agreement with observations made by a num- Shields for uniform size sediments as well as initiation of ber of authors is excellent. φ motion in the case of sediment mixtures. They found that To evaluate the angle of repose 0 of natural particles in for mixed grain sizes the initiation of motion also depends mixed-size beds, the observations made by Miller and Byrne on the relative protrusion of the grains into the fl ows and the (1966) with naturally sorted sediments were used. The fol- particle angle of repose. The relation obtained by Wiberg lowing geometric relationship was found to represent the and Smith (1987) for natural sediment is practically iden- data well, tical to the Ikeda-Coleman-Iwagaki relation (Eq. 2-55b) presented earlier for nearly spherical particles, and can be ⎡ Dk/ z ⎤ written as φ cos 1 ⎢ s * ⎥ (2-62b) 0 / ⎣ Dks 1 ⎦ ()φα α * 4 tan0 cos sin 1 τ (2-62a) where z 0.02 is the average level of the bottom of c 3 ()cc tanφ Fzz2 ()/ * DL0 o an “almost moving” grain and depends on particle sphe-

ricity and roundness. Here k s is the equivalent Nikuradse where roughness length. Equation (2-62b) was found to represent the data of Miller and Byrne (1966) well for D /k 0.5. α bed slope angle; s Computed curves for nondimensional critical shear stress φ angle of repose of the grains; c and c drag and lift 0 D L for a range of ratios of particle diameter to bed roughness, coeffi cients, respectively; / D k s 0.5 5.0, are shown in Fig. 2-22a. A large ratio of / / and the function F u(z) u* is the logarithmic function D k s indicates a larger particle on a smaller bed, and vice (i.e. Eq. 2-4) that relates the effective fl uid velocity act- versa. In this plot, the critical roughness Reynolds num- / ing on the particle to the shear velocity. Wiberg and Smith ber ( R* ) cr (u * )cr ks v is a characteristic of the bed. Thus, (1987) evaluate the logarithmic function with an equation for any bed roughness, the intersections of a vertical line τ for the velocity distribution fi rst proposed by Reichardt in through some ( R* ) cr and the ( * ) cr curves for the appropriate / the early 1950s, which provides a smooth transition between D k s values determine the critical shear stress for the sizes the viscous sublayer and the outer portion of the velocity of material present in the bed. profi le (Schlichting 1979, p. 601). Critical shear veloci- Wiberg and Smith (1987) found that their model (i.e., ties computed with Eq. (2-62a) as a function of nominal Eqs. (2-62a) and (2-62b)) reproduced such observations. grain diameter for quartz density sediment are shown in As in the case of the original Shields diagram, Wiberg and

Fig. 2-21. Calculated critical shear velocity as a function of grain diameter (after Wiberg and Smith 1987).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5533 11/21/2008/21/2008 5:43:125:43:12 PMPM 54 sediment transport and morphodynamics

(a)

(b)

Fig. 2-22. (a) Calculated nondimensional critical shear stress as a function of critical roughness Reynolds number for values of particle diameter to bed roughness scale. (b) Calculated nondimen- sional critical shear stress as a function of nondimensional particle diameter for values of particle diameter to bed roughness scale (after Wiberg and Smith, 1987).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5544 11/21/2008/21/2008 5:43:185:43:18 PMPM threshold condition for sediment movement 55 Smith found it more useful to express the critical shear stress For H / D 744.2 in terms of a parameter that depends only on grain size and density and on fl uid density and viscosity. As shown in Fig. U ⎛ H ⎞0. 1283 2-22b, the abscissa in the critical shear stress diagram is c 1. 630⎜ ⎟ (2-63a) ζ 1/3 ⎝ ⎠ given by a variable K* 0.0047( *) where RgD D

2 / D3 ρρ []()RDk/ For H D 744.2 ζ s g * cr s (2-62c) * 2 ν ρτ() 0. 3221 * cr U ⎛ H ⎞ c 1. 453⎜ ⎟ (2-63b) RgD ⎝ D ⎠ In this fashion, iteration is not needed to fi nd the critical shear stress for a particle of diameter D in a bed with charac- An inspection of Fig. 2-23 suggests that Eq. (2-63a) cor- teristic roughness length k s . responds to fl ow conditions representative of gravel-bed A systematic analysis of eight decades of incipient and -bed streams (i.e., low relative fl ow depth), while motion studies, with special reference to gravel-bed riv- Eq. (2.63b) corresponds conditions commonly found in ers was conducted by Buffi ngton and Montgomery (1997). sand-bed streams (i.e., large relative fl ow depth). Notice Different models available in the literature to estimate that the general form of these relations is very similar to the entrainment into motion of sediments having mixed grain one obtained in Section 2.4.3.1. The Lischtvan-Lebediev sizes and densities are reviewed by Komar (1996). The relations are widely used in Latin America for the design of work of James (1990) with spheres and Carling et al (1992) stable channels and to estimate potential sediment erosion employing regularly shaped particles (rods, cylinders, discs conditions in sand-bed rivers (e.g. Schreider et al. 2001). and cubes) illustrates that grain-shape variability and grain orientation are important to entrainment, resulting in a range of stresses for particles that have otherwise the same weight. 2.4.7 Effect of Bed Slope on Incipient Motion Bridge and Bennett (1992) have developed a mathematical 2.4.7.1 Granular Sediment on a Sloping Bed The model for entrainment and transport, which accounts for dif- work of Shields and others on initiation of motion applies ferent grain sizes, shapes and densities. Niño et al (2003) only to the case of nearly horizontal slopes. Most streams, were able to measure the effect of grain-size variability on particularly in mountain areas, have steep gradients, creating sediment entrainment into suspension with the help of labo- a need to account for the effect of the downslope compo- ratoty experiments. More information about initiation of nent of gravity on the initiation of motion. In fact, the model motion and transport of gravel and sediment mixtures can be of Wiberg and Smith (1987) as given by Eq. (2-62a) does found in Chapter 3. account for the effect of streamwise channel slope. As shown for the case of negligible longitudinal slope, 2.4.6 Lischtvan-Lebediev Diagram for Maximum the effect of the streamwise bed slope on incipient sedi- Permissible Flow Velocity ment motion can be illustrated by considering the forces (lift, drag, buoyancy, and gravity) acting on a particle lying In practice, it is often convenient to estimate the fl ow veloc- in a bed consisting of similar particles over which water ity necessary for initiation of motion and sediment erosion. fl ows. Such analysis yields the equation (Chiew and Parker A number of researchers have conducted fl ume experiments 1994) to collect data relating grain sizes and densities to fl ow velocities, discharges and mean stresses needed to initi- * ate particle movement (e.g. Miller et al. 1977). In the early τ α ⎛ tan α⎞ c cos α ⎜1 ⎟ (2-65a) 1920s, Fortier and Scobey fi rst introduced the concept of τ* ⎝ φ ⎠ co tan maximum permissible fl ow velocity (Chow 1959, p. 165). The maximum permissible fl ow velocity, or the nonerosible fl ow, is the greatest mean velocity that will not cause erosion where of the channel bed. Lischtvan and Lebediev used observa- φ angle of repose; tions made in Russian channels (Lebediev 1959) for wide τ* cα critical shear stress for sediment on a bed with a ranges of quartz sediment sizes (0.005 mm D 500 mm) α longitudinal slope angle ; and and fl ow depths (0.40 m H 10 m) to obtain values of τ* co critical shear stress for a bed with very small slope. the maximum permissible fl ow velocity Uc as a function of / τ* the relative fl ow depth H D (Garcia and Maza 1997). The The value of co can be found from the Shields diagram Lischtvan-Lebediev data are plotted in dimensionless form with Eq. (2-59), or with Eq. (2-61). Eq. (2-65a) is for posi- in Fig. 2-23. Two curves have been found to fi t the data by tive α , which applies for downward sloping beds. For beds García and Maza (1997). with adverse slope, α is negative and the term tan α /tanϕ in

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5555 11/21/2008/21/2008 5:43:195:43:19 PMPM 56 sediment transport and morphodynamics

Fig. 2-23. Lischtvan-Lebediev diagram for maximum permissible fl ow velocity.

Eq. (2-65a) is positive. In terms of shear velocities the rela- equation mathematically equivalent to Eq. (2-65a) of the tion takes the form form

τ* (φα ) u ⎛ tanα⎞ cα sin *c cosα 1 (2-65b) (2-65c) ⎝⎜ φ ⎠⎟ τ* sinφ u*o tan co

An expression similar to Eq. (2-65a) was derived by They found that the condition for inception of motion Lysne (1969), who also performed a set of experiments on depends on the slope angle as well as on the Reynolds num- the effect of the bed slope on the incipient motion of sand ber of the fl ow. Their recommendation is that Eq. (2-65c) can in a closed channel. Lysne’s results agree very well with the be used for slope angles all the way up to the angle of repose φ ° τ* curve given by equation (2-65a) for a value of 47 . A togther with Shields criteria to estimate co . Whitehouse similar result was found by Fernandez-Luque and Van Beek et al. (2000) tested Eq. (2-65c), fi nding good agreement with (1976), who also fi tted a relationship similar to Eq. (2-65a) experimental observations. This relation has been rediscov- to their results for the incipient motion of sand, gravel, and ered many times since Armin Schoklitsch introduced it for magnetite in open channel fl ow on sloping beds. the fi rst time in the early 1900s. Chiew and Parker (1994) conducted a set of labora- Several investigators, such as Stevens et al. (1976), tory experiments with a closed duct, to test the validity of Fernandez Luque and van Beek (1976), Howard (1977), Allen Eq. (2-65b) for both positive and adverse slopes. The (1982), Smart (1984), Dyer (1986), Whitehouse and Hardisty results are plotted in Fig. 2-24. In general, good agreement (1988); Chiew and Parker (1994), Iversen and Rasmussen is observed between the experimental observations and the (1994), Dey (1999), and Dey and Debnath (2000), have values predicted with Eq. (2-65b). used relationships similar to either Eq. (2-65a), Eq. (2-65b), Lau and Engel (1999) used dimensional analysis cou- Eq. (2-65c) to determine the critical shear stress for sedi- pled with the observations made by Fernandez-Luque and ment lying on a nonhorizontal slopes. Stevens et al. (1976) Van Beek (1976) and Chiew and Parker (1994) to obtain an used such relationship to investigate the factor of safety

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Fig. 2-24. Effect of streamwise bedslope on critical shear velocity. Curve correspond to Eq. 2.65b (modifi ed from Chiew and Parker 1994).

for riprap protection, whereas Smart (1984) used them to stable channels in coarse material (see Appendix B). Thus it evaluate sediment transport rates in a steep channel. Kostic is worthwhile to present a more detailed analysis. et al. (2002) used Eq. (2-65a) to study the foreset slope of In the present simplifi ed analysis, the fl ow velocity pro- prograding deltas in lakes and reservoirs. Whitehouse and fi le is again taken to be logarithmic upward normal from Hardisty (1988), Graf et al. (2000), and Damgaard et al. the bed. A force balance is done for a particle located on (2003) used similar concepts to study the inception of bed a side slope, as shown in Fig. 2-25. The fl ow is taken to load transport on steep slopes. The effect of seepage on inita- be in the streamwise direction, parallel to the side slope.

tion of motion has been analyzed by Oldenziel and Brink The vectorial fl uid drag force D f acting on a particle is thus (1974), Cheng and Chiew (1999), and Dey and Zanke given as (2004) . 2.4.7.2 Threshold Condition on Side Slopes The 2 1 ⎛ D⎞ G D ρπ cue2 (2-66a) analyses presented above apply strictly to the case of fl ow fDf⎝⎜ ⎠⎟ 1 on a nearly horizontal or sloping bed in the streamwise 22 direction that is horizontal in the transverse direction (i.e., The gravitational force F has a transverse as well as a negligible transverse bed slope). An important problem in g engineering applications is the case of sediment particles on downward normal component: a side slope (Simons and Senturk 1992). This problem is of GG particular relevance to the design of riprap protection and FFeFegg22 g 33 (2-66b)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5577 11/21/2008/21/2008 5:43:225:43:22 PMPM 58 sediment transport and morphodynamics G G G Further reducing with the aid of Eqs. (2-54) and (2-55), where e 1 , e 2, and e3 are unit vectors in the streamwise, trans- verse, and downward normal to the side directions, respec- it is found that tively.

12/ ⎡ ⎛ ⎞ 2 ⎤ 3 2 4 4 ⎛ D⎞ ⎢()τθ* sin ⎥ ρπ( θθ) c ⎜ 2 ⎟ ()FFgg23,, Rg ⎜ ⎟ sin cos (2-66c) ⎢ ⎝ 3cF ⎠ ⎥ 32⎝ ⎠ ⎣ D ⎦ (2-69b) μ c 4 θμL τ* 2 cos c and θ denotes the local transverse angle of the side slope, as 3cFD cD illustrated in Fig. 2-25. The lift force is given as The case of a transversely horizontal bed is recovered by setting θ 0. The critical Shields stress is found to be given by Eq. (2-55a) for this case. This value is denoted as 2 1 ⎛ D⎞ G τ* , the subscript o denoting that the bed is horizontal in L ρπ⎜ ⎟ cue2 (2-67) co fLf22⎝ ⎠ 3 the transverse direction. Using this value to normalize the τ* θ value c obtained on a side slope of angle , Eq. (2-69b) reduces to The Coulomb resistive force acting on a grain has a G μ / magnitude given by Fegf33 L. As shown in the dia- 12 ⎧⎛ τ* ⎞ 22⎡() μ / ⎤ ⎫ gram, under critical conditions, it must precisely balance ⎪ 1 ccLD ⎪ ⎨⎜ c ⎟ ⎢ sinθ⎥ ⎬ the vectorial sum of the impelling forces due to fl ow ( D ) ⎪⎝ τ* ⎠ ⎣ μ ⎦ ⎪ G f ⎩ co ⎭ (2-69c) and due to the transverse downslope pull of gravity (Fg2e2) . These conditions on magnitude and direction lead to the c ⎛ τ* ⎞ ()1 μcc/ cosθμ L ⎜ c ⎟ following result for threshold conditions: L D ⎝ τ* ⎠ cD co

GG22 2 Equation (2-69c) is a quadratic polynomial in τ * / τ * , and μ2 Fe L D Fe (2-68) c co gffg33 22 as such is easily solved. A solution is shown in Fig. 2-26, which has been evaluated for the case μ 0.84 ( φ 40°) and c 0.85c . It is also assumed that c is given as a Substituting Eqs. (2-66a), (2-66b), (2-66c) and (2-67) L D D function of u D /v according to the standard drag curve for into Eq. (2-68) and reducing, the following relation is f spheres (i.e. Eq. 2-46d). As can be seen there, the Shields obtained: τ* θ stress takes the value co on a horizontal bed ( 0 ). It progressively decreases as the side slope angle θ increases, / ⎡ 2 ⎤12 reaching a value of 0 at the angle of repose. ⎛ 2 ⎞ ⎛ ⎞ 2 ⎢ u f 4 ⎥ Many methods for stable channel design, starting with ⎜ ⎟ sinθ ⎢ ⎝⎜ ⎠⎟ ⎥ the classic work of Glover and Florey in the early 1950s, ⎝ RgD⎠ 3 cD ⎣ ⎦ (2-69a) make use of Eq. (2-69c) to design a channel in coarse allu- ⎛ u2 ⎞ vium that is at the threshold for sediment motion but is 4 cL f μθ⎜ cos ⎟ stable (e.g., Li et al. 1976; Diplas and Vigilar 1992). Parker ⎝ 3c c RRgD⎠ D D (1978) also used this approach to analyze fl ow in self- formed straight rivers with mobile beds and stable banks. More material on stable movable-bed channels can be found in Chapter 7. If the lift force is neglected (i.e., cL 0), Eq. (2-69c) reduces to the well-known Lane (1955) relation,

2 τ* ⎛ tan θ ⎞ c cos θ 1 ⎜ ⎟ (2-70) τ* ⎝ φ ⎠ co tan

Equation (2-70) has also been derived for application to coastal sediment transport problems by Fredsøe and Fig. 2-25. Defi nition diagram for particle located on a side slope. Deigaard (1992, p. 204). Whitehouse et al (2000) tested

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5588 11/21/2008/21/2008 5:43:235:43:23 PMPM threshold condition for sediment movement 59

Fig. 2-26. Variation of normalizedcritical Shields stress for initiation of motion as a function of side slope angle as predicted by Eq. 2.69c.

the values predicted by Eq. (2-70) with the observations sloping bed. Their analysis was extended by Seminara made by Ikeda (1982) and found reasonable agreement. et al. (2002) to include the effect of lift force. While study- Christensen (1972) found out that the critical shear stresses ing coastal sediment transport, Calantoni (2002, p. 77) gen- estimated with Eq. (2-70) have a tendency to be too con- eralized the analysis of Fredsøe and Deigaard (1992) and servative and proposed an alternative method that takes obtained a quadratic equation for the threshold condition into account the ratio between the bed roughness and the for motion, similar to Eq. (2-69c), which shows promise grain size. As this ratio increases, Lane’s Eq. (2-70) and for practical use. The positive root of the equation gives Christensen’s method give identical results. The method of an equation that can be used to estimate the critical shear Wiberg and Smith (1987) presented above also takes into stress for motion of a particle located on a bed surface hav- / account the effect of the relative roughness D k s on the eval- ing a longitudinal (parallel the fl ow direction) slope angle uation of critical shear stress condition for motion. James α and a transverse (perpendicular to fl ow direction) slope (1990) has also provided useful information on the effect of angle θ , such parameter on initiation of motion. Similar approaches, which follow the so-called grain pivoting model, have been ⎛ 2 ⎞ suggested by Slingerland (1977) and Komar and Li (1988) τ* ⎛ tanθ⎞ tanα c cosαθ⎜ cos 1 ⎜ ⎟ ⎟ (2-71a) among others (Komar, 1996). Bridge and Bennett (1992) τ* ⎜ ⎝ tanφ⎠ tanφ ⎟ have also advanced a model that accounts for bedslope co ⎝ ⎠ effects on initiation of motion. 2.4.7.3 Threshold Condition for Motion on an Notice that when α 0, Eq. (2-71a) reduces to Eq. (2-70) Arbitrarily Sloping Bed The general case of an arbitrarily and when θ 0, Eq. (2-71a) reduces to Eq. (2-65a). sloping bed was fi rst treated analytically by Kovacs and Calantoni and Drake (1999) used Eq. (2-71) to develop a dis- Parker (1994), who developed a vectorial equation for sedi- crete-particle model for bed load transport in the surf zone ment threshold on a combined transverse and longitudinal that accounts for variations in bottom slope.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 5599 11/21/2008/21/2008 5:43:255:43:25 PMPM 60 sediment transport and morphodynamics Duan et al. (2001) and Duan and Julien (2005) have used Table 2-4 Sediment Load Classifi cation the following formulation for sediment transport modeling in meandering channels, Classifi cation system Based on Based on 12/ mechanism of particle τ* ⎡ ⎛ θ⎞ 2 ⎤ ⎛ (φα )⎞ c tan sin (2-71b) Total sediment load transport size cos θ ⎢1 ⎜ ⎟ ⎥ ⎜ ⎟ τ* ⎢ ⎝ φ⎠ ⎥ ⎝ φ ⎠ co ⎣ tan ⎦ sin Suspended load Wash load Suspended Suspended load Bed- Notice that when θ 0, Eq. (2-71b) reduces Eq. (2-65c) bed-material load material load α which is also equivalent to Eq. (2-65a). When 0, Bed load Bed load Bed-material Eq. (2-71b) reduces to Eq. (2-65a). It should be clear that Eqs. (2-71a) and (2-71b) are mathematically equivalent. In the early 1960s, Norman Brooks provided an excellent theo- retical analysis of this problem in the context of river bends. It can be found in Vanoni (2006, p.64). example the wash load is many times larger than the load of Other efforts to estimate critical shear stress values for bed material transported as suspended and bed load. Notice sediments on arbitrarily sloping beds include the work of also that there are two peaks for the wash load associated Dey (1999, 2003). More research, including experiments with sediment grain sizes of about 0.001 mm and 0.025 mm. over a wide range of conditions that can be used to test and These might be related to the watershed activities taking improve different formulations, is needed on this important place at the time the observations were made. topic. Stream channel stability, bank erosion, and mean- dering channels are topics where the material covered previously plays a crucial role. This will become appar- 2.5.1 Sediment Transport Modes: Bed-Material Load ent in Chapter 7, Chapter 8, and in Appendix B “RipRap and Wash Load design.” The sediment transport processes that can be characterized with fl uid and sediment dynamics principles are those of bed 2.5 SEDIMENT TRANSPORT load and suspended load. In the former case, the particles roll, slide, or saltate along the bed, never deviating too far Sediment load in this manual refers to the sediment that is in above the bed. In the latter case, the fl uid turbulence comes motion in a river. There are two common ways of classifying into play carrying the particles well up into the water column. the sediment load as shown in Table 2-4. The fi rst divides the In both cases, the driving force for sediment transport is the sediment load according to the mechanism for transport into action of gravity on the fl uid phase; this force is transmitted bed load and suspended load. The second classifi es the load to the particles via drag. While it is possible to quantify the based on particle size into wash load and bed sediment load. mechanics of bed-material transport as suspended load The suspended load, as the term denotes, moves in suspen- and bed load, a similar analysis to assess the wash load sion and is that part of the load which is not bed load. Wash has proved rather elusive. Before considering bed-material load is fi ne sediment moving in suspension which makes up transport in more detail, the wash load will be considered a very small part, usually a few percent, of the sediment on next. Important questions are what is role of sediment in the bed. Wash load is commonly taken as the silt and clay the fl ow energy balance and how the division between wash fraction of the bed sediment, i.e., that fraction with grain load and bed-material load in sand-bed streams can be sizes fi ner than 0.062 mm. The bed sediment load consists of made? particles that are coarser than the wash load. The transport The fl oodplains of most sand-bed rivers often contain rate or discharge of wash load tends not to be correlated with copious amounts of silt and clay fi ner than about 0.062 mm. water discharge while discharge of bed sediment, both in This material is known as wash load because it often moves suspension and as bed load, is usually correlated with water through the river system in suspension without being present discharge. The total sediment load is made up of wash load, in the bed in signifi cant quantities (Colby 1957). Increased suspended (bed-material) load and bed load. Methods and wash load does not cause deposition on the bed, and decreased technologies for measuring sediment transport are covered wash load does not cause erosion, because it is transported at in Chapter 5 . well below capacity. This is not meant to imply that the wash In some rivers the different components of the sediment load does not interact with the river system. Wash load in the load can be clearly differentiated. This is the case of the water column exchanges with the banks and the fl oodplain Niger River, Nigeria, depicted in Fig. 2-27, where the dif- rather than the bed. Greatly increased washload, for example, ferent components of the sediment load were measured in can lead to thickened fl oodplain deposits with a consequent cubic meters per year by NEDECO (1959). In this particular increase in bank-full channel depth. fertility depends

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6600 11/21/2008/21/2008 5:43:265:43:26 PMPM sediment transport 61 cohesive stream banks can contribute to the wash load during bank full fl ow events. Mining activities can also contribute substantially to the wash load of river systems, with poten- tial environmental effects on estuarine and coastal areas (e.g., coral reefs). Despite its importance, a physical characteriza- tion of the wash load is not an easy task. By defi nition, the wash load is not determined by the hydraulic characteristics of a given river reach; hence it can not be computed (Einstein and Chien 1953). At the same time, sediment transport formulae apply only to bed- material transport and do not account for wash load. De Vries (1993) argues that there are at least two reasons why a quantita- tive distinction between bed-material load and wash load is necessary.

(i) For comparison of sediment transport predictions with values measured in the fi eld it is necessary to substract the wash load component. (ii) A reduction of the fl ow velocity in the direction of the current will make a fraction of the wash load become bed-material load (e.g., reservoir sedimentation).

Vlugter (1962) discriminated between sinking material and fl oating material. He argued that fi ne sediment particles (i.e., fl oating material) being moved downstream in a river add part of their potential energy to the fl ow and can be transported in suspension indefi nitetly as long as the fl ow conditions do not change. On the other hand, coarse grains (i.e., sinking material) require kinetic energy from the mean fl ow to remain in suspension. Bagnold (1962) arrived to a similar conclusion while studying turbidity currents, and called this condition “autosuspension.” Interestingly, both Vlugter and Bagnold ideas were very similar to those articu- lated a few years earlier by Knapp (1938) while looking at the energy balance in streams carrying suspended sediment. As pointed out by Jansen et al (1979) in their river engineer- ing book, the energy balance concept underlying the Vlugter and Bagnold arguments has not yet been accepted by every- one (e.g., Parker 1982). In order to better understand some of these ideas it is useful to consider the energy balance in sediment-laden fl ows. Consider a steady, uniform sediment-laden open-channel fl ow in a channel with a bed slope S, as described in Fig. 2-31. The role of fi ne sediment in the energy balance Fig. 2-27. Total Sediment Load in the upper Niger River, Nigeria can be observed by considering the average rate of work P (adapted from Jansen et al. 1979). Ordinates are in hundreds of g thousands of cubic meters per year. (i.e., dot product of momentum and velocity) done by grav- ity on the fl ow which can be approximated as follows, largely on the amount of wash load deposited by fl oods on a given fl oodplain over the years. This fact was well known PgSUHRgSCUHRgHCv≅ ρρ ρ by the Egyptians, who practice agriculture in the fl oodplains gs (2-72a) of the Nile River. Also of relevance, is the fact that contami- (12) () ( 3) nants such as PCBs and heavy metals are often attached to the fi ne-grained sediments that constitute the wash load. The In this simplifi ed energy balance relation, U mean fl ow wash load is controlled by land surface erosion (rainfall, veg- velocity, H fl ow depth, C mean volumetric concentra- ρ /ρ etation, land use) and not by channel-bed erosion. However, tion of suspended sediment, R ( s 1 ) submerged

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6611 11/21/2008/21/2008 5:43:265:43:26 PMPM 62 sediment transport and morphodynamics ρ ρ specifi c gravity of the sediment, s sediment density, This is the classical Bagnold criterion for turbidity water density, v s sediment fall velocity, and g gravita- currents (Bagnold 1962). It ensures that the sediment sup- tional acceleration. plies more energy than it consumes. The Bagnold criterion The physical signifi cance of the terms in Eq. (2-72a) can must be satisfi ed if a self-sustaining is to be identifi ed as follows: occur. This is a necessary condition but is not suffi cient as (1) Mean rate of energy input to the fl uid phase (i.e., described by Parker et al (1986) since the fl ow has to be water). capable of entraining sediment into suspension to sustain (2) Mean rate of energy input to the mean fl ow through itself. the solid phase (i.e. sediment). The analogous energy constraint for a self-sustaining, (3) Mean rate of energy loss by mean fl ow through tur- dilute (RC 0.1), open-channel suspension is found to be bulent mixing required for maintaining sediment in from Eq. (2-72a), suspension. The main input of energy to the mean fl ow through the sedi- US RC ≅ RC (2-72e) ment phase can thus be positive or negative, depending on vs IRC whether or not term (2) is greater than term (3). If term (2) is larger than (3), it means that the suspended sediment con- This condition was fi rst articulated by Knapp (1938) tributes energy to the fl ow. On the other hand, if (3) is larger and expressed mathematically by Vlugter (1942, 1962). An than (2) it means that the fl ow is expenging energy to keep open-channel suspension can guarantee that the Knapp- the sediment in suspension. However for a dilute open-chan- Vlugter criterion is satisfi ed, by lowering the suspended nel suspension (C 1), terms (2) and (3) are very small sediment concentration C , and thus its excess fractional compared to term (1). Thus the fl ow energetics is to a fi rst density RC, via sediment deposition. Vlugter (1962) used approximation independent of sediment concentration. It fol- the above criterion to design stable irrigation channels lows that whether or not term (2) is greater than term (3) has in Indonesia. According to Vlugter, sediment with a fall essentially nothing to do with whether the fl ow has enough velocity v that satisfi es the above condition constitutes the energy to sustain itself, since almost all the energy enters s fl oating material that does not require energy from the fl ow through the water via term (1). to be transported. The fl oating material is equivalent to the In the case of a turbid underfl ow or turbidity current overlain wash load. On the other hand, sediment with fall velocities by clear, still, nonstratifi ed water and fl owing down a submarine that do not satisfy the Knapp-Vlugter condition and that channel with a slope S (Fig. 2-59), the situation is drastically take energy away from the fl ow to be transported is dubbed changed. Clear water will not fl ow down a submarine channel the sinking material. The sinking material can be regarded or due to gravity in the absence of suspended sediment. as the bed-material load. Vlugter states that in practice, An analysis of the equations of motion (see Section 2.11.3) when the mean fl ow velocity U 0.5 m/s, all silt particles shows that the work done by the hydrostatic pressure gradient smaller than 0.07 mm appear to behave as fl oating mate- of the fl uid phase just cancels term (1), so that in fact there is no rial (i.e., wash load). This is close to the grain diameter of positive energy input to the fl uid phase. In the case of turbidity 0.062 mm commonly used to defi ne the wash load (e.g., currents, gravity acts on the solid phase which in turn drags Colby 1957). the fl uid phase downslope forming an underfl ow. The net mean De Vries (1993) has suggested that the Knapp-Vlugter energy input through the solid phase P is simply gs criterion (Eq. 2-72e) could be used to fi nd a tentative divi- sion between wash load and bed-material load in sand-bed PRgSCUHRgHCvρρ streams. There have also been attempts to use Bagnold’s gs s (2-72b) ideas, which are applicable only to turbidity currents as pre- (23) ( ) viously shown, for the analysis of self-sustaining suspen- sions in open-channels fl ows (e.g., Southard and Mackintosh 1981; Wang 1984). As could be expected, this has gener- Thus the only positive energy input into the turbidity cur- ated a substantial amount of discussion in the literature (e.g., rent is via term (2). It follows that term (2) must exceed term Parker 1982; Paola and Southard 1983; Nordin 1985a; Brush (3) for a self-sustaining turbidity current, 1993). It should be clear that Bagnold’s criterion does not correspond to an energy constraint on open-channel suspen- ρρ RgSCUH RgHCvs (2-72c) sions. The fundamental differences and similarities between sediment transport by rivers and turbidity currents are addressed in Section 2.11. This relation can be reduced to While conducting sedimentation studies in the Orinoco River in Venezuela, Nordin and Perez-Hernandez (1985) US 1 (2-72d) defi ned the wash load as the material that can be suspended v / s (i.e., u * vs 1.25) as soon as its motion at the bed is initiated

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6622 11/21/2008/21/2008 5:43:275:43:27 PMPM sediment transport 63 τ τ (i.e., b bc ). Mathematically, this condition can be defi ned gravel bed streams. Sand bed streams typically have values in dimensionless form by the relations of median bed sediment size between 0.1 mm and 1 mm (Fig. 2-12). The sediment tends to be relatively well sorted, τττ***2 111. Rfc when (2- 72f) with values of geometric standard deviation of the bed sedi- ment size varying from 1.1 to 1.5. Gravel bed streams typi- where cally have values of median size of the bed sediment exposed τ * dimensionless Shields stress parameter defi ned by on the surface of 15 mm to 200 mm or larger; the substrate Eq. (2-73a); is usually fi ner by a factor of 1.5 to 3 (Fig. 2-13). The geo- τ* metric standard deviation of the substrate sediment size is c dimensionless critical Shields stress for incipient motion (Fig. 2-19); typically quite large, with values in excess 3 being quite v s common. Although gravel and coarser material constitute the Rf dimensionless fall velocity (Eq. 2-46b) gRD dominant sizes, there is usually a substantial amount of sand stored in the interstices of the gravel substrate (Chapter 3). which is a function of Two dimensionless parameters provide an effective delin- gRDD eator of rivers into the above two types (García 2000). The R and can be evaluated, for example, with ep ν fi rst of these parameters is the dimensionless Shields stress for uniform fl ow conditions, defi ned as: Dietrich’s relation (Eq. 2-47). Nordin (1985b) suggests that the most practical way to τ HS apply this criterion in the fi eld is to plot the largest particle τ* b (2-73a) size that can be suspended and the largest particle size that ρgRD RD can be moved at the bed, as functions of the shear velocity

(u * ). The sediment diameter at which the two curves inter- sect defi nes the upper limiting size of the wash load, and where particles fi ner than this would not be found in appreciable τ b bed shear stress; quantities in the bed because they would go into suspension g gravitational acceleration as soon as their motion is initiated. In the case of the Orinoco ρ ρ and s water and sediment density, respectively; River, Nordin (1985b) found that the upper limiting size for R ( ρ ρ ) / ρ submerged specifi c gravity of the ° s the wash load is 0.095 mm for a water temperature of 25 C. sediment; A limiting size for the wash load in dimensionless form D mean sediment diameter; which implicitly includes the effect of temperature, can be H is the fl ow depth; and found from the dimensionless particle Reynolds number (R ) ep S is the stream slope which for steady, uniform fl ow is the where the curves for initiation of motion and suspension inter- same as the energy gradient. sect, as shown in Fig. 2-28 below. However, the relations pro- posed by Mantz (Eq. 2-59c) or Yalin and Karahan (Eq. 2-61a) The second of these parameters is the particle Reynolds for incipient motion of fi ne-grained sediment should be used number Rep defi ned as: instead of the Shields criterion (Eq. 2-59b) which does not work for the grain sizes found in the wash-load. gRDD While Nordin’s approach can provide an idea of the size R (2-73b) of the particles making up the wash load, because this fi ne- ep ν grained material is transported well below capacity what ultimately determines how much sediment is transported as where ν is the kinematic viscosity of water. This second wash load is the supply of fi ne sediment to a given river from parameter can be considered as a dimensionless surrogate its watershed and not the transport capacity of the river itself. for grain size. Watershed sediment yield is addressed in Chapter 17. Rivers were fi rst introduced into the Shields diagram by In what follows, the emphasis is placed on understanding Gary Parker in the early-1980s. Parker used these two param- the mechanics of bed load and suspended load transport in eters but his diagram, shown in Fig. 2-28, did not include open-channel fl ows, including morphological changes in riv- fi eld data (García 1999). Parker’s diagram, however, gave an ers, lakes and reservoirs, with the goal of providing the knowl- indication of the areas in the modifi ed Shields diagram cor- edge needed for sedimentation engineering. The mechanics responding to sand-bed and gravel-bed streams. of transport by turbidity currents is also considered, and used Motivated by a thorough review of the Shields diagram to analyze delta formation and reservoir sedimentation. done by Buffi ngton (1999), García (2000) used fi eld and laboratoty data to confi rm the early ideas of Parker, result- 2.5.2 Shields-Parker River Sedimentation Diagram ing in Fig. 2-29. This fi gure shows a plot of the values of the Alluvial rivers that are free to scour and fi ll during fl oods Shields stress (Eq. 2-73a) evaluated at bankfull fl ow versus can broadly be divided into two types: sand bed streams and particle Reynolds number (Eq. 2-73b) for six sets of fi eld

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6633 11/21/2008/21/2008 5:43:285:43:28 PMPM 64 sediment transport and morphodynamics

Fig. 2-28. Parker’s River Sedimentation Diagram (García 1999).

data: a) gravel bed rivers in Wales, UK (Wales); b) gravel The critical condition for suspension is given by the ratio bed rivers in Alberta, Canada (Canada); c) gravel bed riv- (Niño and García 1998; Lopez and García 2001) ers in the Pacifi c Northwest, USA (Pacifi c); d) single-thread sand streams (Sand sing); e) multiple-thread sand streams u* (Sand mult); f) large sand-bed rivers (Parana, Missouri, etc.); 1 (2-74) vs and g) large-scale laboratory experiments on bridge-pier scour conducted at St Anthony Falls Laboratory (SAFL), where u * is the shear velocity; and v s is the sediment fall University of Minnesota. velocity. Eq. (2-74) can be transformed into: There are three curves in the Shields-Parker river sedimentation diagram of Fig. 2-29 that make it possible to * 2 τ* τ s R f (2-75) know, for different values of ( , Rep ), if a given bed sediment grain will go into motion, and if this is the case, whether or not the prevailing mode of transport will be suspended load where: or bed load. The diagram can also be used to estimate what

kind of bed forms can be expected for different fl ow condi- 2 * u* tions and sediment characteristics. For example, ripples will τ s (2-76) develop in the presence of a viscous sublayer and fi ne-grained gRD sediment. If the viscous sublayer is disrupted by coarse sedi- ment particles, then dunes will be the most common type of denotes a threshold Shields number for suspension and Rf is bed form. given to be Eq. (2-46b), and can be computed for different As could be expected, the Shields-Parker diagram values of R ep with the help of Dietrich’s fall velocity relation (Fig. 2-29) also shows that in gravel-bed rivers, bed material given by Eq. (2-47a). is transported mainly as bed load. On the other hand, in sand- Finally, the critical condition for viscous effects (ripples) bed rivers, suspension and bed load transport of bed material was obtained with the help of the defi nition for the viscous coexist, particularly at high fl ows. The diagram is valid for sublayer thickness (Eq. 2-6) as follows, steady, uniform fl ow conditions, where the bed shear stress can be estimated with τ ρgHS (Eq. 2-1). The ranges for b ν silt, sand, and gravel are also included. In this diagram, the 11.6 1 (2-77) critical Shields stress for motion was plotted with Eq. (2-59a). u* D

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6644 11/21/2008/21/2008 5:43:295:43:29 PMPM sediment transport 65

Fig. 2-29. Shields-Parker River Sedimentation Diagram (after García 2000).

which in dimensionless form can be written as gravel-bed rivers at bank-full stage, which has implications for movable-bed physical modeling. If one wanted to model ⎛ ⎞2 in the laboratory sediment transport in rivers, the experi- * 11.6 τv ⎜ ⎟ (2-78) mental conditions would be quite different depending on ⎝ Rep ⎠ the river type in question. In order to satisfy similarity in a small-scale, river model, it would be necessary to satisfy the τ* In this equation, v denotes the threshold Shields number identities (García 2000) below which ripples can be expected. Relations (2-59a), (2-75), and (2-78) are the ones plotted ττ* * (2-79a) in Fig. 2-29. The Shields-Parker diagram should be useful model prototype for studies concerning and naturalization (Chapter 9), for it provides the range of dimensionless shear RR (2-79b) stresses corresponding to bankfull fl ow conditions for gravel- ep model ep prototype bed streams (0.01 τ * 0.2) and for sand-bed streams (0.6 τ* 6). Notice that the bank-full dimensionless for bank-full fl ow conditions. In most movable-bed mod- Shields shear stress is in general, an order of magnitude els, Froude similarity is enforced and Eq. (2-79a) is used larger for sand-bed streams than for gravel-bed streams. to achieve sediment transport similarity. However, sediment An interesting observation is that sand-bed streams are transport conditions and the associated bed morphology in in the transition between smooth and hydraulically rough a model, seldom precisely reproduce prototype conditions conditions, while gravel-bed streams are always hydrauli- because the second condition given by Eq. (2-79b) is rarely cally rough. This has implications, for instance, for the use satisfi ed. This leads to the common practice of using light- of Manning’s relation (Eq. 2-23a) which applies only to fully weight material (Table 2-2) to reproduce prototype condi- rough and turbulent hydraulic conditions (Yen 2002). tions in small-scale models (e.g., Shen 1990). However, this The Shields-Parker diagram also shows a very clear dis- does not imply that the observed in the model will tinction between the conditions observed in sand-bed and be the same as those in the prototype. The river sedimentation

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6655 11/21/2008/21/2008 5:43:305:43:30 PMPM 66 sediment transport and morphodynamics diagram provides a tool to quickly determine potential scale In the Bagnoldean formulation, however, a relation for effects in movable-bed model studies by simply plotting the the areal concentration of bedload particles as a function τ* values of ( ; Rep ) for model and prototype conditions in of boundary shear stress derives automatically from the Fig. 2-29. As discussed in Henderson (1966, p. 504), the imposition of a dynamic condition at the bed, according condition given by Eq. (2-79b) can be relaxed for suffi - to which the fl uid shear stress drops to the critical value

ciently large values of R ep (i.e. hydraulically rough fl ow) in for the onset of sediment motion. The physical implication both model and prototype. It is clear from Fig. 2-29, that this is that moving grains will extract enough momentum from would be possible only for the case of gravel-bed streams. the fl uid in the bed load layer, such that the fl uid stress at More information on movable-bed physical models can be the bed remains at the critical shear stress for motion. This found in Appendix C. dynamic condition is referred to as the Bagnold hypothesis or Bagnold constraint. The hypothesis was used by Owen (1964) to calculate sediment transport by saltation for the 2.6 BED LOAD TRANSPORT case of wind-blown sand. It is implicit in the bedload for- mulations of Ashida and Michiue (1972) and Engelund Since the publication of ASCE Manual 54 (Vanoni 1975), a and Fredsøe (1976) for nearly horizontal beds. Wiberg and signifi cant amount of work has been done to understand the Smith (1989), Sekine and Kikkawa (1992) and Niño and mechanics of bed load transport. Two schools of thought can García (1994; 1998), for example, have used the hypothesis be clearly identifi ed and they bear the name of two giants in to derive models of bed load transport on nearly horizon- the fi eld of sedimentation, Brigadier Ralph Alger Bagnold tal beds based on an explicit calculation of grain saltation. and Professor Hans Albert Einstein. Sekine and Parker (1992) used the Bagnold hypothesis to Bagnold (1956) defi ned the bed load transport as that in develop a saltation model for bed load on a surface with which the successive contacts of the particles with the bed a mild transverse slope, and Kovacs and Parker (1994) are strictly limited by the effect of gravity, whereas the sus- extended the analysis of Ashida and Michiue (1972) to the pended load transport is defi ned as that in which the excess case of arbitrarily sloping beds. Bridge and Bennett (1992) weight of the particles is supported by random successions have employed the Bagnold hypothesis to study the bed- of upward impulses imported by turbulent eddies. Einstein load transport of sediment mixtures. (1942, 1950), however, presented a somewhat different view Based on the most recently published formulations of bed of the phenomenon. Einstein defi ned bed load transport as load transport, then, it is possible to say that the fi eld as a the transport of sediment particles in a thin layer about two whole has tended away from the Einsteinean and toward the particle diameters thick just above the bed by sliding, rolling, Bagnoldean formulation. This notwithstanding, doubts have and making jumps with a longitudinal distance of a few par- been expressed from time to time concerning the Bagnold ticle diameters. The bed load layer is considered to be a layer hypothesis. For example, the experimental work of in which mixing due to turbulence is so small that it can- Fernandez-Luque and van Beek (1976) does not support the not directly infl uence the sediment particles, and therefore Bagnold hypothesis. A re-analysis of the data and formula- suspension of particles is impossible in the bed load layer. tion presented in Niño et al. (1994) and Niño and García Further, Einstein assumed that the average distance traveled (1994) caused Niño and García (1998) to cast further doubts by any bed load particle (as a series of successive move- on the hypothesis. Kovacs and Parker (1994) were forced ments) is a constant distance of about 100 particle diameters, to modify the hypothesis in order to obtain a well-behaved independent of the fl ow condition, transport rate, and bed theory of bed load transport on arbitrarily sloping beds. With composition. In Einstein’s view, saltating particles belong to the help of numerical experiments, McEwan et al. (1999) the suspension mode of transport, because the jump heights have found that only in the case of high sediment availability and lengths of saltating particles are greater than a few grain does the fl uid shear stress at the bed equal the critical stress diameters. On the other hand, Bagnold (1956, 1973) regards for intiation of motion. Seminara et al. (2002) and Parker saltation as the main mechanism responsible for bed load et al. (2003) have shown that Bagnold’s hypothesis breaks transport. down when applied to equilibrium bedload transport on beds Most research works that provide a mechanistic descrip- with transverse slopes above a relatively modest value that is tion of bed load transport under uniform equilibrium condi- well below the angle of repose. All of the above suggests that tions have fallen into one or the other school of thought. The formulae that make use of Bagnold’s hypothesis might only centerpiece of the Einsteinean formulation is the specifi ca- be able to predict bed load transport for certain conditions tion of an entrainment rate of particles into bed load trans- (Niño and García 1998). This notwithstanding the ideas of port (pick-up function) as a function of boundary shear stress Bagnold have nevertheless contributed substantially to the and other parameters. The work of Nakagawa and Tsujimoto understanding the physics of the sediment transport prob- (1980), van Rijn (1984a) and Tsujimoto (1991), for example, lem. A collection of hallmark papers by R.A. Bagnold has represent formulations of this type. been published by ASCE (Thorne et al 1988).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6666 11/21/2008/21/2008 5:43:315:43:31 PMPM bed load transport 67 2.6.1 Bed Load Transport Analysis will be overcome, and a grain fl ow will be initiated in the sur- face layer of the bed (Fredsøe and Deigaard 1992). So it is Sediment can be transported in several ways. A grain will important to be able to discriminate between different modes begin to move when the boundary shear stress just exceeds a of sediment transport so that the domain of applicability of bed critical value. At the lowest transport stages, particles move load transport models can be determined (Sumer et al 1996). by sliding and rolling over the surface of the bed, but with In an attempt to interpret different transport modes fol- a small increase in boundary shear stress these grains will lowing initiation of motion, Wiberg (1987) used a mechanis- hop up from the bed and follow ballistic-type trajectories. tic model of bed load transport (Wiberg and Smith 1985) to This latter mode of bed load transport is known as saltation. produce a diagram (Fig. 2-30) of transport stage (T τ / τ ) Gilbert (1914) seems to have been the fi rst to use the term * b bc or bed shear stress (τ ) versus quartz grain diameter(D ), saltation, derived fom the Latin verb saltare , which means b depicting the range of conditions over which sediment moves to leap or dance, to describe the motion of sand particles in strictly as bed load and a saltation-based model can be used water. to describe the phenomenon. Conditions for the initiation Saltation is described as the unsuspended transport of of motion, the transition to suspension and the transition to particles over a granular bed by a fl uid fl ow, in the form of grain fl ow, are also included. consecutive hops within the near-bed region. It is governed As shown in Fig. 2-30(a), at transport stages T 1 no sed- mainly by the action of hydrodynamic forces that carry the * iment moves in a uniform bed of a given grain size. For grain particles through the fl ow, the downward pull of gravity and sizes D 0.08 cm (coarse sand and fi ner), the conditions for the collision of the particles with the bed, which transfers incipient motion occur at transport stages lower than those at their streamwise momentum into upward momentum, thus which the applied stresses at the bed are suffi cient to over- sustaining the saltation motion (Niño and García 1998). This come the bed’s frictional resistance. For sizes D 0.08 cm, differs from an earlier defi nition given by Bagnold (1973) this situation is reversed, and the conditions for potential grain who assumed that the only upward impulses exerted on the fl ow (sheet fl ow) at the bed surface are reached before the par- saltating particles were those resulting from the impact of ticles are signifi cantly affected by the vertical turbulent veloc- particles with the bed. Thus, Bagnold neglected the effect of ity fl uctuations that could entrain the grains into suspension hydrodynamic lift and vertical impulses owing to fl ow tur- (Niño and García 1996). Wiberg (1987, p. 94) indicates that bulence, which have been shown to play an essential role in the advent of either of these processes does not preclude the the saltation phenomenon (e.g. Leeder 1979a; Bridge and possibility of the other, but changes in the bed load dynamics Dominic 1984; Bridge and Bennett 1992; Niño et al., 1994; produced by these processes are certain to infl uence the trans- Niño and García 1995). Experimental studies on saltation of port stage at which the other could occur. This is supported gravel and sand by Niño et al. (1994) and Niño and García by the observations made by Wilson (2005), which show that (1997) have given detailed information on the physics of par- when the ratio between the shear velocity (u ) and the sedi- ticle saltation. In particular, they have provided a description * ment fall velocity (v ) increases over a critical value (u / v of the particle collision with the bed, allowing calibration of s * s 6.5), a rapid increase in both fl ow resistance and sediment a stochastic model for this phenomenon, and have also pro- entrainment into suspension is observed. vided statistics for the geometric and kinematic properties of For all sediment sizes, Fig. 2-30(a) suggests that a trans- the saltation trajectories. port stage of about 20 is an upper limit for saltation-based In addition to its signifi cance for the fl ux of sediment mov- bed load transport. A saltation model might still provide ing as bed load, the bed load layer serves as an exchange zone reasonable transport predictions for incipient grain-fl ow between the bed and sediment transported in suspension; the conditions beyond this limit, but the physics of the phenom- upward fl ux of sediment at the top of the bed load layer pro- enon becomes more complicated as grain-grain interaction vides the boundary condition for suspended sediment transport becomes more intense (e.g., Kobayashi and Seung 1985). calculations. Once sediment starts moving and sliding along Fig. 2-30(b) presents the same results as shown in the bed, the prevalent mode for bed load transport will most Fig. 2-30(a), but in terms of dimensional boundary shear likely be saltation for a range of bed shear stresses. At higher stress, τ (dy/cm 2 ) , to give a better sense of when the transi- values of boundary shear stress, the surface layer of the bed b tion to these transport modes are actually likely to occur. For may deform and move as a grain fl ow or granular fl uid fl ow sediment sizes D 0.018 cm (fi ne sand and fi ner) at initial (Wilson 1987, 1989). Grain fl ow is also known as sheet fl ow motion, the moving particles go directly into suspension fol- (Fredsøe and Deigaard 1992; Sumer et al 1996). Collision of lowing intiation of motion. The corresponding critical shear the moving particles with the bed exerts both a tangential and a stress τ 2 dy/cm 2, is quite low, and material of these sizes is normal stress on the bed surface. The work of Bagnold (1954), b frequently mobilized, provided cohesive effects are not large Hanes and Inman (1985), and Jenkins and Hanes (1998) on (see Chapter 4). Fine to coarse sand ( D 0.018–0.08 cm) high-concentration, granular shear fl ows have shown that if the moves initially as bed load, with particles starting to go into ratio of the applied tangential to normal shear stresses exceeds suspension at higher shears stresses. For example, medium the critical yield criterion, the frictional resistance of the bed

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6677 11/21/2008/21/2008 5:43:315:43:31 PMPM 68 sediment transport and morphodynamics large sand-bed rivers or in high-gradient mountain streams. Thus for the grain sizes commonly encountered, suspended- load transport is an important mode of transport for fi ne sedi- ment, whereas high-concentration grain fl ows are probably relatively uncommon except in a few specifi c environments. For a large range of sediment sizes in the medium to coarse sand range and above, if the sediment is moving at all, it is certainly moving as bed load.

2.6.2 Bed Load Transport Defi nition Bed load particles roll, slide, or saltate along the bed. The transport thus is tangential to the bed. When all of the trans- port is directed in the streamwise, or s direction, the volume bed load transport rate per unit width (n -direction) is given 3 2 by q b ; the units are length /length/time, or length /time. In τ general, qb is a function of boundary shear stress b and other sediment parameters; that is,

( ) qbb q τb , other parameters (2-80)

In general, bed load transport is vectorial, with compo-

nents qbs and q bn in the s (streamwise) and n (lateral) direc- tions, respectively. Basically the bed load transport rate can be defi ned as the product of particle concentration, particle velocity, and bed load layer thickness,

δ qucbbbb (2-81)

2 / in which q b is the volumetric bed load transport rate ( m s), c b is the volumetric sediment concentration (i.e. volume of sedi- Fig. 2-30. Tentative ranges of conditions over which sediment ment/volume of water-sediment mixture), u b is particle veloc- moves strictly as bed load. (a) Initiation of motion, the transition to ity (m / s), and δ is the thickness of the bed load layer (m). suspension and the transition to grain fl ow plotted in terms of trans- b Bagnoldean bed load transport models use this defi nition of the port stage versus grain size. (b) The same curves plotted in terms of dimensional boundary shear stress (in dy/cm2), versus gran size. bedload transport rate (Ashida and Michiue 1972; Engelund The vertical line marks the particle size at the intersection of the and Fredsøe 1976; Van Rijn 1984a; Wiberg and Smith 1987; incipient suspension and incipient grain fl ow curves, D ≅ 0.08 cm Sekine and Kikkawa 1992; Niño and García 1994; Lee and (adapted from Wiberg 1987). Hsu 1994; Niño and García 1998; Lee et al., 2000). The bed load transport rate can also be defi ned as the product of the number of moving particles per unit area, the τ / 2 sand begins to move at b 2 to 3 dy cm and incipient particle volume and the particle velocity (García 2000), τ / 2 suspension begins at b 10 to 30 dy cm Shear stresses of such magnitude can be reached during moderate river fl ows and during storms on continental shelves. qNVubbbb (2-82) In very energetic environments, such as the surf zone in coastal areas or during large river fl oods, it may also be pos- in which N b is the number of particles per unit bed area sible for a grain fl ow (sheet fl ow) to develop (Wilson 1987, ( m 2 ), V is the particle volume (m 3), and u is the particle b b 1989; Sumer et al 1996). For coarse sand and gravel (D velocity (m/s). If the particle velocity is defi ned as the ratio 0.08–6 cm), a relatively large boundary shear stress is required of the saltation or step length λ and the saltation or move- just to initiate the motion of the sediment. For example, for ment period T ( i.e. u λ / T ), then τ b D 0.5 cm (fi ne gravel), the critical shear stress is bc 45 / 2 τ dy cm ) and a grain fl ow is possible at a shear stress b / 2 λλλ/ 550 dy cm ; these conditions are only likely to occur in very qNVTEbbbp D p (2-83)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6688 11/21/2008/21/2008 5:43:315:43:31 PMPM bed load transport 69 Here , E p and D p eroded and deposited volume of particles and Parker 1991; Niño et al. 2003). Erosion into suspension per unit bed area per unit time (m/s), respectively. Equilibrium can be taken to be directed upward normal, i.e., in the posi- bed load transport conditions imply that E D . tive z direction. p p _ u / The idea of a pick-up rate and a step length was fi rst Let denote the mean fl ow velocity (m s)_ at a point proposed by Einstein (1942; 1950) and constitutes the located at a distance z normal to the bed, and c denote the basis of Einsteinian bedload transport models (Nakagawa mean volumetric concentration of suspended sediment (m3 and Tsujimoto, 1980; Tsujimoto, 1992). A comparison of of sediment/m 3 of sediment-water mixture), averaged over several pick-up rate functions and their applications can be turbulence. The streamwise volume transport rate of sus- found in Van Rijn (1984b; 1986). Einstein defi ned the par- pended sediment per unit width is given by ticle step length as the particle travel distance from entrain- ment to deposition (i.e., when the particle stops moving H and comes to rest) and estimated it to be equal to about 100 qucS ∫ dz (2-85) times the particle diameter. Einstein’s particle length can be 0 expected to be several times the saltation length λ previously defi ned. This assumption will be considered below in light Let s denote the streamwise direction and n denote the of the experimental observations made by Wong and Parker lateral direction in a two-dimensional case; then two compo-

(2006a). nents, qSs and qSn result, where

H 2.6.3 Conservation of Sediment Mass: qucSs ∫ dz (2-86a) The and Morphodynamics 0 Before considering bed load transport relations in more H detail, it is useful to formulate the interaction between bed qvcSn ∫ dz (2-86b) sediment and the water column through erosion and deposi- 0 tion, so that the sediment mass conservation can be formu- _ lated. Consider the defi nition diagram for a sediment-laden, where v is the mean lateral ( n -direction) velocity at a dis- uniform, open-channel fl ow shown in Fig. 2-31. The volume tance z above the bed. rate of erosion of bed material into suspension per unit time Deposition onto the bed is by means of settling. The rate 3 at which material is fl uxed vertically downward onto the bed per unit bed area is denoted as E r . The units of E r are length / _ _ 2 (volume/area/time) is given by v c , where c is a near-bed length /time, or velocity. A dimensionless sediment entrain- s b b _ value of the volumetric sediment concentration c . Some ment rate E s can thus be defi ned in terms of the sediment fall velocity v : authors assume that the value of the near-bed concentration is s the same as the sediment concentration in the bed load layer defi ned previously (Einstein 1950; Engelund and Fredsøe Erss vE (2-84) 1976; Zyserman and Fredsøe 1994). The deposition rate D r realized at the bed is obtained by computing the component of this fl ux that is actually directed normal to the bed as In general, E s can be expected to be a function of bound- ary shear stress τ and sediment related parameters (García b Drsb vc (2-87)

which gives the volume of sediment deposited per unit bed area per unit time (García 2001). Now it is possible to formulate the sediment mass conserva- tion for bed material taking into account both bed load trans- port and sediment erosion into and from suspension. Consider a portion of river bottom (Fig. 2-32), where the bed material is λ taken to have a constant porosity p . Mass balance of sediment requires that the following equation be satisfi ed: ∂ [] mass of bed material net mass bedload inflow rate ∂ t nnet mass rate of deposition from suspension

Fig. 2-31. Defi nition diagram for sediment-laden open channel A datum of constant elevation is located well below fl ow. the bed level, and the elevation of the bed with respect to

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 6699 11/21/2008/21/2008 5:43:335:43:33 PMPM 70 sediment transport and morphodynamics itself evolved many decades afterward and is now used for river, coastal, and estuarine problems (e.g., Parker and García 2006). This notwithstanding, Exner deserves recognition as the founder of morphodynamics (Parker 2005). The fi eld of morphodynamics consists of the class of problems for which the fl ow over a bed interacts strongly with the shape of the bed, both of which evolve in time. An introduction to morphodynamics of rivers and turbidity currents is given in Section 2.11 of this chapter. The Exner equation is generalized to the case of sediment mixtures in Chapter 3.

2.6.4 Bed Load Transport Relations Fig. 2-32. Defi nition diagram for sediment mass conservation. A large number of bed load relations can be expressed in the general dimensionless form

the datum is given by η. The mass balance equation trans- qq*** ()τ , RR, (2-89) l ates to ep

∂ Here, q * is a dimensionless bed load transport rate known ρλ()1( ηds dn) ρ() q q dn sp∂ t sbss bs sds as the Einstein bed load number, fi rst introduced by Hans (2-88a) Albert Einstein in 1950, and given by ρ ()qq ds sbnn bn ndn ρ () ssrD E r ds dn q q* b (2-90a) Or upon reduction with Eqs. (2-84) and (2-87), Eq. (2-88a) DgRD takes the fi nal form where qb is the volumetric bedload transport rate, g is the ρ ρ / ρ ∂ η ∂ q ∂ q acceleration of gravity, R ( s ) is the submerged bs bn ()1 λ p vcsb() E s (2-88b) specifi c gravity of the sediment, D is the particle diameter, ∂ t ∂ s ∂ n /ν and RgRDDep is the particle Reynolds number. Einstein’s bed load transport model can be expressed in Bed level changes with time t due to bed load transport, dimensionless form as follows, sediment entrainment into suspension, and sediment deposi- tion onto the bed can be predicted with this partial differen- *** tial equation. To solve this equation, it is necessary to have qELps (2-90b) relations to compute bed load transport (i.e., q and q ), _ bs bn near-bed suspended sediment concentration c and sediment b with entrainment into suspension E s (García and Parker 1991; García 2001). The basic form of Eq. (2-88b), without the E * b suspended sediment component, was fi rst proposed for the Eb (2-90c) case of a one-dimensional fl ow interacting with a sediment- gRD covered bed by Felix Exner (1925). and Felix Maria Exner was an Austrian researcher who was active in the early part of the 20th Century. His main area L L* s (2-90d) of interest was meteorology. At some point he became inter- s D ested in the formation of dunes in rivers (see Leliavsky 1966). In the course of his research on the subject, he derived and In these relations E b volumetric rate of sediment employed one version of the various statements of conserva- entrainment per unit area; and L s particle step length (i.e., tion of bed sediment that are now referred to as “Exner equa- the particle travel distance from entrainment to deposition). tions.” In addition, he made an important early contribution In a study of sand bed instability, Nakagawa and to one-dimensional nonlinear wave dynamics (Exner 1920). Tsujimoto (1980) found that the dimensionless entrainment * τ* Felix Exner was the fi rst researcher to state a morphodynamic rate E b is a function of the Shields parameter and tried to problem in quantitative terms. The term “morphodynamics” develop a probabilistic model for the dimensionless particle

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7700 11/21/2008/21/2008 5:43:355:43:35 PMPM bed load transport 71

* step length Ls but did not fi nd a simple way to characterize when the excess dimensionless shear stress gets larger, as this parameter (Tsujimoto and Nakagawa 1986). Sechet and indicated by Eq. (2-90f). Substituting (2-90e) and (2-90f) Le Guennec (1999) found experimentally that the particle into (2-90b), yields step length is related to the bursting phenomenon (i.e., ejec- tions and sweeps) which is in agreement with observations 3/ q** 2.. 66()τ 0 0549 2 (2-90g) of near-wall particle-turbulence interactions made earlier by Sumer and Deigaard (1981), García et al. (1996), and Niño and García (1996). This might explain some of the dif- This empirical fi t has a structure which is very similar to fi culties encountered in trying to characterize Einstein’s step several formulations presented below. length. Recently, Sumer et al (2003) were able to observe Bagnold’s approach is considered next. A dimension- directly the infl uence of turbulence on bed load transport of less bed load transport equation, such as the one implied by sand (D 50 0.22 mm) with a set of carefully conducted lab Eq. (2-89), can be obtained by simply dividing both sides experiments. The Shields parameter together with the root- of Eq. (2-81) by a characteristic length given by the particle mean-square value (RMS) of the streamwise velocity fl uc- diameter D and a characteristic velocity given by gRD , tuations, were correlated with the sediment transport rate. which yields They found that the sediment transport rate increases mark- edly with increasing turbulence levels. A few years earlier, Drake et al (1988) observed a similar infl uence of the fl ow q c δ u ∗ bbbb turbulence while observing bed load transport of fi ne gravel q (2-91) DgRD D gRD with motion-picture photography. Recently, Wong and Parker (2006a) conducted a tracer study involving transport of uniform-size gravel in a large Bagnold’s (1956) hypothesis makes it possible to esti- fl ume at St. Anthony Falls Laboratory, University of Minne- mate the volumetric sediment concentration in the bedload sota. The sediment used in all the experiments was uni form layer per unit bed area, given by the product of sediment δ gravel, with geometric mean size Dg 7.2 mm, geometric concentration cb and the thickness of the bed load layer b , σ standard deviation g 1.2, median particle size D50 7.1 as follows mm, particle size for which 90% of the sediment is fi ner D 9.6 mm, and a specifi c gravity of 2.55 (R 1.55). 90 c δττ** Based on these observations they found empirical equa- bb c (2-92) μ * D d tions for the dimensionless entrainment rate Eb and particle * τ* step length Ls as functions of the Shields parameter , as μ follows, where d is a dimensionless dynamic friction coeffi cient (Abott and Francis 1977; Sekine and Kikkawa 1992). Niño and García (1998) have used a Lagrangian particle-saltation E** ..τ 197 . p 0 06() 0 0549 (2-90e) μ model to estimate values of d and found that it takes values in the range between 0.25 and 0.4, which are smaller than μ φ d tan 0.63 proposed by Bagnold (1973) but are in and good agreement with laboratory observations of sand trans- μ port (Niño and García 1997). Simulated values of d for the case of sand saltation are much closer to the corresponding L**44.. 33()τ 0 0549 047 . (2-90f) s observed values than in the case of saltation of gravel (Niño and García 1994). Bagnold’s hypothesis used to obtain to Equation (2-90e) predicts values very close to those Eq. (2-92) would be only valid for intense transport condi- observed by Fernandez Luque and van Beek (1976). On the tions involving very high sediment concentrations of sand- other hand, Eq. (2-90f) contradicts the original ideas of size material. The mean velocity of the particles in the bedload

Einstein (1950) who did not include a critical shear stress for layer ub can also be estimated with the help of numerical * motion and assumed that the dimensionless step length L s modeling and experimental observations of particle motion 100 for all fl ow conditions. Also of interest is the fact that the (Reizes 1978; Leeder 1979; Murphy and Hooshiari 1982; step length is found to decrease as the fl ow intensity char- Bridge and Dominic 1984; van Rijn 1984a; Wiberg and τ* * acterized by , increases. L s takes values between 160 and Smith 1985, 1989; Sekine and Kikkawa 1992; Lee and Hsu, 270 for the range of experimental conditions covered in the 1994; Niño and García 1994, 1997, 1998; Lee et al. 2000, experiments. Wong and Parker (2006a) argued that since the 2006; Lukerchenko et al. 2006). chances of a given particle being captured and trapped into Ashida and Michiue (1972) presented a macroscopic anal- a resting position increases with sediment transport rate, it is ysis that does not account for the complexity of the saltation reasonable to assume that the step length becomes smaller process, in particular the treatment of the particle collision

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7711 11/21/2008/21/2008 5:43:365:43:36 PMPM 72 sediment transport and morphodynamics with the bed. In their analysis, a simplifi ed particle equation where the functionality is implicitly defi ned by the relation of motion is used to obtain the following expression for the

dimensionless mean particle velocity in the bedload layer: ()0. 413/τ* 2 * 1435 2 . q 1 ∫edtt (2-96b) π * /τ* 1435. q u 12// 12 ()0. 413 2 b ⎡ ττ** ⎤ 85. ⎢() ()c ⎥ (2-93) gRD ⎣ ⎦ This relation constitutes the fi rst attempt to derive a bed load function. Note that this relation contains no critical shear Upon substitution of Eqs. (2-92) and (2-93) into (2-91) stress. It has been used for uniform sand and gravel. Gomez μ and assuming a value for d of 0.5, the following fi nal form and Church (1989) recommend its use for cases where the for bed load transport is obtained local bed load transport rate needs to be calculated. Yang and Wan (1991) found that it could predict sediment transport rates in large rivers but not in small rivers and fl umes. ⎡ 12// 12⎤ q***** 17()()()ττ τ τ (2-94) cc⎣⎢ ⎦⎥ Yalin (1963):

⎡ () ⎤ τ* **12/ ln 1 as2 Ashida and Michiue recommend a value for c of 0.05 in qs0. 635()τ ⎢ 1 ⎥ (2-97a) their relation. It has been verifi ed with uniform material rang- ⎣ as2 ⎦ ing in size from 0.3 mm to 7 mm. The Ashida-Michiue bed load transport equation is a good example of a Bagnoldean formulation. It is very similar to the one developed indepen- where dently by Engelund and Fredsøe (1976) and more recently by Niño and García (1998) using a Lagrangian particle-salta- ** 12/ ττ tion model for bed load transport. aR245. () 104.*()τ s c (2-97b) 2 c τ* In addition to the relation of Ashida and Michiue (1972), c the following bed load transport relations are of interest. τ* Meyer-Peter and Muller (1948): and c is evaluated from the 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 28.6 mm mate- 3/2 q***8()ττ (2-95a) rial. Wiberg and Smith (1985, 1989) were able to reproduce c Yalin’s relation, with their saltation-based bed load transport model. τ* where c 0.047. This formula is empirical in nature; it has Wilson (1966): been verifi ed with data for uniform coarse sand and gravel. Even though it was developed for alpine streams in Switzerland, it 32/ ***()ττ enjoys wide use in coastal sediment transport (e.g. Soulsby, q 12 c (2-98) 1997). Recently, Wong and Parker (2006b) reanalysed the data used by Meyer-Peter and Muller and found that a better fi t is τ* where c is determined from the Shields diagram. This rela- provided by one of the two alternative forms; tion is empirical in nature; most of the data used to fi t it pertain to very high rates of bed load transport. It has been

16. used extensively to estimate transport of sand and industrial q** 4.. 93()τ 0 047 (2-95b) materials such as nylon in pressurized fl ows (e.g., Wilson 1987). Paintal (1971): 3/2 q**3.. 97()τ 0 0495 (2-95c) q**656. 1018τ 16 (2-99)

Einstein (1950): was obtained though extensive measurements of very low bed load transport rates. It is valid for 0.007 τ* 0.06 and sediment grain sizes between 1 mm (coarse sand) and 25 mm *** qq ()τ (2-96a) (gravel). This relation shows that for low shear stresses, the

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7722 11/21/2008/21/2008 5:43:375:43:37 PMPM bed load transport 73 sediment transport phenomenon is highly nonlinear. That is, equation is based on a dimensionless particle diameter and for small changes in bed shear stress, the rate of bed load the transport stage parameter T , defi ned, respec tively, as transport increases dramatically. 1/3 ⎛ gR⎞ 2/3 Engelund and Freds¿e (1976): DD⎜ ⎟ R (2-103b) * 50 ⎝ ν2 ⎠ ep

⎡ 12// 12⎤ q*****18.. 74()()()ττ τ 0 7 τ (2-100a) and cc⎣⎢ ⎦⎥ ττ** T sc (2-103c) τ* * where c 0.05. τc This formula resembles that of Ashida and Michiue τ* because its derivation, albeit obtained independently, is Here s is the bed shear stress due to skin or grain friction, τ* almost identical. This relation was rederived by Fredsøe and and c is the critical shear stress for motion from the Shields Deigaard (1992, p. 214), resulting in a very similar relation, diagram. Madsen (1991): // *****30 ττ⎡ τ12 τ 12⎤ q ()()()cc07. (2-100b) πμ ⎣⎢ ⎦⎥ **12// *** 12 d ττττ qFMcc()()07. (2-104a)

Fredsøe and Deigaard tested the formula for different where F 8 / tan φ for rolling/sliding sand grains and F values of the dynamic friction coeffi cient μ . For μ 1.0, M M d d 9.5 for saltating sand grains in water. Eq. (2-96b) gives results very close to those of the Meyer- Peter and Muller formula (Eq. 2-95a). However, both formu- Nielsen (1992): lations were found to overpredict bed load transport at hight shear stresses. ****τττ12/ q 12 ()c (2-104b) Fernandez-Luque and van Beek (1976): obtained by fi tting to uniform size sand and gravel bed load 3/2 q***57. ()ττ (2-101) transport data. This relation was also independently derived c by Soulsby (1997). Equations (2-104a) and (2-104b) have been used mainly in . τ* where c varies from 0.05 for 0.9-mm material to 0.058 for 3.3-mm material. The relation is empirical in nature and was Niño and García (1998b): obtained through laboratory observations. 12 // q* ()ττ− () τ12− 07. τ 12 (2-104c) Parker (1979): μ **cc * * d 45. ()τ* 003. * q 11. 2 (2-102) obtained with a Lagrangian description of bed load transport τ*3 by saltating particles and tested with experimental observa- tions of gravel transport (Niño and García 1994) and sand developed as a simplifi ed fi t to the relation of Einstein transport (Niño and García 1998c). A dynamic friction coef- fi cient μ 0.23 was determined, almost three times smaller (1950) for the range of Shields numbers likely to be d encountered in gravel-bed streams. This formula was used than the value proposed for the same coeffi cient by Bagnold to analyze the hydraulic geometry of gravel-bed streams (1973). This relation basically has the same structure as (Parker 1979). Madsen’s Eq. (2-104a). Cheng (2002): Van Rijn (1984a):

**3/2 ⎛ 005. ⎞ 21. q 13τ exp ⎜ ⎟ (2-104d) * T ⎝ *3/2 ⎠ q . (2-103a) τ 0 053 03. D* This relation gives results similar to those obtained with can be used to estimate bed load transport rates of particles Meyer-Peter and Muller (1948) and Einstein (1950) equation with mean sizes in the range between 0.2 and 2.0 mm. This for moderate dimensionless shear τ * values. It also agrees

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7733 11/21/2008/21/2008 5:43:385:43:38 PMPM 74 sediment transport and morphodynamics well with the values predicted with Paintal (1971) for weak These fi ndings indicate a stronger dependence of q * on τ * transport conditions under low shear stresses. at very high transport intensities that one might expect from For intense transport conditions associated with large bed load transport equations. In fact, Abrahams (2003) has values of the dimensionless Shields stress τ *, Eq. (2-104d) recently revisited the concepts advanced by Bagnold (1973) reduces to and found that transport rates under sheet fl ow (grain fl ow) conditions are much higher than previously thought. q**≅ 13τ 3/2 (2-104e) Most of the bed load equations shown above apply to the case of mild slopes or nearly horizontal fl ows. For steep channels, the effect of the dowslope gravitational component In fact, most of the bed load transport equations display cannot be neglected. Smart (1984), Bathurst et al. (1987), the same asymptotic behavior for high values of shear stress Graf and Suszka (1987), Tsujimoto (1989), Rickenmann as can be observed in Fig. 2-33. That is, for τ* τ*, q * ≈ c (1991), Damgaard et al. (1997), and Aguirre-Pe and Fuentes τ* 3/2. However, a word of caution is necessary because this (2002), among others, have proposed equations for bed load does not seem to be the case according to the observations transport in steep slopes. made by several investigators who have found a different Only a few research groups have attempted complete relation for high transport rates. For instance, Rickenmann derivations of the bed load function in water. They include (1991) has indicated that for intense sediment transport con- Wiberg and Smith (1989); Sekine and Kikkawa (1992); ditions, when τ * 0.4, grain fl ow (or sheet fl ow) conditions Niño and García, (1994, 1998); Seminara et al (2002); and develop and q * is actually proportional to τ * 5/2 as found by Parker et al (2003). These results are encouraging because Hanes and Bowen (1985) with a granular fl uid model for they show that bed load transport can be predicted with coastal sediment transport and Takahashi (1987) for debris a mechanics-based approach. More recently, discrete par- fl ows on steep slopes. Hanes (1986) showed that under these ticle simulations of bed load transport which account for conditions transport rates can be approximated with the effect of near-bed turbulence, particle location and particle-particle interaction, have been conducted, among ** τ 5/2 q 6 (2-104f) others, by Jiang and Haff (1993), Calantoni and Drake (2001), Nelson et al. (2001) and Schmeeckle and Nelson (2003). Undoubtedly, the role played by turbulence in bed- load transport is still a subject that deserves more research (Nelson et al. 1995; García et al. 1996; Niño and García 1996; Best et al., 1997; Sechet and Le Guennec 1999; Schmeeckle et al., 2001; Papanicolau et al. 2001; Sumer et al. 2003). It is also clear that direct fi eld observations provide the best information for both developing and test- ing of new formulations (e.g., Almedeij and Diplas 2003). Several bed-load transport relations for gravel and sedi- ment mixtures are considered in Chapter 3.

2.6.5 Two-Dimensional Transport of Bed Load The relations presented above for bed load transport are all one-dimensional in nature. That is, they provide the mag- nitude of a bed load transport vector that is oriented in the direction of the boundary shear stress. That is, if the s coor- dinate is directed along the bed parallel to the boundary shear stress and the n coordinate is directed along the bed and perpendicular to the s coordinate,

G qqqq()()sn,,0 (2-105)

G where q denotes the two-dimensional vector of bed load transport rate and q denotes the magnitude of that vector, Fig. 2-33. Plot of several bed load functions found in the which is computed using one of the relations presented literature. above.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7744 11/21/2008/21/2008 5:43:385:43:38 PMPM bed load transport 75 In point of fact bed load transport is fundamentally A second problem that has played a major role in the two-dimensional in nature. This is illustrated in Fig. 2-34, development of two-dimensional bed load relations is the which illustrates the bed at a river bend. In the diagram s is quantifi cation of the erosion of a river bank composed of a boundary-embedded centerline streamwise component and noncohesive sediment. Banks in noncohesive material form n is a boundary-embedded transverse component. Because a side slopes; as bed load is moved downstream by the fl ow, bend generates secondary fl ow in addition to the downstream gravity pulls it down the side slope, accomplishing bank ero- τG primary fl ow, the boundary shear stress vector b is not paral- sion. Hirano (1973) was among the fi rst to develop a quan- lel to the s direction, but is skewed somewhat inward. This titative formulation for this problem. Streambank erosion is drives a component of bed load transport in the negative n addressed in Chapter 7. direction, i.e., inward (Brooks 1963). The bed itself slopes Gravity can infl uence bed load transport in the down- downward from inside to outside in the transverse direction stream as well as transverse direction. In most cases of inter- with magnitude |∂η / ∂ n |. As a result, gravity pulls bed load est, however, gravity acts only indirectly to drive bed load particles down the slope, driving a component of bed load transport. That is, gravity pulls the fl ow downslope, and the transport in the positive n direction. Depending on the mag- fl ow in turn drags the bed material downslope. If the stream- G nitude of the forces involved, the bed load vector q may have wise slope of the bed is suffi ciently high, however, the direct a positive or negative component in the transverse direction. contribution of gravity acting on the bed load grains can These competing transverse effects play an important increase the bed load transport rate (Fernandez-Luque and role in determining the morphology of rivers in meander van Beek 1976). bends (Engelund 1974; Falcon and Kennedy 1983; Ikeda The fi rst generation of two-dimensional bed load rela- and Nishimura 1986; Bridge 1992). Secondary fl ow tends tions was developed based on a linearized formulation for to drive erosion at the outside of a bend and deposition on small transverse slope and streamwise bed slopes. These all the inside (Johanesson and Parker 1989a, 1989c). This cre- take the general form ates a transverse component to the bed slope, which in turn acts to drive sediment down the slope from inside to outside. ⎡ ∗ no ⎤ τG ⎛ τ ⎞ G An equilibrium condition can be obtained in which second- GG⎢ b mag ⎥ qqG β⎜ ⎟ ∇η (2-106a) ary forces and gravity forces balance. The desire to under- ⎢ τ ⎜ τ∗ ⎟ ⎥ ⎣ b ⎝ c ⎠ ⎦ stand bend morphodynamics has been one of the motivators in the development of two-dimensional bed load transport relations (van Bendegom 1947; Engelund, 1974; Ikeda et al. where the absolute values denote the magnitude of the vector in question, β and n are constants and 1976). Meandering channels are considered in more detail 0 in Chapter 8. τττG bbsbn(), (2-106b)

τG τ∗ b (2-106c) mag ρRgD

G ⎛ ∂ηη∂ ⎞ ∇η ⎜ , ⎟ (2-106d) ⎝ ∂sn∂ ⎠

The derivations of these formulations are based on the following constraints, which allow linearizing an otherwise nonlinear formulation:

τ bn 1 (2-107a) τ bs

∂η 1 (2-107b) ∂n

∂η Fig. 2-34. Illustration of two-dimensional bed load transport in 1 (2-107c) a river bend. ∂s

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7755 11/21/2008/21/2008 5:43:395:43:39 PMPM 76 sediment transport and morphodynamics If in turn the streamwise slope is ∂η /∂ s is so small that Sekine and Parker (1992) modeled the saltation trajec- direct streamwise gravitational forces on bed load can be tories of bed load grains over a bed sloping mildly in the neglected, Eq. (2-106a) can be further cast in the approxi- transverse direction to obtain the evaluations mate form β 1 G 075. , n0 (2-113a), (2-113b) qq 4 s (2-108a) ⎡ ⎛ ∗ ⎞ n0 ⎤ β τ τ ∂η Olesen (1987) has suggested calibrating and n o to site- qq⎢ bn β⎜ s ⎟ ⎥ ns⎢ τ ⎝ τ∗ ⎠ ⎥ ∂n (2-108b) specifi c data. ⎣ bs c ⎦ In applying the above formulation the streamwise sediment τ transport q is evaluated using one of the one- dimensional τ∗ bs s s ρRgD (2-108c) formulations of the previous section and the transverse sedi- ment transport is evaluated from Eq. (2-108b). In recent years fully nonlinear formulations of two- Engelund (1974) proposed the following values of β dimensional bed load transport have become available. and n Kovacs and Parker (1994) used the underlying mechanics o of the one-dimensional formulation of Ashida and Michiue (1972) for uniform material as a basis for a fully nonlinear 1 β , n 0 (2-109a), (2-109b) generalization for arbitrary streamwise and transverse bed μ 0 d slopes. The analysis is involved and is beyond the scope of this chapter. It suffi ces to mention that (1) when applied μ to one-dimensional transport of bed load at low slopes it where d denotes a dimensionless coeffi cient of dynamic Coulomb friction for particles in bed load transport. Koch reduces to Eq. (2-66) of Ashida and Michiue (1972), and (2) and Flokstra (1980) and Struiksma et al. (1985) proposed when it is linearized to the form of Eq. (2-108b) the follow- the form ing evaluations are realized,

11 1 β , n (2-114a), (2-114b) β , n 1 (2-110a), (2-110b) μ 0 0 d 2 f∗ μ with a value of d of 0.5. A treatment allowing the numerical where f* is a calibration coeffi cient between 1 and 2. implementation of the formulation can be found in Kovacs Hasegawa (1984) proposed the form and Parker (1994). The formulations of Ashida and Michiue (1972) and thus Kovacs and Parker (1994) employ forms of a criterion due β 11 , n0 (2-111a), (2-111b) to Bagnold (1956) that contains a conceptual error. Parker μμ 2 sd et al. (2003) have repeated the analysis using a bed load entrainment formulation due to Tsujimoto (e.g., 1991) (and μ where s denotes a dimensionless coeffi cient of static ultimately due to Einstein 1950) that does not rely on the Coulomb friction for bed particles. The Ikeda-Parker rela- Bagnold criterion in question. Their two-dimensional bed tion (Parker 1986, based on Ikeda 1982; see also Kikkawa load formulation is similarly involved; a source for soft- et al., 1976) takes a very similar form (and in fact has a very ware is referenced in their paper. When applied to one- similar derivation) to that of Hasegawa (1984), dimensional transport of bed load at low slopes it reduces to a slightly modifi ed form of Eq. (2-101) of Fernandez Luque 1rμ 1 and van Beek (1976), and when it is linearized to the form of β d , n μ 0 (2-112a), (2-112b) Eq. (2-108b) the following evaluations are realized: d f∗ 2 1 β 07. , n (2-115a), (2-115b) where r denotes the ratio of lift force to drag force on a par- 0 2 ticle in bed load motion and f * is again a calibration coeffi - cient. Johanesson and Parker (1989b) obtained the following Despite very promising recent attempts (Kovacs and evaluations based on bend topography in rivers; Parker 1994; Seminara et al. 2002; Dey 2003; Parker et al. 2003), prediction of bed load transport on arbitrarily slop- ing beds remains a challenge. This is mainly due to the fact μ 043..,,r 085 (2-112c), (2-112d), d that direct observations of transverse bed load transport are f∗ 119. (2-112e) also very challenging (e.g., Talmon et al. 1995). To predict

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7766 11/21/2008/21/2008 5:43:415:43:41 PMPM bed forms 77 morphological changes in rivers, coastal areas, and , geomorphologists (e.g., Ashley 1990; Southard 1990; Julien and such technology is necessary and should be given priority in Klaassen 1995, Julien et al. 1995, Bridge, 2003; Best 2005). sediment transport research (Seminara and Blondeaux 2001; Dunes are one of the most common depositional bed forms, Parker and García 2006). forming in a range of sediment sizes from silt and sand to gravel (Dinehart 1992; Seminara 1995; Best 1996; Carling 1999; Kleinhans 2001, 2002). Recent studies of subsurface 2.7 BED FORMS alluvium also demonstrate that dunes can form the major- ity of the deposits of sandy braided rivers (Best et al. 2003; Sediment waves produced by moving water are, in equal Bridge 2003; Best 2005). The depositional patterns created measure, intellectually intriguing and of great engineer- by dunes can result in heterogeneous and anisotropic permea- ing importance. For example, seasonal water-temperature bility fi elds in the subsurface, thus complicating prediction of changes infl uence stage-discharge and depth-discharge subsurface fl ow in both and hydrocarbon reservoirs relations in rivers (e.g., Missouri River) and in conveyance (Weber 1986). Shoaling rates and dredging times are largely channels and navigation channels where water discharge is controlled by the regeneration time of sand waves (Knaapen constant (Shen et al. 1978). The major infl uence is related and Hulscher 2002), pointing to the practical signifi cance of to changes in bed confi guration following changes in water understanding the dynamics of bed forms. Experimental work temperature (Southard 1989). Large bed forms, such as mega- has also shown the critical role of bed forms, and particularly dunes, can make navigation diffi cult by increasing shoaling dunes, in infl uencing convective fl ow within the bed sediment rates and endangering the stability of pipelines and tun- that is generated by the differential pressure gradient caused nels (Kennedy and Odgaard 1991; Nemeth et al. 2002). The by the bed form (Thibodeaux and Boyle 1987). Packman and threat to infrastructure caused through sediment transport associated with dunes in the Rio Paraná, Argentina is high- lighted by Amsler and García (1997), Amsler and Prendes (2000) and Amsler and Schreider (1999), who describe channel erosion near the city of Paraná as part of the disas- trous fl oods of 1983 and 1992 (Best 2005). In one region of the river, where the Paraná narrows to ෂ1.5 km in width, a 2.4 km long subfl uvial tunnel was built in 1968 and the depth of its placement was determined from a combination of regime theory and physical modeling, with a minimum sand cover thickness of 4 m above the tunnel (Amsler and García 1997. A cross section of the tunnel is shown in Fig. 2-35(a). However, the long duration of high fl oods in 1983 led to the formation of large dunes, up to 6.5 m in height and 320 m in length, that migrated through this river section (Fig. 2-35(b)). These large dunes caused temporary exposure of the tunnel to the fl ow over a distance of 250 m each time the trough of a large dune moved over the tunnel (Amsler and García 1997; Amsler and Schreider 1999), thus threat- ening its stability. Remedial actions were required to ensure the stability of the tunnel, involving placing trucks loaded with sand bags within the tunnel to prevent uplifting during most of the fl ood. This reach is now an area that is receiv- ing much study and monitoring to assess the longer-term changes in bed elevation and the hysteresis effects of dune morphology through fl oods (e.g., Serra and Vionnet 2006). Shifting of the Paraná River makes the protection of both the tunnel itself and the river margin where one of the entrances to the tunnel is located (Santa Fe) even more challenging due to lateral migration experienced by the river (a) (Ramonell et al. 2002). Fig. 2-35. (a) Cross section of tunnel under Paraná River bottom Because of the important role they play in river sedimenta- that links Santa Fe and Paraná city, Argentina. The structure needs tion and their signifi cance in ancient , a sand cover about 4-m-thick on top of it to provide a safety weight bed forms in general and dunes in particular have received Ws to counteract the net buoyancy force FB W (adapted from extensive attention from engineers, sedimentologists and Serra and Vionnet 2006).

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Δ ≈ λ ≈ Fig. 2-35. (b) Echosound bottom profi le of large dunes ( D 6.5m and D 250m) with superim- Δ ≈ λ ≈ posed smaller dunes ( d 1m and d 10m) passing by the tunnel during the fl ood of June 1992 in the middle reach of the Parana River. Each time the trough of a large dune moved over the tunnel, the sand-cover protection was removed and the tunnel was in danger of experiencing uplifting due to buoyancy effects (Echosound courtesy of Mario Amsler). (Continued)

Brooks (2001), Packman and Mackay (2003) and Packman of fl ow for which the would be stable, that is the et al. (2004) have also demonstrated the importance of bed fl ow conditions under which a small disturbance of an ini- forms in hyporheic exchange (i.e., the mixing of stream water tially plane bed would be dampened. The formation of sand with pore water beneath the sediment bed). They show the waves is considered by most scientists as an instability prob- importance of dunes for both the clogging of the porous bed lem (e.g., Fredsøe 1996). For instance, if a plane sand bed is by fi ner grains, such as kaolinite, and that the exchange of sol- slightly perturbed, the fl ow and sediment transport will be utes and is also linked to the rate of bed form migra- also affected. Two possibilities exist: tion and scour, with faster- moving bedforms reworking the bed and leading to a greater turnover of sediment. Rutherford (1) The changes in fl ow pattern and sediment transport (1994, p. 316) has also analyzed the role of dune turnover on will attenuate the amplitude of the perturbation and benthic oxygen uptake in the Tarawera River, New Zealand. eventually the bed will go back to the original plane This river receives treated pulp mill effl uent and experiences bed state (i.e., the bed is stable); or, high rates of deoxygenation. The role of dunes in contami- (2) The fl ow and sediment transport changes cause the nant exchange at sediment-water interfaces is also considered perturbation of the bed to grow in time, resulting in in Chapter 21. the formation of ripples, dunes, and/or antidunes (i.e., the bed is unstable). 2.7.1 Background Knowledge and Recent Advances In stability analysis, a plane bed is usually upset with a small A major advance in the theory of alluvial channel fl ows has sinusoidal bed perturbation having a certain amplitude and been the application of stability theory to identify the regions wave length. The goal of the analysis is to observe if the

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7788 11/21/2008/21/2008 5:43:425:43:42 PMPM bed forms 79 perturbation will grow (unstable) or decay (stable) with and length are both mainly functions of the fl ow depth and time. In his pioneering work on morphodynamics, Exner display a more complex dependency on particle size. The (1920, 1925) studied the behavior of sand waves with a co-existance of bed forms makes it more diffi cult to con- simple model that related the local sediment transport to duct a stability analysis capable of distinguishing between the depth-averaged fl ow velocity, and predicted instability ripples and dunes. This problem was addressed by Richards of the bed but no stable regions. Exner’s work is described (1980), who improved the eddy-viscosity description of the in Graf (1984, p. 289). Anderson (1953) improved upon fl ow over a hydraulically-rough wavy bottom by introduc- Exner’s fl ow description and was the fi rst to apply stabil- ing a one-equation turbulence model. His analysis discov- ity theory to the analysis and prediction of bedforms (rip- ered two peaks in the growth rate of the perturbations. The ples, dunes, antidunes), although his analysis predicted higher peak corresponds to the maximum found indepen- only neutral instability. Kennedy (1961, 1963) was the fi rst dently by Engelund and Smith in 1970, and later corrobo- to advance a comprehensive theory to account for dunes, a rated by Fredsøe (1974) when including the bed-slope (i.e., plane bed, and antidunes by means of linear stability analy- gravity) effect. Richards (1980) found that the location of sis. Kennedy used a potential fl ow model with an empirical this peak depends strongly on the fl ow depth indicating that relation between fl ow velocity and sediment transport; in it is related to the instability of dunes. The lower peak in later studies the theory was extended to the formation of rip- perturbation amplifi cation is not affected by changes in the ples in closed ducts (Kennedy 1964) and to ripples formed fl ow depth, and according to Richards this maximum can in oscillatory fl ows (Kennedy and Falcon 1965). Potential be related to the instability of ripples. Sumer and Bakioglu fl ow models for bed stability analysis were also advanced by (1984) considered only the ripple mode and extended the Reynolds (1965) and Gradowczyk (1968). Kennedy (1963) work of Richards to the case of hydraulically smooth and emphasized the need to include a lag between the local sedi- transition bed-roughness conditions. This extension is ment transport rate and the local fl ow velocity. The main two important since experimental observations have shown that factors causing Kennedy’s lag in sand bed rivers are (1) fl uid ripples disappear and dunes are the prevailing bed form friction, and (2) delayed response of sediment transport to instability when the fl ow transitions from hydraulically- spatial changes in fl ow velocity. Hayashi (1970) made theo- smooth to hydraulically-rough conditions. retical studies on bed form development using the St. Venant The effect of sediment transport on bed stability was fi rst equations and futher interpreted the lag distance introduced included by Engelund (1970) for suspended load and by by Kennedy. A decade later, Engelund (1980) developed a Parker (1975) for bed load to assess the role of sediment iner- model which did not require the lag distance and accounted tia on the development of antidunes. Reynolds (1976) indi- for velocity and suspended sediment distribution as well as cated that Kennedy’s lag was indeed related to the relaxation turbulent diffusion. time associated with settling of suspended sediment and The instability due to bed friction was recognized inde- with less obvious adjustments in bed load. The stabilizing pendently by Frank Engelund (1970) and J. Dungan Smith effect of sediment was used by Engelund and Fredsøe (1974) (1970). Both applied an eddy-viscosity model to compute to explain the transition from dunes to plane bed in the lower the fl ow over a wavy bottom. The Engelund-Smith approach fl ow regime. For low Froude numbers, bed load transport succeeded in predicting the instability of the bed, but could is the dominant mode due to low dimensionless Shields’ not predict the wave length for which the perturbation numbers, thus giving place to the formation of dunes. At growth rate was largest. In both of these cases the perturba- higher Shields’ numbers, suspended load will become the tion increases monotonically with decreasing wave length. predominant transport mode, resulting in a stable plane bed Fredsøe (1974) improved this description by including a for suffi ciently high shear stresses. The linear stability analy- bed-slope effect (i.e., gravity) on the bed load movement. sis of Engelund and Fredsøe (1974) was applied by Chen When gravitational effects are included, the sediment trans- and Nordin (1976) to explain the transition from dunes to port rate becomes smaller on the upslope part of the per- plane bed in the Missouri River. Recently, Coleman and turbation and larger on the downslope part. The net result Fenton (2000) have revisited potential fl ow analysis of bed is that less sediment makes it to the top of the perturbation instabilities. and more sediment is transported downslope from the top of Linear stability theories apply strictly to the inception of the perturbation, thus the bed-slope effect stabilizes the bed bed forms (Nakagawa and Tsujimoto 1984; Coleman and (Fredsøe 1996). Melville 1996). They are only able to predict whether or not Different types of sand waves may occur simultaneously sand waves are generated. Another limitation is that the out- on a fl at bed, but as the fl ow evolves only one type may reach come is independent of the initial perturbation amplitude. fully-developed conditions. As will be described below, Unsteady perturbations amplify or decay forever. Ji and usually ripples and dunes are the most common bed forms Mendoza (1997) applied weakly nonlinear stability theory observed for low Froude-number conditions (i.e., lower to the set of equations governing the motion of turbulent regime). Ripples are steeper and shorter than dunes and their fl ow and transport of sediment in rivers proposed by Fredsøe length depends on particle diameter. Whereas dune height (1974). They found that nonlinearities play an important

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 7799 11/21/2008/21/2008 5:43:435:43:43 PMPM 80 sediment transport and morphodynamics role in the formation of dunes. A comparison between the and Kinzelbach 2004), their stability and transformations results of linear and nonlinear models indicated that for a (Leeder 1983; Bennett and Best 1996; Robert and Ulhman nonlinear model, dunes are less unstable and less sensitive 2001; Schindler and Robert 2004), uses in estimating bed to changes in the intervening variables. Mendoza (1996) load transport (e.g. Engel and Lau 1980; Mohrig and Smith conducted a weakly nonlinear stability analysis to derive an 1996; Vionnet et al. 1998; Zhou and Mendoza 2005) and equation of the Landau-Stuart type for the amplitude time their role in determining fl ow resistance (e.g., Ogink 1988; evolution in steady, unidirectional turbulent fl ow over a Yoon and Patel 1996; Fedele and García 2001; Julien et al. movable boundary. It was found that nonlinear effects affect 2002; Wilbers 2004). Additionally, these studies have been both dune growth and celerity, and the dune height reaches conducted both in increasingly sophisticated and quantita- an equilibrium value at large time. A nonlinear model was tive laboratory studies (e.g. van Mierlo and de Ruiter 1988; also used by Zhou and Mendoza (2005) for analyzing the Shen et al. 1990; Lyn 1993; McLean et al. 1994, 1996, 1999; growth of sand wavelets from an initially fl at bed. The results Nelson et al. 1993; Bennett and Best 1995a ; Bennett and of the model were supported by the observations made by Venditti 1997; Kadota and Nezu 1999; Nelson et al. Coleman and Melville (1994, 1996). Stability analyses have 2001; Best and Kostaschuk 2002; Maddux 2002; Maddux also been used by Colombini et al. (1987) to study fi nite- et al 2003a, 2003b; Coleman et al. 2003; Fernandez et al. amplitude bars and Seminara and Tubino (1989) to assess 2005) and a growing quantifi cation of bed forms within the the role played by sediment bars on the initiation of river natural environment (e.g., Kostaschuk 1989; Gabel 1993; meandering, and by Seminara (1995) to study the effect of Julien and Klaassen 1995; Kostaschuk and Church 1993; sediment on the evolution and characteristics of bed Kostaschuk and Ilersich 1995; Kostaschuk and Villard 1996, forms. Komarova and Newell (2000) have also studied the 1999; Roden 1998;Villard and Kostaschuk 1998; Carling nonlinear dynamics of sand banks and sand waves in the et al. 2000a,b; Best et al. 2001; Williams et al 2003; . Coastal bed forms have been reviewed by Blondeaux Sukhodulov et al. 2004; Parsons et al. 2005). (2001). The simplifi ed problem of fl ow over fi xed (nonerodible) Recent years have seen great progress in our knowledge bedform shapes, motivated by the equilibrium problem, of bed form dynamics that has often been linked to sig- has received intense attention. Several laboratory studies nifi cant advances in our ability to monitor fl ow and dune have been conducted to elucidate the mean fl ow and tur- morphology in the laboratory and fi eld, and the increasing bulence characteristics above dunes (e.g., van Mierlo and sophistication of numerical modeling to capture not only de Ruiter 1988; Lyn 1993; Nelson et al. 1993; McLean the characteristics of the mean fl ow fi eld but realistically et al. 1994; Nelson et al 1995; Bennett and Best 1995; Best simulate the origins and motions of coherent fl ow struc- and Kostaschuck 2002; Fernandez et al. 2006). Flow resis- tures above dune beds (Best 2005). Most of this work has tance measurements over fi xed bedforms have been made by been summarized in several review articles, reports, and Shen et al. (1990) and Maddux et al. (2003a, 2003b). Such books, which have appeared since the publication of ASCE studies have been restricted to fl ows without mobile-bed Manual 54 (Vanoni 1975). They include those by Reynolds material; only recently studies of sediment-transporting fl ows (1976), Engelund and Fredsøe (1982), Ikeda and Parker over, though still artifi cial, bed forms have appeared (Cellino (1989), McLean (1990), Southard (1991), Kennedy and and Graf 2000; Venditti and Bennett 2000). Observations by Odgaard (1991), Seminara (1995), Best (1996), Blondeaux Ikeda and Asaeda (1983) of a sediment-laden fl ow over a and Seminara (2001), Yalin and da Silva (2001), ASCE rippled bed are an exception. (2002), Bridge (2003), Best (2005), and Parker and García Numerical simulations of fl ow over fi xed-bed forms (2006). have been performed by, among others, Mendoza and As detailed in Best (2005), in recent years there has been Shen (1990); Yoon and Patel (1996); and Zedler and Street great progress in our knowledge of bed form dynamics, (2001). Recently, Tjerry and Fredsøe (2005) replaced the which has often been linked to signifi cant advances in our semi-empirical fl ow model used earlier by Fredsøe (1982) ability to monitor fl ow and dune morphology in the labora- with a two-equation turbulence model, to compute numeri- tory and fi eld, and the increasing sophistication of numerical cally the morphology of dunes. More information on modeling to capture not only the characteristics of the mean fl ow and turbulence modeling over dunes can be found in fl ow fi eld but realistically simulate the origins and motions Chapter 16. Coherent turbulent structures in fl ows over bed of coherent fl ow structures above dune beds. Some of these forms have also received considerable attention (Ashworth advances are considered next. et al 1996). Their dynamics, even in simpler fl ows such Signifi cant advances in our understanding have been as uniform fl at-bed fl ows, is still complicated (e.g. Nelson achieved through studies that have been concerned both with et al. 1995; García et al. 1996; Niño and García 1996). the origin of bed forms (e.g. Yalin 1992; Nelson and Smith Some interpretations of suspended sediment fl ux measure- 1989; McLean 1990; Southard 1991; Bennett and Best 1996; ments in the presence of dunes have been formulated within Coleman and Melville 1996; Nikora and Hicks 1997; Gyr a coherent-structures conceptual framework (Lapointe

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8800 11/21/2008/21/2008 5:43:435:43:43 PMPM bed forms 81 1992, 1996; Bennett et al. 1998; Nikora and Goring 2000; dune morphology have shown promising results for labora- Venditti and Bennett 2000). Jackson (1976) was among tory-scale dunes. Particularly lacking, however, are similar the fi rst to bring attention to the sedimentological effect bed form predictors for large alluvial rivers (Schumm and of coherent structures, in particular the turbulent bursting Winkley 1994; Sambrook-Smith et al. 2006). Research phenomenon, in rivers. needs in this area have been recently addressed by an ASCE Field studies (e.g., Kostachuk and Villard 1996; Holmes Task Committee (ASCE 2002), Best (2005), and Hulscher 2003; Kostachuk et al. 2004; Parsons et al. 2005) with and Dohmen-Jansen (2005). detailed measurements not only of dune characteristics, but also of fl ow and transport, are valuable and daunting for 2.7.2 Dunes, Antidunes, Ripples, and Alternate Bars the same reason: they indicate the complexity of the real problem, which, together with practical constraints on fi eld The ripples, dunes, and antidunes illustrated in Fig. 2-36 measurements, make the analysis and interpretation of the are the classic bed forms of erodible-bed, open-channel data more diffi cult (van den Berg and van Gelder 1993). fl ow. On the one hand, they are the product of fl ow and Field measurements present additional diffi culties in inter- sediment transport, and on the other hand, they profoundly pretation and raise the question of how meaningful are direct infl uence fl ow and sediment transport. In fact, all of the bed comparisons between fi eld and laboratory/theoretical results. load equations quoted previously are strictly invalid in the Keulegan (1978) had to address this issue while attempting presence of bed forms. The adjustment necessary to render to estimate the channel roughness of interoceanic and them valid (i.e., removal of form drag) is discussed later in recommended that reliance should be placed on fi eld calibra- the chapter. tion of bed form predictors. Ripples, dunes, and antidunes are undular (wavelike) fea- Accurate prediction of stage and fl ow developments during tures that have wavelength λ and wave height Δ that scale a fl ood event must recognize the transient nature of erodible- with the fl ow depth H , as defi ned below. boundary roughness, implying clear knowledge about bed- 2.7.2.1 Dunes Well-developed dunes tend to have wave form generation and development processes as fl ows increase heights Δ scaling up to about one-sixth of the depth; i.e., and decrease in intensity (Julien and Klaasen, 1995; Julien et al. 2002). Amsler and García (1997) cite that large dunes Δ 1 (2-116) on the Parana River, Argentina, decreased in magnitude with H 6 increasing discharge, however, smaller superimposed dunes increased in size (Fig. 2-46). Large dunes with superimposed Dune wavelength can vary considerably. A fairly typical smaller dunes are shown in Fig. 2-35b. range can be quantifi ed in terms of dimensionless wavenum- For sedimentation engineering purposes, a minimum ber k , where contribution desired of a theory or model for bed form 2 π H development would be a reliable means of determining k (2-117) which equilibrium bed confi guration would be established, λ i.e. delineating stability boundaries, as well as a reliable predictor for bed form dimensions under equilibrium condi- This range is given by tions (i.e. wavelength and amplitude). As explained above, many attempts have been made to base such boundaries on 0.25 k 4.0 (2-118) theoretical stability models, but engineering approaches have been primarily based on dimensional analysis and empiricism. Dunes invariably migrate downstream. They are typi- Recent experimental and theoretical work (e.g., Coleman cally approximately triangular in shape and usually (but not and Melville 1996; Coleman and Fenton 2000; Coleman always) possess a slip face, beyond which the fl ow is sepa- et al. 2003; Zhou and Mendoza 2005) has focused on the rated for a certain length. bed form initiation process, in particular the conditions lead- A dune progresses forward as bed load accretes on the slip ing to the development of wavelets, the precursors of ripples face. Generally, very little bed load is able to pass beyond the (Coleman and Eling 2000). Although the mechanics of bed face without depositing on it, whereas most of the suspended form development (e.g., Coleman and Melville 1994) and load is not directly affected by it. rates of bed form growth for steady fl ows (e.g., Nikora and Dunes are characteristic of subcritical fl ow in the Froude Hicks 1997) have been clarifi ed, practical implications sense. In a shallow-water (long-wave) model, the Froude and models accessible to the engineer remain to be elabo- criterion dividing subcritical (Fr 1 ) and supercritical rated. For example, how and at what rates bed forms change ( Fr 1 ) fl ow is with increasing and decreasing fl ows remains to be quanti- fi ed. Recent computations by Tjerry and Fredsøe (2005) of Fr 1 (2-119a)

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Fig. 2-36. Schematic of different bedforms. Note: F ; d sediment size.

1 Fr 2 tanh (k) (2-120b) where the Froude number is given by, k

U must be satisfi ed. Both dunes and antidunes cause the water Fr (2-119b) gH surface to undulate as well as the bed. In the case of dunes, the undulation of the water surface is usually of much smaller ° Dunes, however, do not qualify as long waves, in that amplitude than that of the bed; the two are nearly 180 out their wavelength is of the order of the fl ow depth. A detailed of phase. potential fl ow analysis over a wavy bed yields the following Let c denote the wave speed of the dune. The bed load (wave number-dependent) criterion for critical fl ow over a transport rate by dunes can be estimated as the volume of bedform (Kennedy 1963), material transported forward per unit bed area per unit time by a migrating dune (Simons et al. 1965b). If the dune is 1 approximated as triangular in shape, the following approxi- Fr 2 tanh (k) (2-120a) mation holds (Engel and Lau 1980; Havinga 1983) k

Note that as k → 0 (λ → ∞), tanh ( k ) → k , and condition 1 q ≅ Δ c ()1 λ (2-122) (2-119a) is recovered in the long-wave limit. For dunes to 2 p occur, then, the condition

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8822 11/21/2008/21/2008 5:43:435:43:43 PMPM bed forms 83 where Values lower than the above are associated with upstream- migrating antidunes. Δ and c amplitude and celerity, respectively, of the bed Relations (2-120a), (2-120b), (2-124) and (2-125) were form; and obtained from a potential fl ow analysis over a wavy bed. A λ porosity of the sediment bed. p phase diagram showing these conditions is shown in Fig. Rubin and Hunter (1982) proposed that the transport rate 2-37. The boundaries they defi ne are modifi ed when turbu- given by Eq. (2-122) be called the bed form transport rate lent shear fl ows transporting sediment are considered (e.g., instead of the bed load transport rate. An elegant derivation Engelund and Fredsøe 1982). For instance, Wan and Wang of this equation can be found in Ten Brinke (1999). The (1994) have observed the absence of dunes for volumetric celerity of dunes c as a function of the Froude number Fr can sediment concentrations greater than about 0.08. be estimated with an empirical relation proposed by Kondap 2.7.2.3 Ripples Ripples are dune-like features that and Garde (1973), occur most of the time in the presence of a viscous sublayer. The existence of a viscous sublayer does not imply that the fl ow is either laminar or turbulent. Rather, when the fl ow is c turbulent, the existence of a well-defi ned viscous sublayer 0. 021 Fr3 (2-123) U implies fl ow in the turbulent smooth regime rather than the turbulent rough regime. Ripples look very much like dunes in that they migrate downstream and have a pronounced slip The celerity of dunes is a small fraction of the mean fl ow face. They generally are much more three-dimensional in velocity. Kondap and Garde found that for grain sizes in the structure than dunes, however, and have little effect on the range between 0.18 mm to 2.28 mm, sediment size seems water surface. to have a negligible effect on the celerity of dunes. The data As mentioned earlier, many authors have suggested that were from laboratory experiments and low gradient streams a criterion for the existence of ripples is the existence of a having depths of less than 1 m. For the case of large sand viscous sublayer. Recalling that the thickness of the viscous bed rivers, Fedele (1995) obtained an empirical relation to δ ν / sublayer is given by ν 11.6 u * , it follows that ripples estimate the velocity of dunes in the Paraná and Paraguay will form when Rivers in South America. Vionnet et al (1998) have also pro- posed a methodology to compute sediment transport from dune celeritiy and amplitude based on kinematic-wave the- uD R * 11. 6 (2-126) ory. More recently, Serra and Vionnet (2006) extended the p ν analysis to account for the transport of smaller dunes super- imposed on larger ones (see Fig. 2.35b). 2.7.2.2 Antidunes Antidunes are distinguished from Raudkivi (1990; 1997) indicates that ripples only develop dunes by the fact that the water surface undulations are in fi ne grained sediments having mean sizes of less than 0.7 nearly in phase with those of the bed. They are associated to 0.9 mm, and for shear velocity Reynolds number (Re * / with supercritical fl ow, in the sense that u * D v ) of less than 10 to 27 (see Fig. 2-39b).

1 Fr 2 tanh (k) (2-124) k

Antidunes may migrate either upstream or downstream. Upstream-migrating antidunes are usually rather symmetrical in shape and lack a slip face. Downstream-migrating anti- dunes are rather rarer; these have a well-defi ned slip face and look rather like dunes. The distinguishing feature is the water surface undulations, which are very pronounced in the case of antidunes. The potential-fl ow criterion dividing upstream- migrating antidunes from downstream-migrating antidunes is (Kennedy 1963)

1 Fig. 2-37. Phase diagram for dunes and antidunes based on liner Fr 2 (2-125) kk tanh ( ) potential theory over a wavy bed (after Parker 2005).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8833 11/21/2008/21/2008 5:43:455:43:45 PMPM 84 sediment transport and morphodynamics Richards (1980) gave the following criteria for ripple In straight streams, the minimum channel slope S neces- formation sary for alternate- formation is given by (Jaeggi 1984) 0.0007 kz0 0.16 , where k is the wave number (2π/λ) and z is the Smith (1977) 0 ⎡ 0.15 ⎤ roughness-length parameter defi ned as ⎛ B ⎞ exp ⎢ 1.07 ⎜ ⎟ M ⎥ ⎢ ⎝ ⎠ b ⎥ ⎣ D g ⎦ (2-129a) S ττ ⎛ ⎞ z 26. 3 0 c k (2-127) B 0 ()ρρ g n 12.9 ⎜ ⎟ s ⎝ D g ⎠

with k n 0.033 k s , where k s is the Nikuradse roughness height. B is the channel width, D g is the geometric mean size of the Karim (1999) proposed a relation for predicting ripples bed sediment as given by Eq. (2-41a), and M b is a parameter from laboratory data reported by Guy et al. (1966). He found that varies from 0.34 for uniform-size bed material to 0.7 for

that ripples would only occur if poorly-sorted material. Scour depth (S d ) due to alternate bar formation can be estimated with:

* 80 (2-128a) B Δ S d 0.76 AB 0.15 uD U ⎛ B ⎞ (2-129b) * 50 6 where N* , U is the mean fl ow velocity, ⎜ ⎟ ν ⎝ D g ⎠ gRD50 and D50 is the median grain size. Coleman and Melville (1994) studied the relation between Δ celerity of small bed forms (i.e., wave lets) as a function of where AB is the total height of the alternate bar. bed-form height and found that such relation can be approxi- Using dimensional analysis and experimental observa- mated by the expression tions, Sukegawa (1973) found that the condition for the for- mation of alternate bars in straight channels is given by the following: c(Δ 35. )1.3 40 (2-128b) 2/3 u2 ⎛ gB ⎞ * S (2-130) 2 5⎜ ⎟ where u*c ⎝ u*c ⎠ c' dimensionless celerity / ⎡()ττ**⎤ ; cu⎣ ** ucc()⎦ The wavelength of alternate bars is approximately six to H' dimensionless bed-form height Δ / D ; and g ten times the channel width (i.e., λ 6 to 10B ; Yalin 1992). D geometric mean sediment diameter. g Channel will affect the celerity of alternate bars When plotted this relation results in a hyperbolic parab- (García and Niño 1993). As the sinuosity increases, the ola, which indicates that as the bed form increases in height, migration speed of alternate bars decreases with respect to its celerity decreases according to Eq. (2-128b). that observed in a straight channel. In general, the celerity 2.7.2.4 Alternate Bars Alternate bars are bed forms of alternate bars is always less than 0.01% of the mean fl ow most commonly found in straight alluvial channels (Bridge velocity in the channel. 2003). Their geometry is three-dimensional. Navigation Most of the experimental and theoretical work done conditions and streambank stability can be affected by to characterize alternate bars in straight channels has been alternate bars. When alternate bars are present, pools done for steady fl ow conditions (e.g. Sukegawa 1973; Ikeda develop on alternate sides of the channel and the fl oor 1984, Jaeggi 1984; Kuroki and Kishi 1985, Seminara and from pool to pool. Under these conditions, the Tubino 1989; García and Niño 1993; Lanzoni 2000a, 2000b; fl ow might start to attack the stream banks, eventually caus- Knaapen et al. 2001). One exception is the work of Tubino ing bank erosion (e.g., Jang and Shimizu 2005) and lead- (1991), who analyzed the growth of alternate bars in unsteady ing to the initiation of stream meandering (Blondeaux and fl ow. Tubino (1991) noted that the characteristic time scale Seminara 1985; Rhoads and Welford 1991). The pools for bar development and the time scale of natural unsteady formed by alternate bars also provide habitat and play an fl ow events are typically of the same order of magnitude. The important role in stream ecology. effect of sediment sorting on both the growth and dynamics

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8844 11/21/2008/21/2008 5:43:465:43:46 PMPM bed forms 85 of alternate bars has been analyzed by Seminara (1995) and Lanzoni and Tubino (1999).

2.7.3 Progression of Bed Forms Various bed forms are associated with various fl ow regimes. In the case of a sand-bed stream with a characteristic size less than about 0.5 mm, a clear progression is evident as fl ow velocity increases. This is illustrated in Fig. 2-36 presented above. The bed is assumed to be initially fl at. At very low imposed velocity U , the bed remains fl at because no sedi- ment is moved. As the velocity exceeds the critical value, ripples are formed. At higher values, dunes form and coex- ist with ripples. For even higher velocities, well-developed dunes form in the absence of ripples. At some point, the velocity reaches a value near the short-wave critical value in the Froude sense, i.e., Eq. (2-120a). Near this point, the τ Fig. 2-38. Variations of bed shear stress b and Darcy-Weisbach dunes are often suddenly and dramatically washed out. This friction factor f 8 Cf with mean velocity U in fl ow over a fi ne- results in a fl at bed known as an upper-regime (supercriti- sand bed (after Raudkivi, 1990). cal) fl at bed. Further increases in velocity lead to the forma- tion of antidunes, and fi nally to the chute and pool pattern. The last of these is characterized by a series of hydraulic jumps. antidunes appears; it is usually not as pronounced as that The effect of bed forms on fl ow resistance can be of dunes. explained as follows. As noted earlier for equilibrium fl ows In the case of a bed coarser than 1.0 mm, the ripple regime in wide straight channels, the relation for bed resistance can is replaced by a zone characterized by a lower-regime (sub- be expressed in the form critical) fl at bed. Above this lie the ranges for dunes, upper- regime fl at bed, and antidunes.

τρ C 2 (2-131) bfU 2.7.4 Dimensionless Characterization of Bed Form Regime Based on the preceding arguments, it is possible to identify where C f is bed friction coeffi cient. If the bed were rigid and the fl ow rough, C would vary only weakly with the fl ow, at least three dimensionless parameters that govern bedforms f * according to the logarithmic law embodied in Eq. (2-19). at equilibrium fl ow. These are Shields stress parameter τ , τ shear Reynolds number R u D / v , and Froude number As a result, the relation between b and U is approximately p * parabolic for a fl at rough bed. Fr U/ gH . A characteristic feature of sediment trans- The effect of bed forms is to increase the bed shear stress port is the proliferation of dimensionless parameters to values often well above that associated with the skin fric- (Vanoni 2006). This feature notwithstanding, Parker and τ tion of a rough bed alone. In Fig. 2-38, a plot of b versus U Anderson (1977) have shown that equilibrium relations is given for the case of an erodible bed. At very low values of of sediment transport for uniform material in a straight U , the parabolic law is followed. As ripples and then dunes channel can be expressed in terms of just two dimension- are formed, the bed shear stress rises to a maximum value less hydraulic parameters, along with a particle Reynolds (Robert and Uhlman 2001). At this maximum value, the /ν number (e.g., R p or RgRDDep ) and a measure of den- value of C is seen to be as much as fi ve times the value with- ρ ρ / ρ f sity difference (e.g., R ( s ) ). out dunes. It is clear that dunes play a very important role as In the case of bed forms, then, the following classifi cation regards bed resistance. The increased resistance results from can be proposed: form drag in the lee of the dune. As the fl ow velocity increases further, dune wavelength ππ gradually increases and dune height diminishes, leading to bedform type f ()12,;RRep , (2-132) a gradual reduction in resistance. At some point, the dunes are washed out, and the parabolic law is again satisfi ed. Here, any independent pair of dimensionless hydraulic π π At even higher velocities, the form drag associated with variables 1 , 2 applicable to the problem may be specifi ed,

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8855 11/21/2008/21/2008 5:43:465:43:46 PMPM 86 sediment transport and morphodynamics because any one pair can be transformed into any other inde- pendent pair. For example, the pair (τ * , F r ) might be used, or alternatively the pair (S , H / D ). One of the most popular discriminators of bed form type is not expressed in dimensionless form at all. It is the dia- gram proposed by Simons and Richardson (1966), shown in Fig. 2-39a. In this diagram, regimes for ripples, dunes, tran- sition to upper-regime plane bed, and upper-regime plane bed and antidunes are shown. The two hydraulic param- τ eters are abbreviated to a single one, stream power b U , and particle Reynolds number is replaced by grain size D . The diagram is applicable only to sand-bed streams of relatively small scale. A similar bed phase discriminator which is popular in the geology community was proposed by Boguchwal and Southard (1990). An adaptation of the Boguchwal-Southard predictor made by Ashley (1990) is shown in Fig. 2-39b. It is a plot of laboratory obser- vations of mean fl ow velocity against median sediment size covering the range of fi ne to coarse sand for fl ow depths between 0.20 and 0.40 m. Because the Boguchwal- Southard diagram uses a logarithmic scale for grain size instead of a normal scale like the Simons-Richardson dis- criminator, it shows that for fi ner grain sizes a marked increase in fl ow velocity can wash out existing ripples and transition abruptly to upper plane bed conditions with- out the appearance of dunes. The transition to the upper- regime is shown to take place for Froude numbers smaller (a)

(b) Fig. 2-39. Bed form discriminators proposed by (a) Simons and Richardson (1966) and (b) Boguchwal and Southard (adapted from Ashley 1990).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8866 11/21/2008/21/2008 5:43:475:43:47 PMPM bed forms 87 than one ( Fr 1) depending on the sediment size. The streams having mean fl ow depths less than 3 m (10 feet) and fact that the bed phases shown in this diagram have been for the design of movable-bed laboratory experiments (e.g., standardized to a given water temperature, highlights both Zwamborn 1966; 1981). the importance of accounting for temperature effects as In fact, dunes can occur over a limited range in the case well as the need to use dimensionless parameters to dis- of coarse material (e.g., Dinehart 1989). This is illustrated in criminate between bed-stability phases. Several dimen- Figs. 2-41(a) and 2-41(b), which shows the discriminators sionless bed form and fl ow regime discriminators are originally advanced by Chaubert and Chauvin (1963) and presented next. Bonnefi lle-Pernecker (Vollmers and Giese 1970; Bechteler The discriminator originally proposed by Liu (1957) and et al. 1991). The Chaubert-Chauvin diagram plots the Shields τ* later extended by Simons and Richardson (1961), is shown parameter ( ) versus the shear stress Reynolds number Re * . in Fig. 2-40. Liu’s discriminator uses one dimensionless A third parameter in this diagram is the Valenbois-Bonnefi lle / τ* hydraulic parameter, u * vs (a surrogate for ), and the par- dimensionless particle diameter given by ticle shear Reynolds number Re u D / v. The diagram is 13/ * * ⎛ gR⎞ of interest in that it covers sizes much coarser than those DD* ⎜ ⎟ (2-133) ⎝ ν2 ⎠ of Simons and Richardson and Boguchwal and Southard (Fig. 2-39). It is seen that the various regimes become com- The Chaubert-Chauvin diagram shows that D * must be pressed as grain size increases. For the case of very coarse less than about 15 for ripples to form. Employing R 1.65, material, the fl ow must be supercritical for any motion to g 981 cm/s2 and ν 0.01 cm 2 /s, it is seen that the condi- occur. As a result, neither ripples nor dunes are to be expected. tion D * 15 corresponds to a value of D of approximately According to Simons and Senturk (1992), the Liu diagram 0.6 mm. does not give acceptable results for fi eld conditions because For coarser grain sizes, the dune regime is preceded by few fi eld data were used in the analysis. Nevertheless, the a fairly wide range consisting of lower-regime fl at bed. diagram should be useful for predictions of bed form type in Many gravel-bed rivers never leave this lower-regime fl at

Fig. 2-40. Criteria for bed forms originally proposed by Liu (1957) and later extended by Simons and Richardson (1961).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8877 11/21/2008/21/2008 5:43:495:43:49 PMPM 88 sediment transport and morphodynamics bed region, even at bankfull fl ow. The Chaubert-Chauvin A complete set of bed form diagrams for the case of sand diagram in Fig. 2-41(a) is not suited for the description of is shown in Figs. 2-42(a) to 2-42(f); they are due to Vanoni upper-regime fl ow. (1974) and were not published in ASCE Manual 54 . The two The Bonnefi lle-Pernecker diagram displayed in hydraulic parameters are the Froude number Fr and the rela- * / / Fig. 2-41(b) plots values of D versus Re* . This diagram tive fl ow depth H D ( d d 50 in the diagram); the particle / shows the transition from the lower regime (ripples) to fl at Reynolds number used in the plot is equal to the ratio R ep bed condition and then to the upper regime (antidunes), for R 1/2 , and the submerged specifi c gravity R is set constant at * * D 20 as Re * increases. For coarser sediment, D 20, 1.65. Note how the transition to upper regime occurs at pro- the diagram shows the lower regime conditions (dunes) fol- gressively lower values of Fr for relatively deeper fl ow (in lowing a narrow range of fl at-bed transport conditions. An the sense that H/D becomes large). Shen et al. (1978) con- interesting aspect of this diagram is that it shows a narrow fi rmed that Vanoni’s bedform discriminator fairly well cap- set of conditions, roughly defi ned by 20 Re * 45 and tures temperature effects on river bottom confi guration, such 7 D * 20, for which ripples are superimposed on dunes. as those commonly observed in the Missouri River. Bechteler et al. (1991) have used this diagram to analyze One of the most complete bed form classifi cation sediment transport conditions in alpine rivers. schemes that includes both the lower and the upper regime

(a)

Fig. 2-41. Bed form classifi cation diagrams (a) after Chabert and Chauvin (1963) and Bonnefi lle- Pernecker (after Bechteler et al. 1991).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8888 11/21/2008/21/2008 5:43:505:43:50 PMPM bed forms 89

(b) Fig. 2-41. Bed form classifi cation diagrams (a) Chabert and Chauvin (1963) and Bonnefi lle- Pernecker (after Bechteler et al. 1991). (Continued)

(a) Fig. 2-42. Bed-form chart (a) Rg 0.11 and 2.7 to 4.2 (D50 0.1 and 0.88 mm); (b) Rg 4.5 to 10 (D50 0.12 to 0.20 mm); (c) Rg 10 to 16 (D50 0.15 to 0.32 mm); (d) Rg 16 to 26 (D50 0.228 to 0.45 mm); (e) Rg 24 to 48 (D50 0.40 to 0.57 mm) ; (f) Rg 82 to 92, 130, 140 to 200 (D50 0.9, 1.20, 1.35 mm) (after Vanoni 1974).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 8899 11/21/2008/21/2008 5:43:525:43:52 PMPM 90 sediment transport and morphodynamics

(b)

(c) Fig. 2-42. Bed-form chart (a) Rg 0.11 and 2.7 to 4.2 (D50 0.1 and 0.88 mm); (b) Rg 4.5 to 10 (D50 0.12 to 0.20 mm); (c) Rg 10 to 16 (D50 0.15 to 0.32 mm); (d) Rg 16 to 26 (D50 0.228 to 0.45 mm); (e) Rg 24 to 48 (D50 0.40 to 0.57 mm) ; (f) Rg 82 to 92, 130, 140 to 200 (D50 0.9, 1.20, 1.35 mm) (after Vanoni 1974). (Continued)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 9900 11/21/2008/21/2008 5:43:555:43:55 PMPM bed forms 91

(d)

(e) Fig. 2-42. Bed-form chart (a) Rg 0.11 and 2.7 to 4.2 (D50 0.1 and 0.88 mm); (b) Rg 4.5 to 10 (D50 0.12 to 0.20 mm); (c) Rg 10 to 16 (D50 0.15 to 0.32 mm); (d) Rg 16 to 26 (D50 0.228 to 0.45 mm); (e) Rg 24 to 48 (D50 0.40 to 0.57 mm) ; (f) Rg 82 to 92, 130, 140 to 200 (D50 0.9, 1.20, 1.35 mm) (after Vanoni 1974). (Continued)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 9911 11/21/2008/21/2008 5:43:595:43:59 PMPM 92 sediment transport and morphodynamics

(f) Fig. 2-42. Bed-form chart (a) Rg 0.11 and 2.7 to 4.2 (D50 0.1 and 0.88 mm); (b) Rg 4.5 to 10 (D50 0.12 to 0.20 mm); (c) Rg 10 to 16 (D50 0.15 to 0.32 mm); (d) Rg 16 to 26 (D50 0.228 to 0.45 mm); (e) Rg 24 to 48 (D50 0.40 to 0.57 mm) ; (f) Rg 82 to 92, 130, 140 to 200 (D50 0.9, 1.20, 1.35 mm) (after Vanoni 1974). (Continued)

has been advanced by van Rijn (1984c, 1993). Both labora- rivers where the Froude number is never larger than 0.2 to tory and fi eld data were used to develop van Rijn’s diagram. 0.3, even during large fl oods, and the fl ows never reach upper The scheme is based on the Bonnefi lle dimensionless par- regime conditions (Julien 1992; Julien and Klaassen 1995;

ticle diameter D * introduced earlier and the transport-stage Amsler and García 1997). For instance, in the Mississippi parameter T , defi ned, respectively, as: River large bed forms are observed for values of T as high as 50 (van Rijn 1996). More fi eld observations in large rivers 13/ are needed to further test van Rijn’s method. ⎛ gR⎞ 2/ DD⎜ ⎟ R 3 (2-134) Brownlie (1983) studied the transition regime and sug- * 50 ⎝ ν2 ⎠ ep gested that it can be delineated by the value of the grain / Froude number or the sediment number, Ns U gRD50 , and and the ratio of grain diameter to viscous sublayer thick- / δ δ / ness, D 50 v , where v 11.6v u * . For slopes greater than ττ** 0.006, Brownlie found that all the bed forms were in the T sc (2-135) upper regime, while for slopes less than 0.006, he suggested τ* c the following relationships for the lower limit of the upper regime based on both fl ume and river data: τ* Here s is the bed shear stress due to skin or grain friction, τ* and s is the critical shear stress for motion from the Shields diagram. 2 N D ⎛ D ⎞ Van Rijn’s diagram indicates that ripples form when both log.s 0 02469 0 . 1517 log50 0 . 8381⎜ log 50 ⎟ * δ δ N v ⎝ ν ⎠ D* 10 and T 3, as shown in Fig. 2-43. Dunes are present s elsewhere when T 15, dunes wash out when 15 T 25, D for 550 2 and upper fl ow regime starts when T 25. However in large δ v rivers, large dunes are found to exist for T 25, casting doubts on the applicability of van Rijn’s predictor to large alluvial (2-136a)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 9922 11/21/2008/21/2008 5:44:035:44:03 PMPM bed forms 93 These equations for the transition zone are shown in Ns D50 log log125 . for 2 (2-136b) Fig. 2-44. It can be seen that the variable D / δ , the ratio N* δ 50 v s v of grain size to viscous sublayer thickness, refl ects the infl u- ence of viscous effects near the bed, thus indicating tempera- Here N * 1.74 S 1/3 and S is the slope. For the upper s ture dependence of the bedforms. A very similar fl ow-regime limit of the lower regime, Brownlie proposed the following predictor, which accounts explicitly for viscous effects best-fi t equations: though the sediment fall velocity, was proposed earlier by Cruickshank and Maza (1973) and is presented below. 2 N DD⎛ ⎞ Karim (1995) developed bed form regime predictors in logs 0 . 2026 0 . 07026 log50 0 . 9330⎜ log 50 ⎟ * δδthe form of limiting Froude numbers, defi ned as N s vv⎝ ⎠ D for 550 2 (2-137a) 025. δ ⎛ H ⎞ v Ft 2. 716⎜ ⎟ N D ⎝ D50 ⎠ logs log08 . for 50 2 * (2-137b) 027. (2-138a,b) N δ ⎛ ⎞ s v H Fu 4. 785⎜ ⎟ ⎝ D50 ⎠

where Ft is the beginning of the transition regime (from the lower fl ow regime), and F u gives the beginning of the upper regime. Based on these defi nitions for limiting Froude ( Fr U/ gH ) numbers proposed by Karim (1995), the bed form geometry type can be determined as follows: Lower regime (ripples, dunes) Fr Ft Transition regime (washed out dunes) Ft Fr Fu Upper regime (plane bed, antidunes) Fr Fu Karim (1995) also used Fr 0.8 as a limit for the exis- tence of antidunes.

Fig. 2-44. Delineation of bedform transition zone from lower regime to upper regime (after Brownlie, 1983). This diagram gen- Fig. 2-43. Bedform clasifi cation (after van Rijn 1984c, 1993). erally agrees with the more detail diagrams of Vanoni (Fig. 2-42).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 9933 11/21/2008/21/2008 5:44:075:44:07 PMPM 94 sediment transport and morphodynamics As part of a study on the transition of ripples to plane bed with bed material particle sizes ranging from 80 to 5,100 μ m in fl ows over fi ne sand and silt, resembling Chinese rivers, and fl ow depths ranging from 1 to 15 m were also included. van den Berg and van Gelder (1993) proposed a stability This diagram shows the transition from lower regime (ripples) diagram (Fig. 2-45) that considers the bed shear stress com- to upper-stage fl at-bed conditions, taking place for D * 20. It ponent responsible for sediment transport. In the Van der appears that the position of the upper boundary of dune exis- Berg-Van Gelder diagram a mobility parameter related to tence in fi ne sand as revealed from fl ume data is equally appli- grain roughness θ ', as proposed by van Rijn (1984c), appears cable to greater depths of natural river fl ows. on the ordinate given by It is clear though that many more fi eld observations using newer technology are needed in order to obtain more reli- U 2 θ (2-139a) able bed stability diagrams (e.g., Parsons et al. 2005). The 2 gRD50 () C challenges associated with the production of a universal bed form stability diagram are discussed by Ashley (1990) and with Best (1996). 4H C18log (2-139b) D90 2.7.5 Prediction of Equilibrium Bed Form Dimensions and the Valenbois-Bonnefi lle dimensionless particle diam- The equilibrium dimensions of ripples have been studied for eter D in the abscissa a wide range of sizes by a number of authors. Most studies * indicate that ripple dimensions are controlled by sediment / ⎛ gR⎞ 13 size and are independent of fl ow depth. DD ⎜ ⎟ (2-139c) * 50 ⎝ ν2 ⎠ Baas (1993) proposed the following equations for ripple wavelength and height at equilibrium where U is the mean fl ow velocity, H is the fl ow depth, R ( ρ ρ ) / ρ is the submerged specifi c gravity of the sediment, s λ 75. 4log D50 197 (2-140a) D50 and D 90 are the sediment sizes for which 50% and 90% of the bed sediment is fi ner, respectively, and ν is the kine- Δ matic viscosity of water. In all, 372 fl ume experiments with 34. log D50 18 (2-140b) median particle diameters ranging from 11 to 4,080 μm and fl ow depths up to 1 m, were used to produce the bed stability diagram shown in Fig. 2-45. A total of 262 fi eld observations or

Δ 0. 097 18. 16 D50 (2-140c)

Raudkivi (1997) found that data on ripple length show a dependence on grain size as follows

λ 245 D035. (2-141a)

where the grain diameter D is in mm. According to Raudkivi (1997), the ripple steepness decreases with increasing grain size,

Δ/λ 0. 074 D0. 253 (2-141b)

Additional relations to estimate the characteristics of ripples can be found in Yalin (1985) and Mantz (1992). Coleman and Melville (1996), Nikora and Hicks (1997), Coleman and Eling (2000) and Coleman et al. (2003) provide details about the characteristics of sand wave development, in particular of wavelets, the precursors of ripples and dunes. Coleman Fig. 2-45. Bed form in relation to grain mobility number and et al. (2003) report that length of wavelets in open channels grain size parameter (after van den berg and van Gelder 1993). and closed conduits is a function only of particle size and

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 9944 11/21/2008/21/2008 5:44:105:44:10 PMPM bed formst 95 can be estimated with, λ 175D 0.75 , where both Δ and λ The bed form length obtained by dividing these two equa- are in mm. tions, λ 7.3H, is close to the value, λ 2πH , proposed For the case of dunes, Julien and Klaassen (1995) analyzed by Yalin (1964). The agreement with laboratory data was a large number of laboratory and fi eld data and proposed the found to be quite good, but both curves tend to underestimate simple relation the bedform height and steepness of fi eld data (Julien 1992; Julien and Klaassen 1995). As mentioned above, lower-regime λ bedforms are observed in the at values of 625. (2-143) H T well beyond 25. Van Rijn’s curves largely underpredict the dimensions of bed forms in large rivers (van Rijn 1996). as a reasonable fi rst approximation for dune length. They Field measurement of bed forms in large sand-bed riv- proposed the following equation for dune height; ers such as the Paraná in South America, the branches of the Rhine in the The Netherlands, and the Mississippi in the USA, show that the fl ows never leave the lower fl ow regime ⎛ D ⎞ 03. Δ 25. H ⎜ 50 ⎟ (2-144) and while large dunes might elongate and disappear as the ⎝ H ⎠ fl ow discharge increases, the smaller bed forms, originally superimposed on the larger sand waves, do remain along the In Julien and Klaassen’s analysis, the dune length is bottom. This behavior can be observed in Fig. 2-46 for the Δ described by case of the Paraná River, where the amplitude D and steep- Δ / λ Δ ness D of large dunes as well as the amplitude d and 03. Δ / λ ⎛ H ⎞ steepness of smaller dunes d are plotted as a function of λ 25. Δ ⎜ ⎟ (2-145) mean fl ow discharge Q for the year 1987 (Amsler and García ⎝ D ⎠ 50 1997). The smaller dunes are seen to readily re spond to variations in fl ow discharge, reaching a maximum am plitude In a discussion of the work of Julien and Klaassen (1995), and steepness around the peak fl ow discharge. On the other Amsler and García (1997) noted that Julien and Klaassen’s hand, the larger dunes display more inertia with respect data indicate that large dune heights increased monotoni- to fl ow change, responding with an increase in amplitute cally with increasing discharge (i.e., bed shear stress), which just before the peak discharge and a minimum in steepness is contrary to what had been observed on the Paraná River, when the fl ow discharge reaches its largest value. It is clear Argentina, and in other large rivers around the world. The that bed form predictors that have been mainly developed Julien-Klaassen formulation implicitly assumes hydraulically with laboratory data cannot be expected to capture the non- rough fl ow conditions and does not account for the existence linear behavior displayed in Fig. 2-46. More research using of viscous effects. Schreider and Amsler (1992) proposed an emerging technologies (e.g., ADCP, multibeam sonars’ empirical set of curves for dune steepness which includes etc.) is needed to be able to understand the morphodynam- viscous effects. They used laboratory observation as well ics of bed forms in large alluvial rivers (e.g., Parsons et al. as data from the Missouri River, USA, the Paraná River, 2005). Argentina, and the ACOP channels in Pakistan. Their anal- Karim (1999) used an approach similar to that of Kennedy ysis implicitly accounts for the existence of smaller dunes and Odgaard (1991) whereby energy loss because of form superimposed on large dunes (see Fig. 2-35b). drag is related to the head loss across a sudden expansion In the lower regime, the geometry of bed forms refers to in an open channel. Essentially this is Carnot’s formula for Δ λ representative dune height and wavelength as a function head loss, which had been applied erlier to bed forms by of the average fl ow depth H , median bed particle diameter Engelund and Hansen (1966) and by Fredsøe (1989) to esti- D , and other fl ow parameters such as the transport-stage 50 mate stage-discharge relations in sand bed streams. Karim T parameter , and the grain shear Reynolds number Rp . The (1999) presents the following equation for the geometry of bedform height and steepness predictors proposed by van ripples, dunes, and transition bedforms as Rijn (1984c) are 073. ⎡⎪⎧ ⎛ ⎞ 033. ⎪⎫ λ 120. ⎤ 0.3 D50 2 ⎛ ⎞ Δ ⎛ ⎞ ⎢⎨S 0. 0168 F ⎬ ⎥ D50 () 0.5 T( ) e ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ 0.11 ⎝⎜ ⎠⎟ 1 e 25 T (2-146a) Δ ⎢⎩⎪ H ⎭⎪ H ⎥ H H ⎢ ⎥ H ⎢ 047. F 2 ⎥ (2-147) ⎢ ⎥ and ⎣⎢ ⎦⎥

Δ ⎛ ⎞ 0.3 D50 0.5 T where S is the energy slope. Karim further recommends 0.015 ⎜ ⎟ () 1e ( 25 T ) (2-146b) e λ ⎝ H ⎠ solving for λ / H using: the equation of Julien and Klaassen

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Fig. 2-46. Variation of bed form characteristics, large dunes with superimposed smaller dunes, with fl ow discharge for a reach of the middle Paraná River (After Amsler and García 1997).

(1995), i.e., λ / H 6.25 for dunes; Yalin’s (1964) relation to fully developed stage. Using observations of fl ow over a for ripples, i.e., λ 1000 D; and Kennedy’s equation for the negative bottom step, Fredsøe (1982) made an analogy with wavelength of antidunes, i.e., λ / H 2 π Fr 2, where Fr is the fl ow separation that takes place over the top of a dune. the Froude number. Karim (1999) compares his approach Fredsøe’s model accounts for the effect of suspended load (Eq. 2-147) with the relations of Yalin (1964), Ranga Raju and bed load on the dimensions of dunes as the fl ow inten- and Soni (1976), Allen (1978), van Rijn (1984), Kennedy sity increases. Using computations of bed shear stress along and Odgaard (1991), Julien and Klaassen (1995), and Karim a dune profi le (e.g., McLean and Smith 1986; Mendoza (1995). Karim’s method performs as well as or better than and Shen 1990), Fredsøe’s model was used to solve an the rest of the relations tested. However, although it per- approximate analytical expression for dune shape (Fredsøe forms fairly well when tested with laboratory data, it does and Deigaard 1992, p267). Recently, Tjerry and Fredsøe not capture the behavior observed in the fi eld. At the same (2005) estimated the bed shear stress needed in Fredsøe’s time, Karim’s approach is one of the few available meth- dune model with a two-equation turbulence model, and ods for predicting sand wave equilibrium dimensions over used these results to investigate the shape and dimensions a wide range of conditions. It can be applied to various bed of dunes caused by a turbulent fl ow over an erodible bed. forms, i.e., ripples, dunes, antidunes/standing waves, and Other attempts at developing analytical expressions similar transitional bed regimes. However, it needs to be tested with to Fredsøe’s for the shape of dunes and ripples include the more observations from large alluvial rivers. work among others of Haque and Mahmood (1983, 1985). Most of the work in the literature has concentrated on The evolution from wavelets to ripples to dunes and to obtaining equilibrium dimensions of ripples and dunes. One megadunes seems to suggest that there is self-similarity in exception is the approach proposed by Fredsøe (1982) to the mechanics of sand waves, as implied by the work of estimate the evolution of dunes, all the way from inception Raudkivi and Witte (1990), Coleman and Melville (1994),

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 9966 11/21/2008/21/2008 5:44:115:44:11 PMPM bed forms 97 Nikora and Hicks (1997), and Jerolmack et al. (2006). This standard Manning-type relation for a nonerodible bed, the hypothesis is supported by the work of Flemming (2000) following should hold: who prepared a log-log plot of height (Δ ) versus wavelength ( λ ) for 1,491 observations of subaqueous bedforms. The col- 1 2/ /2 lapse of the data was rather remarkable and a single “dis- U HS31 (2-149a) n continuity” could be observed in the continuum of lengths at slightly less than 1 m, which provides support for the dis- tinction between current ripples and large-scale bed forms. Here, the channel is assumed to be wide enough to The best fi t to the data yields allow the hydraulic radius to be replaced with the depth H . According to Eq. (2-149a), if energy slope remains relatively constant, depth should increase monotonically with increas- 081. Δ 0. 0677λ (2-148) ing velocity. This would indeed be the case for a rigid bed. In a sand-bed stream, however, resistance decreases as U where both the wavelength λ and the wave height Δ are in increases over a wide range of conditions. meters. The sandwave length covers the range from wave- At equilibrium, lets (a few centimeters long) observed in the laboratory to megadunes (several hundred meters long) observed in large τρ2 ρ bfC U gHS (2-149b) alluvial rivers (Fig. 2-35b). The development of a reliable bed form predictor for large alluvial fl ows is one of the outstanding problems in This decrease in resistance implies that depth does not river sedimentation (e.g., Schumm and Winkley 1994). In increase as rapidly in U in a movable-bed stream as it would order to more completely understand the morphodynamics for a rigid-bed open channel. In fact, as the transition to upper of river dunes, a fuller appreciation is needed of the complex regime is approached, the bed forms can be wiped out quite links between fl ow turbulence, bed morphology, and sedi- suddenly, resulting in a dramatic decrease in resistance. The ment transport (ASCE 2002; Best 2005; Parker and García result can be an actual decrease in depth as velocity increases. 2006). Future research on relations between sediment trans- This phenomenon was documented for the case of the Padma port mechanics and dune morphology should focus on: River, Bangladesh, by Stevens and Simons (1973) and pre- dicted numerically by Chollet and Cunge (1980), as shown • Numerical modeling applied to a wide range of fl ow in Fig. 2-47. This plot shows how Manning’s n decreases as and bed-material conditions that would allow an analy- the fl ow discharge increases and the dunes are fi rst elongated sis of the adjustment of dune morphology, particularly and fi nally washed out. Notice that the numerical model pre- steepness and lee slope angle, to both bed load and sus- dictions agree very well with the observations for the lower pended load (e.g., Zedler and Street 2001; Tjerry and regime conditions but overestimate the value of Manning’s n Fredsøe 2005). in the upper regime where most of the fl ow resistance should • Detailed fi eld studies of fl ow, sediment transport and be mainly due to grain friction. Even for one-dimensional the evolving morphology of individual dunes using computations, this remains a challenging problem for river modern technology such as acoustic Doppler current engineers as shown in Chapter 14. profi lers and velocimeters, in-situ particle size trans- The effect of the transition phenomenon on fl ow-stage missometers and multibeam echosounders to capture discharge is best illustrated with a fl ow resistance dia- the three-dimensional river-bed morphology shown in gram fi rst proposed by Cruickshank and Maza (1973) and the cover of this manual (e.g., Parsons et al. 2005). shown in Fig. 2-48(a). and river data were used to • Laboratory experiments on and numerical simulations develop this dimensionless diagram showing the transition of fundamental processes such as sediment entrainment from the lower regime to the upper regime. In the transition into suspension and particle-turbulence interaction in region the fl ow depth is seen to decrease as the fl ow veloc- the presence of bed forms (e.g. Maddux et al. 2003a; ity increases. The straight part of the curves in the diagram Coleman et al. 2003; Schmeekle and Nelson 2003). can be expressed with exponential curves obtained by a least • Relations amongst dune celerity, crest planform, dune square regression analysis. The original Cruickshank-Maza profi le shape, water temperature, and sediment trans- empirical relations have been adapted to follow the notation port mechanics (e.g., Kostaschuk 2006; Serra and used in this chapter. Vionnet 2006). For the lower regime, the mean fl ow velocity U can be estimated with

2.7.6 Effect of Bed Forms on River Stage ⎛ ⎞ 0. 634 U v H 0. 456 The presence or absence of bed forms on the bed of a river 603. s50 ⎜ ⎟ S (2-150a) ⎝ D ⎠ can lead to curious effects on river stage. According to the gRD50 gRD50 84

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Fig. 2-47. Variation of Manning’s n with fl ow discharge for the Padma River, Bangladesh. (o) ob- servations of Stevens and Simmons and (x) computations of Chollet and Cunge (1980).

(a) (b)

Fig. 2-48. (a) Cruickshank-Maza Flow Resistance Predictor, and (b) Stage-Discharge curve for Rio Grande near Bernalillo, New Mexico, estimated with Cruickshank-Maza predictor.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 9988 11/21/2008/21/2008 5:44:125:44:12 PMPM bed forms, fl ow resistance, and sediment transport 99 which is valid for, Fig. 2-48(b). The same behavior does not seem to occur in large, low-slope sand-bed streams, because such streams 0. 350 never enter upper-regime conditions, even at or above bank- 1 ⎛ H ⎞ 70 (2-150b) full discharge (e.g., Amsler and Prendes 2000; Wright and ⎝⎜ ⎠⎟ S D84 Parker 2004b). In the case of the Paraná River, Argentina, shown in Fig. 2-46, it is observed that the larger dunes elon- For the upper regime, the mean fl ow velocity U can be gate and their steepness decreases as the fl ow discharge estimated with increases. On the other hand, the smaller dunes, originally superimposed on the larger ones (see Fig. 2-35b), persist ⎛ ⎞ 0. 644 along the bottom of the river and their steepness reaches a U v H 0. 352 545. s50 ⎜ ⎟ S (2-150c) maximum for the peak fl ow discharge. The Froude num- ⎝ D ⎠ gRD50 gRD50 84 ber Fr is in general less than 0.25, even during large fl oods, hence the fl ow never leaves the lower regime (Amsler and García 1997) . 0. 382 1 ⎛ H ⎞ In the case of the Missouri River, temperature effects seem 55 ⎝⎜ ⎠⎟ to control the transition from plane bed to a bed with dunes S D84 (2-150d) (Shen et al 1978). Southard (1989) used the observations of Shen et al to produce a fl ow velocity versus sediment size where, diagram, showing that as the water gets colder and the tem- perature drops, dunes vanish and plane bed conditions are U mean fl ow velocity; observed in the Missouri River. Temperature effect on the H fl ow depth; dynamics of bed forms is a topic that needs to be addressed S energy slope (same as channel slope for uniform in the future so that fl ood stages can be estimated for differ- fl ow); ν ent climate conditions. s 50 fall velocity for sediment size D 50 ; D50 median grain size used to determine the fall veloc- ν ity s50 ; and 2.8 BED FORMS, FLOW RESISTANCE, D84 sediment size for which 84% of bed-material is AND SEDIMENT TRANSPORT fi ner; 2.8.1 Form Drag and Skin Friction These relations are recommended for median grain sizes

( D50 ) in the range from 0.2 to 2 mm. The range of geometric As has been seen, bed forms can have a profound infl uence σ standard deviation ( g ) of the bed-material size distribution on fl ow resistance, and thus on sediment transport in an allu- data, went from 1.2 to 2.5. The submerged specifi c gravity vial channel. To characterize the importance of bed forms in was the same for all the data (R 1.65). The exponents for this regard, it is of value to consider the forces that contribute the fl ow depth ( H) and energy slope (S ) are similar to those to the drag force on the bed. reported by Simmons and Albertson (1960) for stable allu- Consider, for example, the case of normal fl ow in a wide, vial channels. In fact, the exponents are not very different rectangular channel. In the presence of bed forms, Eq. (2-1) from those found in Manning’s equation (Eq. 2-149a). must be amended to Cruickshank and Maza used there relations to compute τρ the stage-discharge relation shown in Fig. 2-48(b). The data b gHS (2-151) shown correspond to the Rio Grande near Bernalillo, New Mexico. It is seen that the Cruickshank-Maza relations cap- − where τ is an effective boundary shear stress, where the ture the behavior of the hydraulic radius, which increases b overbar denotes averaging over the bed forms, and can be with fl ow velocity along the lower regime (ripples and defi ned as the streamwise drag force per unit area, where H dunes), remains almost constant for a wide range of fl ow now represents the depth averaged over the bed forms. velocities during the transition (fl at bed), and continues to In most cases of interest, the two major sources of the increase again in the upper regime due to the development − effective boundary shear stress τ are skin friction, which of antidunes. b is associated with the local shear stresses, and form drag, It is often found that the discharge at which the dunes which is associated with the pressure. That is, are obliterated is a little below bank-full in sand-bed streams with medium to high bed slopes. As a result, flooding is ττ≅ τ not as severe as it would be otherwise. The precise point of bbsbf (2-152) transition is generally different, depending on whether the τ discharge is increasing or decreasing. This behavior can where bs is the shear stress component due to skin friction τ lead to double-valued stage-discharge relations as shown in and bf is the shear stress component due to form drag.

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The important thing to realize is that form drag results without any loss of generality. The parameter H s denotes from the net pressure distribution over an entire bed form. the depth that would result in the absence of bedforms (but At any given point along the surface of the bed form, the with U held constant). This depth is necesarilly less than H , pressure force acts normal to the body. For this reason, form because the resistance is less in the absence of bed forms. The

drag is directly effective neither in moving sediment as bed remaining problem is how to calculate H s . Einstein restricts load nor in entraining sediment into suspension. In the case his arguments to the case of normal (steady, uniform) fl ow. In of dunes in rivers, the fl ow usually separates in the lee of the this case, Eq. (2-151) holds; that is, crest, so that form drag is often substantial. The part of the effective shear stress that governs sediment transport is thus τρ2 ρ b C f U g H S (2-157a) seen to be the skin friction. To render any of the bed-load formulas presented in Section 2.6.4 valid in the presence of bed forms, it is neces- 2 τρ C U ρ g S τ* τ* bs f s H s (2-157b) sary to replace the Shields stress by the Shields stress s associated with skin friction only: Now between Eqs. (2-154) and (2-157b), the following τ relation is obtained for H : τ* bs (2-153) s s ρgRD 2 U 2 ⎡ 1 ⎛ H ⎞ ⎤ The fact that the form drag needs to be excluded for the ⎢ ln 11 s ⎥ (2-158) H s κ ⎝⎜ ⎠⎟ purposes of computing sediment transport does not by any gS ⎣ k s ⎦ means imply that it is unimportant. It is often the dominant

source of boundary resistance, and thus plays a crucial role For given values of U , ks , and S (averaged over bedforms), in determining the depth of fl ow (e.g., Brownlie 1983). This Eq. (2-158) is easily solved iteratively for Hs . Once H s is will be considered in more detail below. known, it is not diffi cult to complete the partition. From Eq. (2-152), it follows that 2.8.2 Shear Stress Partitions τττ bf b bs (2-159) 2.8.2.1 The Einstein Partition Einstein (1950) was the fi rst to recognize the necessity of distinguishing between skin friction and form drag. He proposed the following sim- In analogy to Eqs. (2-152), (2-153), and (2-155), the fol- ple scheme to partition the two. Eq. (2-131) is amended to lowing defi nitions are made, represent an effective boundary shear stress averaged over bed forms, τρ2 ρ bf C ff U gH f S (2-160) τρ 2 b C f U (2-154) from which it follows that

where C now represents a resistance coeffi cient that includes f CCff sf C f (2-161a) both skin friction and form drag. For given fl ow velocity U , Einstein computes the skin friction as HHsf H (2-161b) τρ 2 bs C f s U (2-155)

Here C ff denotes the resistance coeffi cient associated with where C fs is the frictional resistance coeffi cient that would form drag, and H f denotes the extra depth (compared to the result if bed forms were absent. For example, in the case of case of skin friction alone) that results from form drag.

rough turbulent fl ow, Eq. (2-19) may be used: Up to this point, it is assumed that U , S , and ks are given. τ − If, for example, H is also known, b can be calculated from ⎡ ⎛ ⎞ ⎤ 2 Eq. (2-151). After H , C , and τ are computed from Eqs. 1 H s s fs bs C fs ⎢ ln ⎜11 ⎟ ⎥ (2-156) (2-156) to (2-158), it is possible to compute τ , H , and C ⎣κ ⎝ k ⎠ ⎦ bf f ff s from Eqs. (2-159) and (2-161). For example, consider a sand-bed stream at a given cross In fact, Einstein presents a slightly different formula, section with a slope of 0.0004, a mean depth of 2.9 m, a which allows for turbulent smooth and transitional fl ow as median bed sediment size of 0.35 mm, and a discharge per well. Here the analysis is only done for rough fl ow conditions unit width q U H 4.4 m2 /s. Assume that the fl ow

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110000 11/21/2008/21/2008 5:44:155:44:15 PMPM bed forms, fl ow resistance, and sediment transport 101 τ τ is under near-normal conditions. Compute values of bs , bf , They estimate the drag coeffi cient as c D 0.21 from Cfs , Cff , Hs , and H f . measurements taken in the Columbia River (Smith and The mean fl ow velocity is given by U 4.4/2.9 1.52 m/s. McLean 1977). An appropriate estimate of k s for a sand-bed stream is ks It follows that 2.5 D50 . By solving Eq. (2-158) by successive approximations, it is Δ τρ1 2 D ffs found that H 1.047 m. The following values then hold: bf c D U r (2-164) s 2 λλB τ / 2 τ* bs 4.11 N m ( s 0.725) The reference velocity Ur is defi ned to be the mean veloc- τ / 2 τ* Δ bf 7.27 N m ( f 1.283) ity that would prevail between z ks and z if the bed forms were not there. From the logarithmic profi le repre- − τ / 2 τ* b 11.38 N m ( 2.008) sented by Eq. (2-17a), this is found to be given by

C 0.00178 fs U 1 ⎡ ⎛ Δ ⎞ ⎤ r ⎢ln ⎜30 ⎟ 1⎥ (2-165) / ρ κ ⎝ ⎠ τb s ⎣ k s ⎦ Cff 0.00315

C 0.00493 (C 1/ 2 14.5) It is now assumed that a rough logarithmic law with rough- f f Δ ness ks prevails from z ks to z , and a different rough Δ logarithmic law with roughness k c prevails from z to z Hs 1.047 m H. Here kc represents a composite roughness length, includ- H 1.842 m ing the effects of both skin or grain friction and form drag. f The two fl ow velocity distribution laws are thus given by H 2.9 m u ( z ) 1 ⎛ z ⎞ In these relations, ln ⎜ 30 ⎟ , k z Δ (2-166a) τρ/ κ ⎝ ⎠ s bs k s τ τ* bf (2-162) f ρgRD u ( z ) 1 ⎛ z ⎞ ln ⎜30 ⎟ , Δ z H / ρ κ ⎝ ⎠ (2-166b) ()ττb s b f k c denotes a form-induced Shield stress. In this case, only some 30% of the total Shields stress (skin form) contributes to Nelson and Smith match the above two laws at the level the transport of sediment. z Δ (i.e., the top of the dune). After some manipulation, it The Einstein method provides a way of partitioning the is found that boundary shear stress if the fl ow is known. It does not pro- vide a direct means of computing form drag. A method pro- ⎡ ⎤ 2 posed by Nelson and Smith (1989) overcomes this diffi culty ττb s b f ln (30 Δ/ k ) ⎢ s ⎥ (2-167a) when dune height and wavelength are known. Δ/ τb s ⎣ln (30 k c)⎦ 2.8.2.2 The Nelson-Smith Partition Nelson and Smith (1989) consider fl ow over a dune; the fl ow is taken to separate in the lee of the dune. This method builds on The partition requires a prior knowledge of total bound- −τ τ τ the work of Smith and McLean (1977) and is very simi- ary shear stress b bs bf , as well as roughness height k s , Δ λ lar to the method proposed independently by Kikkawa and dune height , and dune wavelength . Between Eqs. (2-163) Ishikawa (1979). Based on experimental observations in the and (2-164),

Columbia River, Nelson and Smith use the following rela- 2 tion for form drag: 1 ΔΔ⎡ ⎛ ⎞ ⎤ τττ c ⎢ln ⎜30 ⎟ 1⎥ τ bf b bs Dλ 2 bs (2-167b) 2 κ ⎣ ⎝ k s ⎠ ⎦ 1 DcB ρΔ U 2 (2-163) ffs2 D r τ τ This equation can be solved for bs, and thus bf . The value of the composite roughness kc is then obtained from Here D ffs denotes that portion of the streamwise drag Eq. (2-167a). force D fs that is due to form drag, B is the channel width, For example , chosen to be rather similar to the previous and Ur denotes a reference velocity to be defi ned below. one, let H 2.9 m, S 0.0004, k s 2.5 D50 , D50 0.35 mm,

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110101 11/21/2008/21/2008 5:44:185:44:18 PMPM 102 sediment transport and morphodynamics Δ 0.4 m, and λ 15 m. The technique, which requires no resistance in open channel fl ows (e.g., Ikeda and Asaeda iteration, yields the following results: 1983). 2.8.2.3 The Engelund-Fredsøe Partition This method τ 4.45 N/m2 ( τ * 0.785) bs s is based on the ideas of Engelund, who fi rst suggested that the head loss due to the expansion of the fl ow right after a τ / 2 τ * bf 6.93 N m ( f 1.223) dune’s crest, and thus form drag, could be estimated with − Carnot’s head loss formula (Fredsøe and Deigaard 1992, τ / 2 τ * b 11.38 N m ( 2.008) p. 280). Similar to the Einstein partition method, the total dimensionless bed stress due to skin friction plus form drag k c 0.0311 m (Fredsøe 1982), is expressed as

C 0.00130 fs τττ*** sf (2-169a) Cff 0.00203 In this method, the component of the dimensionless shear C 0.00333 (C 1/2 17.3) f f stress due to skin (grain) friction is

H 1.134 m 2 s u 2 U 2 ⎛ u ⎞ τ* **ss⎜ ⎟ (2-169b) s gRD RgD ⎝ U ⎠ Hf 1.766 m ρ / ρ H 2.9 m Here, R ( s ) 1 submerged specifi c gravity of / 2 sediment (R 1.65 for sand), and C fs (u * s U ) skin In computing friction coeffi cients, the Keulegan relation- friction coeffi cient (Eq. 2-156). The contribution to the ship was used for the depth-averaged fl ow velocity: dimensionless shear stress by the form drag is estimated with Carnot’s formula, U 1 ⎛ H ⎞ ln ⎜11 ⎟ (2-168a) κ ⎝ ⎠ 2 () ττ / ρ k c 1 U ΔΔ b s b f τ* (2-169c) f 2 RgD H λ From this equation, an expression for the composite roughness is obtained, Combining Eqs. (2-169a), (2-169b), and (2-169c), yields 11H k c ()κ/Uu (2-168b) e * ΔΔ *** *⎡ 1 1 ⎤ τττ τ⎢1+ C ⎥ (2-169d) sf s⎣ 2 H λ fs⎦ Here the shear velocity includes the effect of grain fric- tion and form drag, i.e., u ()ττρ/ . * bs bf Fredsøe (1982) tested the relationship implied in The Nelson-Smith method does not require the assump- Eq. (2-169d) between total dimensionless shear stress tion of quasi-normal fl ow; the user must, however, have τ * and contribution made by skin friction τ * with the information about the bedform dimensions. This method bs experimental observations made by Guy et al. (1966) at has been extended by García and Parker (1993) to the case Fort Collins, Colorado. Fredsøe (1989) has also used this where the boundary is hydraulically smooth (i.e., k δ s v approach to compute stage-discharge curves in small sand- 11.6 v / u ) and viscous effects are present. In this case, * bed streams. the bed shear stress associated with form drag can be esti- Fredsøe (1982) and Fredsøe and Deigaard (1992) have mated with formulated equations to estimate the dune parameters (i.e., Δ / H and Δ / λ ) in Eq. (2-169d). Following the example 2 Δ ⎡ ⎛ u Δ ⎞ ⎤ presented earlier, let H 2.9 m, C 0.00178 (from τττ 1 *s τ fs bf b bs c D ⎢ln ⎜91⎟ ⎥ bs (2-168c) Δ λ 2 λ κ2 ⎣ ⎝ ν ⎠ ⎦ Eq. 2-156), 0.4 m, and 15 m. Substituting these τ* / τ* values into Eq. (1-169d), yields s 2.03. This value is close to those estimated for the same ratio with the Einstein τ* / τ* τ* / τ* This expression was used to remove the effect of ripples ( s 2.77) and Nelson-Smith ( s 2.56) partitions. in laboratory experiments on eroding density currents. It This is not surprising because the Engelund-Fredsøe parti- can also be applied to remove the effect of ripples on fl ow tion combines elements of both of these approaches.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110202 11/21/2008/21/2008 5:44:195:44:19 PMPM bed forms, fl ow resistance, and sediment transport 103 2.8.3 Empirical Techniques for An adaptation of the original Einstein-Barbarossa Stage-Discharge Relations bar-resistance diagram, given implicitly by Eq. (2- 170a), is shown in Fig. 2-49(a). Note that it implies that To use either the Einstein, Nelson-Smith, or Engelund-Fredsoe the friction coeffi cient for the bed forms C declines for partition, it is necessary to know in advance the total effective ff − increasing τ* . That is, the relation applies in the range boundary shear stress τ . In general, this is not known. As a s35 b for which increased intensity of fl ow causes a decrease in result, the relations in themselves cannot be used to predict the form drag (i.e., the transition from dunes to fl at bed). A boundary shear stress (as well as the contributions from skin fl ow resistance diagram like the one shown in Fig. 2-49(a) friction and form drag), and thus depth H , for a fl ow of, say, can be developed for any sand-bed or gravel-bed stream given slope S and discharge per unit width q . w provided that fl ow stage and discharge observations for a A number of empirical techniques have been proposed wide range of conditions, as well as bed-material charac- to accomplish this. Only a few selected ones are presented teristics, are available. herein; they are known to perform at least reasonably well In the Einstein-Barbarossa method, C is computed from for sand-bed streams with dune resistance. fs a relation very similar to Eq. (2-156). That relation is used 2.8.3.1 The Einstein-Barbarossa Method The method here to illustrate the method, which employs the Einstein of Einstein and Barbarossa (1952) is applicable for the case partition for skin friction and form drag. of dune resistance in sand-bed streams. It is of historical The method is now used to synthesize a depth-discharge value and this is the main motivation for presenting it here. relation; that is, a relation between H and water discharge This method assumes an empirical relation of the following Q is obtained. It is assumed that the river slope S and the form: sizes D50 and D35 are known. The river is taken to be suf- ≅ fi ciently wide so that the hydraulic radius R h H ; otherwise, C f ()τ* (2-170a) ff s35 R h should be used in place of H. In addition, cross-sectional shape is known, allowing for specifi cation of channel width Here as a function of fl ow depth:

τ τ* b s s35 ρ (2-170b) gR D35 B B ()H (2-171)

(a) Fig. 2-49. Flow resistance diagrams (a) Einstein and Barbarossa (1952) and (b) Engelund and Hansen (1967).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110303 11/21/2008/21/2008 5:44:195:44:19 PMPM 104 sediment transport and morphodynamics proposed by Parker and Peterson (1980) for gravel-bed streams. They obtained an empirical bar-resistance equa- tion that displays the same behavior as the bedform resis- tance of Fig. 2-49a developed for sand-bed streams. The 6 τ *1.744 Parker-Peterson equation reads C ff 2.33 10 s , and as expected it yields values that are at least an order of magnitude smaller than those predicted with the Einstein- Barbarossa relation for sand-bed streams. This is mainly due to the absence of dunes in most gravel-bed streams. Parker and Peterson (1980) found that for high fl ows, little of the resistance is due to alternate bars, but for the low- est stages bar resistance can reach 56% of the total. More information on fl ow-resistance predictors for gravel-bed (b) streams can be found in Chapter 3. 2.8.3.2 The Engelund-Hansen Method The method Fig. 2-49. Flow resistance diagrams (a) Einstein and Barbarossa of Engelund and Hansen (1967) also specifi cally applies (1952) and (b) Engelund and Hansen (1967). (Continued) to sand-bed streams. It is generally more accurate than the method of Einstein and Barbarossa, to which it is closely It is also assumed that auxiliary relations for area A , wet- allied. The method assumes quasi-uniform material size; it D ted perimeter P and hydraulic radius R h as functions of H are is necessary to know only a single grain size . Roughness k D known. A range of values of H s is arbitrarily assumed, ranging height is estimated as s 2.5 50 . from a very shallow depth to near-bankfull depth (recall that Hs The method uses the Einstein partition. Skin friction is < H ). For each value of Hs , the calculation proceeds as follows: computed via Eq. (2-153). Form drag is computed from the empirical relation → H s C fs Eq. (2-156) ττ** f () → τ → τ* s (2-174) H s bs s35 Eqs. (2-157b), (2-170b) τ * b C , H → U Eq. (2-158) τ (2-175a) fs s ρ R g D → C ff , U H f Eq. (2-160) H H H τ s f Eq. (2-161b) τ* bs s (2-175b) ρ RgD τ* → s35 C ff Eq. (2-170a); use the diagram in Fig. 2-49a Relation (2-174) is shown graphically in Fig. 2-58 in Q U H B Eq. (2-171) Vanoni (2006, p.81). It has two branches, correspond ing to lower-regime and upper-regime fl ows. The two do not meet The result may be plotted in terms of H versus Q for the smoothly, implying the possibility of a sudden transition. The desired depth-discharge relation. point of transition is not specifi ed, suggesting the possibility The analysis may be continued for bed load transport of double-valued rating curves as shown in Fig. 2-48(b). τ rates. That is, the parameter bs may be computed from The lower-regime branch is given by τ bs τ* * * 2 s ρ (2-172) τs 0.06 0.4 ()τ (2-176a) gRD50

and this parameter may be substituted into an appropriate The upper branch satisfi es the relation

bed load transport equation to obtain qb . The volumetric bed- load transport rate Q b is then computed as ** ττs (2-176b) QqBbb (2-173) over a range; this implies on upper-regime plane bed. For τ* τ* Most depth-discharge predictors have been devel- higher values of Shields stress, again exceeds s , implying oped for sand-bed streams. One exception is the predictor form drag due to the development of antidunes.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110404 11/21/2008/21/2008 5:44:225:44:22 PMPM bed forms, fl ow resistance, and sediment transport 105 The procedure rather closely parallels that of the Einstein- that the velocity profi le over a rough bed with dunes has Barbarossa method. It is assumed that values of S and D are approximately the same shape as that over a fl at bed. Instead

known, as well as cross-sectional geometry. Values of H s are of using the logarithmic velocity distribution, a power law is selected, ranging from a low value to near bank-full. The used for the mean fl ow velocity, calculation then proceeds as follows: ⎛ ⎞ 16/ U 832. H H → C → U Eqs. (2-156) and ⎜ ⎟ (2-178a) s fs u α ⎝ k ⎠ (2-158) * st c

* H → τ → τ Eqs. (2-157b) and 16/ s bs s U 832. ⎛ H ⎞ (2-175a) s (2-178b) α ⎝⎜ ⎠⎟ u*sstks τ* → τ* s Eq. (2-174); use Equation (2-176a) or Fig. 2-49b Here − τ * → τ → H Eq. (2-175b) and b ; (2-157a) ugHS*

Q U H B Eq. (2-171) ugHS*ss ; τ* The value of s can be used to calculate bed load transport rates, in a fashion completely analogous to the procedure ks 3 D90 roughness height due to skin friction; and outlined for the Einstein-Barbarossa method. k c composite roughness accounting for both skin fric- 2.8.3.3 The Wright-Parker Method The method of tion and form drag. Engelund and Hansen (1967) was developed using data α from laboratory fl umes (Guy et al. 1966), and verifi ed in Also, st is a stratifi cation parameter that can be estimated the fi eld using relatively small sand-bed streams (Fredsøe as a function of near-bed sediment concentration and channel 1989). Posada (1995) found that Eq. (2-176a) did not per- slope as follows (Wright and Parker, 2004b), form well for the large river data collected in her study. It was found that the relation tends to overpredict the skin- ⎛ ⎞ 077. cbbc friction shear stress for large rivers at high fl ows, indicat- α 1 0. 06⎜ ⎟ for 10 (2-179a) st ⎝ S ⎠ S ing that large rivers do not make the transition from dunes to a fl at bed at shear stresses as low as those observed in laboratory fl umes. A recent reanalysis of the prob- ⎛ c ⎞ c lem by Wright and Parker (2004a, 2004b) indicates that α 0.. 67 0 0025bbfor 10 st ⎝⎜ ⎠⎟ (2-179b) the Engelund-Hansen method does indeed provide good S S results for laboratory fl umes and small- to medium-scale sand-bed streams, but does not perform well for large, Here low-slope sand-bed streams such as the lower Mississippi River. They found, as mentioned earlier, that such streams – c b volumetric sediment concentration at a distance b rarely if ever enter the upper regime. The following modi- 0.05 H above the bed; and fi ed relation not only provides good results for small and S channel slope. medium-sized sand-bed streams such as the Niobrara In the case of a sediment mixture, this concentration River, Middle Loup River, and Rio Grande, but also per- would be the total concentration for all sizes. Wright and forms well for such large, low-slope sand-bed streams as Parker (2004b) used a somewhat modifi ed version of the the Mississippi River, Atchafalaya River and Red River. equation proposed by García and Parker (1991) to esti- It reads – mate c b (Eq. 2-224a), as given in Eq. (3-143e) of Chapter . 3. Using the water continuity equation, q UH , Eq. ττ* * 07. 08 w s 005.. 07()Fr (2-177) (2-178a) can be re-arranged into the dimensionless depth- predictor form

In this relation Fr denotes the Froude number of the fl ow, / ⎡ 16/ ⎤35 given as Eq. (2-119b ). α  ⎛ k ⎞ H ⎢ stq c ⎥ (2-180) The relationship between depth and discharge appropriate ⎜ ⎟ D ⎢832. S ⎝ D ⎠ ⎥ for lower-regime conditions is developed by fi rst assuming 50⎣ 50 ⎦

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110505 11/21/2008/21/2008 5:44:235:44:23 PMPM 106 sediment transport and morphodynamics Here σ denotes the geometric standard deviation of the ~ / g ~ Here qqw gRDD50 50 is a dimensionless water dis- bed material, and q denotes dimensionless water discharge charge. The composite roughness is related to the sand-grain per unit width, given by roughness by eliminating U between Eqs. (2-178a) and (2-178b), ~ q w q (2-182b) g D50 D 50 4 ⎛ ⎞ 4 ⎛ τ* ⎞ kc H ⎜ ⎟ ⎜ ⎟ (2-181) For known values S , D , and σ , q , and thus Q q B k ⎝ H ⎠ ⎝ τ* ⎠ 50 g w w ss s can be computed directly as functions of depth H with the help of Eqs. (2-182a) and (2-182b). The water discharge per τ* τ* unit width is given by q UH where U is the mean fl ow The relationship between and s is given by the modi- w fi ed Engelund-Hansen formulation, Eq. (2-177), which velocity and H is the fl ow depth. provides all the information needed to solve for the fl ow Because of its similarity with Brownlie’s relation, the depth. empirical relation for lower regime proposed by Cruickshank The required known parameters are the unit fl ow dis- and Maza (Eq. 2-150a), is reintroduced here. It can be used as a stage-discharge predictor by rewriting it as charge qw , the river slope S, and the grain size distribution. To do the computations, guess the depth H and calculate τ∗ and the Froude number Fr . Then calculate τ * from ⎛ ⎞ 1. 634 s ~ H 0. 456 Eq. (2-177) and k from Eq. (2-181), using k 3D . Then qR775. f ⎜ ⎟ S (2-182c) c s 90 ⎝ D ⎠ compute the near-bed concentration with Eq. (3-143e) of 84 Chapter 3, which allows computation of the stratifi cation α where the dimensionless specifi c fl ow discharge (i.e., fl ow correction st from Eqs. (2-179a) and (2-179b). Finally, compute the fl ow depth H with Eq. (2-180) and iterate to discharge per unit width) is convergence. The method has been tested with the large river data set ~ qw UH q (2-182d) of Toffaleti (1968) with excellent results (Wright and Parker gD50 D 84gD 50 D 84 2004b). The depth predictor was then used to support the development of a suspended-load relation for mixtures and the dimensionless fall velocity is which will be presented in Chapter 3. The partition of bed shear stress into skin-friction and v form-drag components is largely motivated by the hypothesis s50 Rf (2-182e) that sediment transport is directly linked to the former com- gRD50 ponent. Recently McLean et al. (1999) have used detailed

laboratory measurements to challenge the assumed linkage Notice that the dimensionless fall velocity R f is a function between sediment transport and bed shear stress due to skin of the particle Reynolds number RgRDD ( )/ν , friction. They suggest that the fl ow velocity at the crest of a ep 50 50 bedform may be a better parameter with which to correlate and can be estimated with Dietrich’s fall velocity relation sediment transport. (Eq. 2-47a), the Jimenez-Madsen relation (Eq. 2-48a), or 2.8.3.4 The Brownlie & Cruickshank-Maza Methods from Fig. 2-11. An empirical method offered by Brownlie (1981) has For known values S , D50 , D84 , and kinematic viscosity of proved to be quite accurate (Brownlie 1983). Like the water v , q w, and channel width B, Q qw B can be com- Cruickshank-Maza (1973) method presented earlier (Section puted directly as functions of depth H with the help of Eqs. 2.7.6), Brownlie’s approach does not involve a decomposi- (2-182c), (2-182d), (2-182e), and (2-47). The water dis- tion of bed shear stress, but rather gives a direct predictor charge per unit width is given by qw UH where U is the of depth-discharge relations based on nonlinear regression mean fl ow velocity and H is the fl ow depth. analysis of hundreds of data points from laboratory and fi eld Notice that the Cruickshank-Maza relation accounts for observations. water temperature effects through the dimensionless fall The complete method can be found in Section 2.10.3. Here velocity parameter (Eq. 2-182e). This approach can also be the relation is presented only for the case of lower-regime used to obtain a stage-discharge predictor for the upper-fl ow dune resistance in sand-bed streams. It takes the form regime by rewriting Eq. (2-150c),

⎛ ⎞ 1. 644 HS 0.6539 ~ H 0. 352 0.3724 (q S) 0.09188 σ0.1050 qR 70. S (2-182f) S g (2-182a) f ⎝⎜ ⎠⎟ D50 D84

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110606 11/21/2008/21/2008 5:44:255:44:25 PMPM suspended load 107 Equations (2-182c) for lower-fl ow regime and (2-182f) For a given fl ow depth, the mean fl ow velocity can be for upper-regime apply to the range of conditions defi ned computed directly from Eq. (2-183d). The bed forms are by Eqs. (2-150b) and (2-150d), respectively. Although the identifi ed as being in the lower regime for τ * 1.2, transi- Cruickshank-Maza method has not been tested as thoroughly tion for 1.2 τ * 1.5, and upper regime for τ * 1.5. as Brownlie’s, it has been found to provide an effective 2.8.3.6 Other Stage-Discharge Predictors There are approximation for both the design of movable-bed models almost as many empirical resistance predictors for rivers as well as for computing stage-discharge relations in several as there are sediment transport relations. A fairly compre- Mexican sand-bed rivers with fl ows depths in the range from hensive summary of older methods can be found in ASCE 1 to 8 m, and with bed-material size in the range from 0.2 to Manual 54 (Vanoni 2006). Other fl ow-resistance predictors 2.0 mm (Berezowsky and Lara 1986). for sand-bed streams worth considering, besides the ones 2.8.3.5 The Karim-Kennedy Method An approach presented above, are those of van Rijn (1984c), Camacho similar to the one used by Brownlie (1983) was employed and Yen (1991), and Bennett (1995). by Karim and Kennedy (1981) to develop a depth-discharge predictor for alluvial streams. Nonlinear regression analysis was applied to a database consisting of 339 river fl ows and 2.9 SUSPENDED LOAD 608 fl ume fl ows to determine the most signifi cant dimension- less variables affecting depth-discharge as well as sediment 2.9.1 Mass Conservation of Suspended Sediment transport relationships. The fl ow resistance was formulated in terms of the ratio of friction factors f / f , in which f is the A phenomenon of considerable interest in river mechanics is o the erosion, transport, and deposition of noncohesive mate- friction factor for fl ow over a moving sediment bed, and f o is a reference friction factor for fl ow over a fi xed sediment bed rial by turbulent open-channel fl ow. Suspended sediment given by a Keulegan-type resistance relation as differs from bed load sediment in that it may be diffused throughout the vertical column of fl uid via turbulence. As long as the suspended sediment under consideration is suf- 8 f (2-183a) fi ciently coarse not to undergo Brownian motion, molecular o ⎡ ()/ ⎤ ⎣575.log 12Hks ⎦ effects can be neglected. Suspended particles are transported solely by convective fl uxes. These sediment fl uxes have two in which ks 2.5 D50 . It was assumed, based on Engelund’s components: one associated with the mean fl ow motion and / analysis of fl ow in the lower regime that f f o varies linearly one associated with the turbulence of the fl ow. A complete with the ratio of ripple or dune height to fl ow depth as derivation of the mass conservation equation for suspended follows: sediment can be found in García (2001). More material can also be found in Chapter 16. Here, only the main elements f Δ of the theory of equilibrium suspensions needed to estimate 120.. 892 (2-183b) suspended load are presented. fH o The equation describing mass conservation of suspended sediment of uniform size and constant material density Karim and Kennedy used an expression proposed by in a turbulent fl ow can be written as follows (García and Allen (1978) for the ratio of bed form height to fl ow depth, Parker 1991) given by ∂ ∂ c Fi 2 0 (2-184a) Δ ⎛ ττ**⎞ ⎛ ⎞ ∂t ∂x 008.. 224⎜ ⎟ 18 . 13⎜ ⎟ i H ⎝ 3 ⎠ ⎝ 3 ⎠ (2-183c) ⎛ τ* ⎞ 3 ⎛ τ* ⎞ 4 where 70. 9⎜ ⎟ 888. 33⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 3 3 δ Fuvcuc iisii()3 (2-184b)

* * for τ <1.5 and Δ / H 0 for τ 1.5. They applied regres- xi Cartesian coordinate system such that x 3 is directed sion analysis to the data to obtain an relationship for dimen- upward vertically; sionless fl ow velocity as a function of relative roughness H / t time; D , slope S , and f / f , given by ui fl uid velocity fi eld averaged over turbulence, 50 o u ' turbulent fl uctuations; _ i . . ⎛ ⎞ 0 626 ⎛ ⎞ 0 465 c volume suspended-sediment concentration averaged U H 0. 503 f 6. 683⎜ ⎟ S ⎜ ⎟ (2-183d) over turbulence; ⎝ D ⎠ ⎝ f ⎠ gRD50 50 o c ' instantaneous fl uctuation in concentration;

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110707 11/21/2008/21/2008 5:44:265:44:26 PMPM 108 sediment transport and morphodynamics 2 / Fi volume fl ux vector of suspended sediment aver- The kinematic eddy diffusivity D d has dimensions of L T aged over turbulence; Common practice is to treat the eddy diffusivity as a scalar. vs fall velocity of sediment in quiescent water; and However, this is the case only for isotropic, homogeneous δ δ δ i 3 Kronecker delta (i.e., i3 1 for i 3 and i3 0 turbulence, as realized in the pioneering experiments con- otherwise). ducted by Rouse (1938) on equilibrium suspensions with grid-generated turbulence in a jar (Ettema 2006). More Equations (2-184a) and (2-184b) are valid only for dilute insight into turbulence in sediment-laden fl ows can be found suspensions (i.e., c– 1) of particles that are not too coarse in Chapter 16. To solve Eq. (2-185), appropriate boundary (i.e., size is less or equal than 0.5 mm). The mean volumet- _ conditions are needed. These are presented next. ric sediment concentration c is defi ned as the ratio between the volume of sediment and the volume of sediment-water mixture. 2.9.2 Boundary Conditions for Sediment If x 3 z is upward vertical, Eqs. (2-184a) and (2-184b) Advection-Diffusion Equation reduce to Equation (2-185), coupled with a Fickian closure approxi- ∂c ∂c ∂c ∂c mation such as Eqs. (2-186a) or (2-186b), represents an u v ()w v ∂t ∂s ∂n s ∂z advection-diffusion equation for suspended sediment (García (2-185) 2001). The condition of vanishing fl ux of suspended sedi- ∂ ∂ ∂ uc vc wc ment across (normal to) the water surface defi nes the upper ∂s ∂n ∂z boundary condition. If steady, uniform, turbulent fl ow over a fl at bed (when where averaged over bed forms) is considered, the water surface u–, v–, and w– mean fl ow velocities in the s , n , and boundary condition for the net vertical fl ux of sediment z directions, respectively; and reduces to —— —— —— u 'c', v'c', and w'c' sediment fl uxes due to turbulence, also known as Reynolds fl uxes. Fsz zH 0 (2-187a) The main assumption here is that the sediment particles follow the fl uid particles (i.e., have the same velocity as the where fl uid), except in the vertical z direction, where the effect of gravity introduces a slip velocity denoted by the sediment fall velocity v in quiescent water (see section 2.3.8). s Fvcwcsz s (2-187b) It is seen from Eq. (2-184b) that the mean fl ux of sus-

pended sediment Fi is composed of two components, i.e., a mean convective fl ux and a Reynolds fl ux. The Reynolds gives the net vertical fl ux of sediment. —— The boundary condition at the bed differs from the one fl ux u' i c' in the above relation is clearly diffusive in nature. The simplest closure assumption to represent the Reynolds at the water surface, in that it must account for entrainment sediment fl uxes is to assume these fl uxes proportional to gra- of sediment into the fl ow from the bed and deposition of dients in sediment concentration sediment from the fl ow onto the bed. For a fl at (averaged over bedforms) bed, the mean depositional fl ux of suspended ∂c sediment onto the bed is given by Dr , which needs to be uc D id∂ (2-186a) evaluated at a distance z b near the bed, xi Dvcrsb (2-188) In this equation, the kinematic eddy diffusivity D d is assumed to be a scalar quantity. For the case of nonisotropic turbulence, Equation (2-186a) must be generalized to the denotes the volume rate of deposition of_ suspended sediment form (García 2001) per unit time per unit bed area. Here c b denotes a near-bed value of mean volumetric sediment concentration. ∂c The component of the Reynolds fl ux of suspended sedi- uc D idij∂ (2-186b) ment near the bed that is directed upward normal to the x j bed may be termed the rate of erosion, or more accurately, entrainment of bed sediment into suspension per unit bed Here D is a tensor quantity. It is often assumed to repre- dij E ≠ ≠ area per unit time. The entrainment rate r is thus given by sent a diagonal matrix, such that D dij 0 if i j, and Dd 11 D ≠ D . Notes on tensor notation and turbulence can be d22 d33 found in the appendix at the end of Chapter 16. Er w c (2-189)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110808 11/21/2008/21/2008 5:44:265:44:26 PMPM suspended load 109 where 2.9.3 Equilibrium Suspension in a Wide Channel w' and c' turbulent fl uctuations around both the mean Consider normal fl ow in a wide, rectangular open channel. vertical fl uid velocity and the mean sediment concentra- The bed is assumed to be erodible and has no curvature when tion, respectively. averaged over bed forms such as ripples and dunes. The The ‘overbar’ denotes averaging over turbulence. The ter- z -coordinate is quasi-vertical, implying low channel slope minology “near-bed” is employed to avoid possible sin- S. The suspension is likewise assumed to be in equilibrium. gular behavior at the bed (located at z 0). That is, c– is a function of z alone, as shown in Fig. 2-50. The fl ow and suspension are uniform in s (streamwise direc- It is seen from these equations that the net-upward, nor- tion) and n (transverse direction) and steady in time, so that mal fl ux of suspended sediment at (or rather just above) the Eq. (2-185) reduces to bed is given by

d () ()wc´´ vs c 0 (2-193a) F s z |zb= EDrrvs E s c b (2-190a) dz

—— _ w c v c where indicating that for equilibrium conditions, ( ' ' s ) constant for all values of z in the range z b to z H . It fol- lows from the boundary condition at the water surface, given E ≡ r by Eqs. (2-187a) and (2-187b), that the constant 0, thus E s (2-190b) vs Eq. (2-193a) reduces to

denotes a dimensionless rate of entrainment of bed sediment w′′ c vc 0 into suspension (i.e., volume of entrained sediment per unit s (2-193b) bed area per unit time). The required bed boundary condi-

tion, then, involves a specifi cation of E s , which can be used It is appropriate to close this equation with the assumption to estimate the sediment entrainment fl ux at the bed (i.e., of an eddy diffusivity as in Eq. (2-186a) so that Eq. (2-193b) Eq. 194b). Typically a relation is assumed of the form becomes (Rouse 1937, Vanoni 1946)

dc E ss E (τb s ,other parameters) (2-191) Ddsvc 0 (2-194a) dz τ where bs denotes the boundary shear stress due to skin friction. According to the literature, this equation was fi rst devel- If it is furthermore assumed that an equilibrium steady, oped by Wilhelm Schmidt in the mid 1920s for studies of uniform suspension has been achieved._ It follows that there transport in atmospheric fl ows and by M. P. O’Brien in the early 1930s for studies of suspended sediments in should_ be neither net deposition_ on (Fsz 0) nor erosion from (F 0) the bed. That is, F 0, yielding the result streams. It has a simple physical interpretation. The term sz sz|zb _ vsc represents the rate of deposition of suspended sedi- ment under the infl uence of gravity; it is always directed E c sb (2-192) downward. If all of the sediment is not to settle out, there

This relation simply states that the entrainment rate equals the deposition rate at equilibrium; thus there is no net normal fl ux of suspended sediment at the bed. García and Parker (1991) took advantage of this relation and developed a sedi- ment entrainment formulation presented below. Sediment entrainment and deposition fl uxes were directly measured in Central Long Island Sound by Bedford et al. (1987) with the help of acoustic transducers. A control vol- ume approach was used to estimate settling fl uxes as well as Reynolds sediment fl uxes. The sediment entrainment mea- surements correlated well with near-bed turbulent kinetic energy. Sediment resuspension has also been backcalculated from in-situ fl ow observations in the off Fig. 2-50. Defi nition diagram for sediment entrainment from and California by Wiberg et al. (1994). deposition on channel bed.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 110909 11/21/2008/21/2008 5:44:305:44:30 PMPM 110 sediment transport and morphodynamics must be an upward fl ux that balances this term. The upward existence of bed forms, the turbulent rough law embodied in fl ux is provided by the effect of turbulence, acting to yield Eq. (2-166b) is employed. a Reynolds fl ux. According to _Eq. (2-186a), this fl ux will be directed upward as long as dc/dz 0 . It follows that the u 1 ⎛ z ⎞ equilibrium suspended sediment concentration decreases for ln 30 (2-195) κ ⎝⎜ ⎠⎟ increasing z, so that turbulence diffuses sediment from zones u* k c of high concentration (near the bed) to zones of low con-

centration (near the water surface). Thus the general bound- Here k c is a composite roughness chosen to include ary conditions for Eq. (2-194a) are given by (Parker 1978; the effect of bed forms, as outlined in Section 2.8.2.2. Fredsøe and Deigaard 1992, p. 246) Furthermore, according to Eq. (2-2), the bed shear stress is given by dc Dd vEss (2-194b) dz zb τρ 2 b u* (2-196)

dc Ddsvc 0 (2-194c) where b is chosen to be very close to the bed, i.e., dz zH

b The fi rst of these specifi es the near-bed rate of entrain- 1 (2-197) ment of sediment into suspension, and the second specifi es H the condition of vanishing upward normal sediment fl ux

at the water surface. It follows that at equilibrium, the fl ux Now the kinematic eddy viscosity Dd is defi ned such that boundary condition given by Eq. (2-194b) is equivalent to the identity given by Eq. (2-192). However, for nonequilib- τ du rium conditions, Eq. (2-194b) should be used as the near- Dd (2-198) dz bed boundary condition (Parker 1978; Fredsøe and Deigaard 1992; Admiraal and García 2000). where the distribution of fl uid shear stresses τ is given by Eq. (2-3) 2.9.4 Form of Eddy Diffusivity (Prandtl analogy) ⎛ z ⎞ Further progress requires an assumption for the kinematic ττ1 (2-199) b ⎝⎜ ⎠⎟ eddy diffusivity Dd . The simple approach taken here is that H of Rouse (1937). It involves the use of the Prandtl analogy. The argument is as follows. Fluid mass, heat, momentum, From the above equations, the following equation is etc., should all diffuse at the same kinematic rate due to tur- obtained bulence and thus have the same kinematic eddy diffusivity, because each is a property of the fl uid particles, and it is ⎛ z ⎞ the fl uid particles that are being transported by Reynolds Duzκ ⎜1 ⎟ (2-200) fl uxes. d * ⎝ H ⎠ The Prandtl analogy is by no means exact but has been found to be a reasonable approximation for many turbulent If Dd is averaged in the vertical, the following result is fl ows. Its application to sediment is more problematic (e.g., obtained: Coleman 1970; Kerssens et al. 1979; Nielsen 1992). Inertial effects might cause the sediment particles to lag behind the κ ≅ 1 fl uid, resulting in lower eddy diffusivity for sediment than Dd uu** H H (2-201a) for the fl uid (Niño and García 1998a). Furthermore, the mean 6 15 fall velocity of sediment grains should reduce their residence time in any given eddy, again reducing the diffusive effect This relation provides a good approximation to estimate (Nielsen 1992). If the particles are not too large, however, the longitudinal dispersion of fi ne-grained sediment and it may be possible to equate the vertical diffusivity of the contaminants in rivers and streams (e.g., Rutherford 1994; sediment with the vertical eddy viscosity (eddy diffusivity Huang and García 2000). of momentum) of the fl uid, as a fi rst approximation. This is In the early 1940s, Lane and Kalinske tried to obtain a done here. simple method to estimate suspended sediment load in the The velocity profi le is approximated as logarith- fi eld. To this end, they used the average value for the eddy mic throughout the depth. To account for the possible diffusivity given by Eq. (2-201a) to integrate Eq. (2-194a),

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 111010 11/21/2008/21/2008 5:44:315:44:31 PMPM suspended load 111 resulting in the following equation for suspended sediment suspended sediment transport to determine parameters such distribution as the eddy diffusity, can be found in Muste (2002). Different approaches to estimate numerically the eddy diffusivity in ⎧ v ⎛ ⎞⎫ sediment-laden fl ows are discussed in Chapter 16. In large, c 15 s zb exp ⎨ ⎜ ⎟⎬ (Eq. 201b) low-gradient, sand-bed streams, the diffusion coeffi cient has c ⎩ u ⎝ H ⎠⎭ b * to be adjusted for stratifi cation effects. This adjustment to account for stratifi cation effects is presented below. – where c b is a near-bed reference concentration measured at a distance z b from the bed. This simple exponential equa- tion provides a reasonable approximation to estimate the sus- 2.9.5 Rouse-Vanoni-Ippen Suspended pended sediment distribution in wide rivers (i.e., H / B Sediment Distribution 1). It also yields a fi nite value of sediment concentration at To integrate Eq. (2-194a) in the vertical, the nominal “near the water surface (i.e., z H ), which is one weakness of bed” elevation in applying the bottom boundary condition the Rousean distribution presented below. The exponential is taken to be z b , where b is a distance taken to be very decay in suspended sediment concentration with distance close to the bed (i.e., satisfying condition Eq. (2-197)). In the from the bed given by the Lane-Kalinske relation was the Rousean analysis, this value cannot be taken as z 0, because solution originally obtained by Rouse in his turbulence jar Eq. (2-195) is singular there. experiments. In Rouse’s jar, however, the eddy diffusivity Equation (2-200) is now substituted into Eq. (2-194a), was also constant in the vertical direction but the turbulence which is then integrated from the nominal bed level to dis- was generated by an oscillating grid and not by a velocity tance z above the bed in z . The resulting form can be cast as gradient (i.e., shear stress) as in the case of Eq. (2-201a).

Equation (2-200) is known as the Rousean formulation for z the vertical kinematic eddy viscosity. The form predicted is H dc z Hdz ⎡⎛ Hz ⎞ ZR ⎤ Z ln⎢ ⎥ parabolic in shape. Although strictly applying to the turbulent ∫ R ∫ ⎝⎜ ⎠⎟ (2-202) c zH()− z ⎣⎢ z ⎦⎥ diffusion of fl uid momentum, it is equated to the eddy dif- b b b fusivity of suspended sediment mass below. Coleman (1970)

was among the fi rst to estimate the variation of Dd with dis- where ZR denotes the dimensionless , given tance from the bed z , from laboratory and fi eld observations as of suspended sediment. He found that its variation is para- bolic only in the lower portion of the fl ow and then it remains v constant up to the water surface. Van Rijn (1984b), among Z s (2-203) R κ others, has argued that the ratio between the diffusivity of u* sediment and the kinematic eddy viscosity is slightly larger than 1 and has proposed an empirical coeffi cient to adjust Integration yields the profi le

the values of D d accordingly. On the other hand, Bennett et al. (1998) have found with the help of turbulence measure- Z ments in sediment-laden fl ow that D is a good surrogate for c ⎡ (H z) / z ⎤ R d ⎢ ⎥ (2-204) the sediment diffusivity as long as it is directly measured ( ) / cb ⎣ H b b ⎦ and not back-calculated from Eq. (2-194a). Nielsen (1992) argues that diffusion models do not capture the physics of the problem in coastal sediment transport. Muste and Patel Equation (2-204) is commonly recognized as the Rousean (1997) have done detailed laboratory measurements that can distribution for suspended sediment and is one of the mile- be used to understand sediment diffusion in open-channel stones in the mechanics of sediment transport (Vanoni fl ows. Through an analysis of the turbulent kinetic energy 1984). Profi les of suspended sediment obtained from labora- budget, Niño and García (1998a) have found that near the tory observations are plotted in Rousean form in Fig. 2-51 bed, fi ne particles reduce the kinetic energy of the fl ow but (Vanoni 2006). It has also been found to work well in sev- coarser particles enhance turbulence because of the produc- eral large alluvial rivers. Sediment concentration profi les of tion of turbulent kinetic energy through vortex shedding suspended sediment observed in the Amazon River, Brazil, mechanisms. Cellino and Graf (2000) have analyzed the are plotted in Rousean coordinates (i.e., (H-z ) / z ( d -y ) / y )) effect of bed forms on open channel suspensions, showing in Fig. 2-52 (Vanoni 1980). The slope of the straight lines

that the turbulent diffusion is enhanced in the presence of in log-log paper corresponds to the Rouse number Z R for a bed forms. Greimann and Holly (2001) have used a two- given grain size range. If the mean fall velocity of the par- phase fl ow model to estimate the role of cross trajectories ticles in each size range can be estimated, it is possible to

on eddy diffusivity. A review of most of the experimental backcalculate the shear velocity u * . Despite the wide range work, including sources of error, which has been done on of sediment sizes present in the water column and the large

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 111111 11/21/2008/21/2008 5:44:335:44:33 PMPM 112 sediment transport and morphodynamics water depth (44 m), it is clear that the concentration pro- fi les follow the vertical distribution given by Eq. (2-204). The Rousean distribution has also been found to work well for the analysis of sediment transport in vegetated channels (Lopez and García 1998). In this particular case, the eddy diffusivity is enhanced by the presence of the vegetation, resulting in vertical sediment concentration distributions that are slightly more uniform than those predicted by the Rousean formulation. Although strictly speaking the credit for the above rela- tion should go to Hunter Rouse, Kennedy (1983) argued that Eq. (2-204) should be named the Rouse-Vanoni-Ippen equation for suspended sediment distribution, because all of these researchers contributed either directly or indirectly to its development and subsequent testing. The seminal idea of using the logarithmic fl ow velocity distribution and the Prandtl analogy to estimate the eddy diffusivity was sug- gested independently to both Hunter Rouse and Arthur Ippen Fig. 2-51. Laboratory observations of suspended sediment distri- by Professor Theodore von Karman at the California Institute bution (Vanoni 2006). of Technology. Rouse used the Karman-Prandtl logarithmic velocity distribution that led to the development of the clas- sic relation that now bears his name, while Ippen used the velocity distribution proposed earlier by Krey instead of the Karman-Prandtl relation. Thus, the Rousean formula- tion predicts sediment_ concentration relative to a near-bed concentration cb, which has to be estimated as a function of fl ow parameters and sediment characteristics, as will be shown below. Starting with his PhD dissertation completed at Caltech in 1940, Vito Vanoni spent a substantial amount of effort study- ing open-channel suspensions, in particular the predictions made by the Rousean formulation both in the laboratory (Fig. 2-51) and in the fi eld (Fig. 2-52). By the time ASCE Manual 54 was fi rst published, the Rouse-Vanoni-Ippen for- mulation had already reached prominence and since then it has become one of the most important milestones in the fi eld of sediment transport (Vanoni 1984). 2.9.5.1 Modifi cation of the Rousean Formulation to Include Flow Stratifi cation Effects River fl ows carrying suspended sediment are self-stratifying. As illustrated in Fig. 2-51, the Rousean profi le predicts a concentration of suspended sediment that decreases with increasing elevation above the bed. It follows that the density of the water-sedi- ment mixture also decreases with increasing elevation above the bed. This stable stratifi cation inhibits turbulent mixing of both fl ow momentum and sediment mass in the vertical. The result is a modifi cation of the vertical distributions of both streamwise momentum and suspended sediment con- centration. More specifi cally, stratifi cation effects lead to a streamwise velocity profi le that increases more rapidly in the Fig. 2-52. Distribution of Suspended Sediment plotted in vertical than the logarithmic profi le, and a suspended sedi- Rousean Coordinates for the Amazon River at Manacapuru, where ment profi le that decreases more rapidly in the vertical than the fl ow depth is 44 m. Each line shows the particle size range the Rousean profi le. and the corresponding value of the Rouse number (adapted from Starting with his seminal contribution to the subject of Vanoni 1980). sediment transport by currents and waves in continental

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 111212 11/21/2008/21/2008 5:44:345:44:34 PMPM suspended load 113 shelves (Smith 1977), J. Dungan Smith has been a tireless earlier. The corresponding boundary condition on Eq. advocate of the importance of self-stratifi cation by fl ows with (2-194a) is suspended sediment. Smith and McLean (1977a), Smith and McLean (1977b), Gelfenbaum and Smith (1986), McLean dc (1990), and McLean (1992) offer quantitative formulations Dd vEss (2-209) of stratifi cation effects based on simple algebraic closures. dz b

The kinematic eddy diffusivity D d given in Eq. (2-200) is D o now denoted as do , where the subscript “ ” denotes the Solving Eqs. (2-194a) and (2-207) subject to the boundary D absence of stratifi cation effects. The value of d in the pres- conditions (2-209) and (2-208) and the relations (2-205a), ence of stratifi cation effects is given as (2-205b), and (2-205c) results in the forms DDFRId do strat() g (2-205a) ⎡ ⎤ ⎢ ⎥ ⎛ z ⎞ z κ ⎢ vs ⎥ (2-210) Duzdo ∗ ⎜1 ⎟ (2-205b) cEexp ∫ dz ⎝ H ⎠ s b ⎢ ⎛ ⎞ ⎥ κ z ⎢ uz∗ ⎜1 ⎟ FRIstrat() g ⎥ ⎣ ⎝ H ⎠ ⎦ dc Rg and RI dz g 2 (2-205c) ⎛ du ⎞ ⎜ ⎟ ⎝ dz ⎠ ⎡ ⎤ u ⎛ b ⎞ z 1 u ∗ ⎢ln⎜ ⎟ ∫ dz ⎥ (2-211) κ ⎢ z b ()⎥ Here RI g denotes a gradient and ⎣ ⎝ o ⎠ zFst RI g ⎦ Fstrat is a function that decreases with increasing gradient Richardson number, thus capturing the effect of damping of These two equations do not constitute an explicit solu- the turbulence due to fl ow stratifi cation. Smith and McLean tion for the concentration and velocity profi les, because RI (1977) offer the following form for the function F : _ g strat is a function of the concentration gradient dc /dz in accor- → dance with Eq. (2-205c). In the limit, as RI g 0, how- FRI()147. RI (2-206) ever, Eq. (2-210) converges to the Rousean solution of Eq. strat g g (2-204) and Eq. (2-211) converges to the logarithmic pro- fi le of Eq. (2-195). These two unstratifi ed profi les can be used as base forms for an iterative solution of Eqs. (2-210) Note that F strat equals unity for a gradient Richardson number of zero (no stratifi cation effects) and decreases to and (2-211). The fi rst step in this iterative process can be illustrated as zero as RI g increases to a value of 0.21, at which turbulent _ follows. Let c (z) denote the Rousean solution for the profi le mixing is extinguished. o _ Smith and McLean (1977a) approximate the equation of of suspended sediment concentration and let u o denote the streamwise momentum balance for the case of equilibrium logarithmic profi le of streamwise velocity. The fi rst itera- fl ow in a wide channel Eq. (2-199) to the form tion,_ including_ the effect of stratifi cation yields the forms c1(z) and u 1(z), where ⎛ ⎞ du 2 z (2-207) Dd u∗ ⎜1 ⎟ ⎧ ⎫ dz ⎝ H ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ An appropriate near-bed boundary condition on ⎪ ⎪ Eq. (2-207) at z b is obtained by matching the velocity z ⎪ v ⎪ − s profi le to the logarithmic law (Eq. 2.17a) there: cE1 s exp ∫ ⎨ dz⎬ b ⎪ ⎡ ⎤ ⎪ ⎢ dco ⎥ ⎪ ⎛ ⎞ Rg ⎪ u ⎛ ⎞ ⎪ z ⎢ dz ⎥ ⎪ zb 1 b κuz∗ ⎜1147⎟ . ln⎜30 ⎟ (2-208) ⎪ ⎝ H ⎠ ⎢ ⎛ ⎞ 2 ⎥ ⎪ κ ⎢ duo ⎥ u∗ ⎝ kc ⎠ ⎪ ⎜ ⎟ ⎪ ⎩ ⎣⎢ ⎝ dz ⎠ ⎥⎦ ⎭

Here k c is the composite roughness length accounting for both grain resistance and form-induced drag introduced (2-212)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 111313 11/21/2008/21/2008 5:44:365:44:36 PMPM 114 sediment transport and morphodynamics

⎧ ⎫ Z ⎪ ⎪ 1 ⎡ ()1δδ/ ⎤ R CcJc⎢ ⎥ dδ ⎪ ⎪ bb1 ∫ δδ/ δ ⎣()1 bb⎦ (2-215) ⎪ ⎪ b ⎪ ⎪ u ⎪ ⎡ b ⎤ z 1 ⎪ u ∗ ⎨ln ⎢ ⎥ ∫ ddz ⎬ 1 κ b ⎡ ⎤ (2-213) ⎪ ⎣z o ⎦ ⎪ Einstein (1950) proposed a relation for the depth- averaged ⎢ dco ⎥ ⎪ Rg ⎪ sediment concentration as ⎪ ⎢ dz ⎥ ⎪ z ⎢147. 2 ⎥ ⎪ ⎛ du ⎞ ⎪ ⎪ ⎢ ⎜ o ⎟ ⎥ ⎪ ⎩⎪ ⎣⎢ ⎝ dz ⎠ ⎦⎥ ⎭⎪

A sample calculation illustrates the procedure. The / following values are used: R 1.65, vs 0.748 mm s ν -6 2/ ( D 0.1 mm for 1x10 m s), H 2 m, k c 5 mm, u ∗ / _ 0.02 m s, b 0.05 H, and Es c b 0.0001. Fourteen iterations are needed to converge to a solution based on a convergence criterion of less than 0.1 % of error. The Rousean and stratifi cation-modifi ed profi les of suspended sediment concentration are shown in Fig. 2-53(a). The loga- rithmic and stratifi cation-modifi ed profi les of streamwise velocity are shown in Fig. 2-53(b). Recently, Wright and Parker (2004) have shown that (a) sediment-induced stratifi cation effects are important in large, low-gradient, sand-bed streams. Using an approach similar to the one presented here, they have estimated velocity pro- fi les for the Red and the Atchafalaya Rivers, showing the difference between clear-water and stratifi ed fl ow velocity profi les as shown in Figure 2-54. Other approaches in the literature to account for the effect of sediment-induced stratifi cation on velocity profi les not mentioned earlier include the work of Itakura and Kishi (1980), Adams and Weatherly (1981), Coleman (1981), and Soulsby and Wainright (1987). With the exception of Coleman (1981) who also used the Richardson number to (b) quantify the effect of stratifi cation, the rest of the formu- lations have made an analogy with stratifi ed atmospheric Fig. 2-53. (a) Concentration profi les and (b) streamwise velocity fl ows, introducing the so-called Monin-Obukhov length into profi les without and with stratifi cation. their analyses.

2.9.6 Vertically Averaged Concentrations: Suspended Load Often times it is useful to know the average concentration in the water column. Assuming that a value of near-bed ele- vation b is selected, Eq. (2-204) can be used to evaluate a depth-averaged volume suspended sediment concentration C , defi ned by 1 H Ccz ∫ () d z (2-214) H b

δδz b Using and b , where b denotes a location H H Fig. 2-54. Velocity profi les for Red and Atchafalaya Rivers with just above the bed, Eq. (2-214) can be expressed as: and without stratifi cation effects (Wright and Parker 2004).

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 111414 11/21/2008/21/2008 5:44:385:44:38 PMPM suspended load 115 δ uniform, equilibrium fl ow conditions, it is necessary to know b (2-216) _ Ccb I1 0. 216 the near-bed concentration c b , the total friction velocity ττρ/ u* ()bs bf , the fl ow depth H , the value of the Here, I is given in graphical form by Fig. 2-55(a). It fol- 1 composite roughness which can be computed from lows from Eqs. (2-215) and (2-216) that ⎧ κU ⎫ Eq. (2-168b) as kH11 exp ⎨ ⎬ , and the values of c ⎩ u ⎭ Z * δ 1 ⎡ ()1δδ/ ⎤ R JIb ⎢ ⎥ dδ (2-217) the integral parameters J and J . Assuming that the fl ow dis- 11∫ δδ/ 1 2 0. 216 δ ⎣()1 bb⎦ b charge per unit qw width is known, the mean fl ow velocity / can be estimated as U qw H . J is defi ned above and J is given by The streamwise suspended load q s was seen in Eq. (2-85) 1 2 to be given by the relation Z 1 ⎡ ()1 δδ/ ⎤ R H Jd ⎢ ⎥ ln()δδ (2-220a) 2 ∫ δδ/ q s ∫ cub dz δ ⎣()1 ⎦ (2-218) b bb b Again, Einstein (1950) stated another relation for this Reducing with the aid of Eqs. (2-195) and (2-204), it is integral as found that Eq. (2-218) can be expressed as (García 1999) δ JI b (2-220b) 1 ⎡ ⎛ H ⎞ ⎤ 22 q cu H ⎢ J ln 30 J ⎥ 0. 216 s κ b * 12⎝⎜ ⎠⎟ (2-219) ⎣ k c ⎦ where -I 2 is also given in tabular form by Fig. 2-55(b). There have been a number of attempts at fi nding analyti- This equation indicates that to compute the rate of volu- cal expressions to estimate the integrals, fi rst presented by metric suspended sediment transport per unit width under Einstein (1950), and shown graphically in Figures 2-55a and

(a) (b) Fig. 2-55. Einstein integrals (a) I1 and (b) I2.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 111515 11/21/2008/21/2008 5:44:425:44:42 PMPM 116 sediment transport and morphodynamics

Table 2-5 Coeffi cients from Regression Analysis for (a) J1 and (b) J2 δ b C0 C1 C2 C3 C4 C5 C6 0.001 8.0321 26.273 114.69 501.43 229.51 41.94 2.7722 0.005 2.1142 3.4502 12.491 60.345 29.421 5.4215 0.3577 0.01 1.4852 0.2025 14.087 20.918 10.91 2.034 0.1345 0.05 1.1038 2.6626 5.6497 0.3822 0.6174 0.1315 0.0091 0.1 1.1266 2.6239 3.0838 0.3636 0.0734 0.0246 0.0019

δ b C0 C1 C2 C3 C4 C5 C6 0.001 2.5779 12.418 47.353 17.639 13.554 2.8392 0.2003 0.005 1.2623 1.033 13.543 0.7655 1.6646 0.3803 0.0275 0.01 1.151 2.1787 7.6572 0.2777 0.57 0.1424 0.0105 0.05 1.2574 2.3159 1.9239 0.3558 0.0075 0.0064 0.0006 0.1 1.4952 2.2041 1.0552 0.2372 0.0265 0.0008 0.00005

2-55b (e.g., Nakato 1984; Guo and Wood 1995; Guo and case). Table 2-6 summarizes most of the available relations. Julien 2004). Recently, Abad and García (2006) have pro- It includes the formulations proposed by Einstein (1950); posed an approach that should be of practical use because it Engelund and Fredsøe (1976); Smith and McLean (1977a), is easy to implement in computational models. Itakura and Kishi (1980), van Rijn (1980), Engelund and With the help of series analysis and appropriate software, Fredsøe (1982); Celik and Rodi (1984), Akiyama and Fukushima (1986), García and Parker (1991), Zyserman Abad and García (2006) obtained expressions for J 1 and J2 that are well approximated by the following regression- and Fredsøe (1994), and Cao (1997). García and Parker analysis equations: (1991) performed a detailed comparison of eight such rela- tions against data. The relations were checked against a carefully selected set of data pertaining to equilibrium sus- 1 J pensions of uniform sand. In such case, it is possible to mea- 1 2 3 4 5 6 _ ccZcZcZcZcZcZ01RR 2 3 R4 R5 R6 R sure cb directly at some near-bed elevation z b , and to equate the result to E according to Eq. (2-192). (2-221a) s The data consisted of some 64 sets from 10 different and sources, all pertaining to laboratory suspensions of uniform ρ ρ / ρ sand with a submerged specifi c gravity R ( s ) 1 near 1.65. Information about the bed forms was typically not J 2 2 3 4 5 6 suffi cient to allow for a partition of boundary shear stress in ccZcZcZcZcZcZ01RR 2 3 R4 R5 R6 R accordance with Nelson and Smith (1989). As a result, the τ (2-221b) shear stress due to skin friction alone bs , and the associated shear velocity due to skin friction u * s , given by All the coeffi cients in Eqs. (2-221a) and (2-221b) are func- tions of the parameter δ b / H z / H where the near-bed τρ u2 (2-222a) _ b b bs* s concentration c bis evaluated. The values of such coeffi cients for both equations are presented in Tables 5a and 5b. were computed with Eq. (2-156) with the following relation It is apparent from Eq. (2-219) that further progress is for roughness height k s predicated on a method for evaluating the near-bed reference _ kDs 2 50 (2-222b) concentration cb, or equivalently (for the case of equilibrium suspensions) the sediment entrainment rate E s (García and Parker 1991; Zyserman and Fredsøe 1994). Such a relation The data covered the following ranges: is necessary to model transport of suspended sediment in Es : 0.0002 to 0.06 non-equlibrium situations (Celik and Rodi 1988; Alonso and Mendoza 1992; Admiraal and García 2000). / u* s vs : 0.70 to 7.50

2.9.7 Functions for Sediment Entrainment or H/D : 240 to 2400 Equilibrium Near-Bed Sediment Concentration R 3.50 to 37.00 A number of relations are available in the literature for esti- ep mating the entrainment rate of sediment into suspension E The range of values of RgRDD ()/ν corresponds _ s ep to a grain size range from 0.09 mm to 0.44 mm. Except for (and thus the reference concentration c b for the equilibrium

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 111616 11/21/2008/21/2008 5:44:455:44:45 PMPM 6690-003-2pass-002-r03.indd 117 9 0 - 0 0 3 - 2 p a Table 2-6 Existing Relations to Estimate Sediment Entrainment or Near-Bed Concentration Under Equilibrium Conditions s s - 0 0

2 Author Formula Parameters Reference height - r 0 3

.

i Einstein b 2D n q∗ s d d (1950) cb 05.

∗

1 τ 1 23.2()s 7 Engelund and b 2Ds 065. 05. ⎡ 4 ⎤ 025. Fredsøe cb ⎡ ∗ βπp ⎤ ⎛ βπ ⎞ 1 3 τ ⎢ ⎥ ()λ ⎢ s 006. ⎥ (1976; 1982) 1 b 6 ⎜ 6 ⎟ λ ⎢ ⎥ ; p ⎢ 1 ⎜ ⎟ ⎥ ; β 1.0 b ( τ∗∗) ⎢ τ ⎥ ⎢ 0.. 027R 1 ss⎥ ⎜ 0 006⎟ ⎢ ⎝ ⎠ ⎥ ⎣ ⎦ ⎣ ⎦

Smith and γ ∗ * ∗∗ 065. oT ττ ατ τ sc 3 bD()k McLean cb T ; γ 2.4 10 os c s s 1γ T * o (1977) o τ α c o 26.3 Itakura and ⎛ u Ω ⎞ ⎛ ⎡ ⎤⎞ b 0.05H ∗ ∗ 2 Kishi (1980) cb12kk⎜ ∗ 1⎟ τ ⎜ ⎢ exp()Ao ⎥⎟ k ⎝ v τ ⎠ Ω 3 s ⎜ k4 ⎢ ∞ ()⎥⎟ 1 ; Ao ∗ k;4 k 2 τ 3 ⎜ ⎢∫ exp ξξd ⎥⎟ ⎝ ⎣ Ao ⎦⎠ k1 0.008; k2 0.14; k3 0.143; k4 2.0; vs fall velocity Van Rijn D T1.5 13/ Δ c 0.015 s ⎛ gR⎞ b (1984) b 0.3 DD ⎜ ⎟ ; Δ is the mean dune height b b D *s⎝ 2 ⎠ b 2 Δ * ν if b known else b ks. bmin 0.01H / Celik and kC ⎡ 006. ⎤ 2 vs* 0.4u b 0.05H om ⎛ k ⎞ u U 1 ⎛1η η ⎞ Rodi (1984) cb ⎢ s ⎥ ∗ m ⋅ b η I Cm 0.034 1 ⎜ ⎟ ; I ∫ ⎜ ⎟ d; ⎝ ⎠ 0.05 ⎝ η η ⎠ ⎣⎢ H ⎦⎥ gRH vs 1 b / h z H; hb 0.05; ko 1.13

Akiyama and u b 0.05H E;sc0 ZZ * 0.5 Fukushima Z R;p Zcm 5; Z 13.2 ⎛ Z ⎞ vs (1986) ⋅ 12 10 c E310Z1s ⎜ ⎟ ; Zcm Z Z ⎝ ⎠ suspended load Z Esm 0.3; Z Z García and AZ5 u 05. ⎛12R ⎞ b 0.05H u ∗s n g ⋅ b ⋅ -7 Parker (1991) E Z Ru;;∗ U8logC′ 1⎜ ⎟ ; n 0.6; A 1.3 10 s A u p s ′ m ⎝ ⎠ 5 vs C 3Ds 1 Zu 0.3

175.

Zyserman and b 2D 117 τ∗ 2 s 0.331()s 0.045 ∗ ()u∗ 11/21/2008 5:44:46 PM Fredsøe s

/ τ 2 c s 1 b ()

/ (1994) 175. 2 0.331 ∗ RgDs 0 τ 0 1 s 0.045 8

0.46

5 : 4 4 : 4 6

P M 118 sediment transport and morphodynamics the somewhat small values of H/D, the values cover a range that includes typical_ fi eld sand-bed streams. Three of the relations for E s (or c b at equilibrium) performed particularly well and are presented here. The fi rst of these presented is the relation of García and Parker (1991). The reference level is taken to be 5 % of the depth; that is,

b δ 0.05 (2- 223) H b

The relation takes the form

A Z 5 E u s ⎛ A ⎞ (2-224a) ⎜ 1 Z 5 ⎟ ⎝ 0.3 u ⎠ where

A.13 107 (2-224b)

u* s 0.6 Z u R e p (2-224c) vs

García and Parker (1993) have found that for fi ne-grained, non-cohesive sediments, Eq. (2-224a) performs well, provided that the similarity variable given by Eq. (2-224c) is modifi ed to

u * 06. Zu 0.. 708RRep for ep 3 5 (2-225) vs

A plot of the García-Parker entrainment function, includ- ing data from open-channel fl ows as well as turbidity cur- Fig. 2-56. Sediment entrainment function of García and Parker rents, is shown in Fig. 2-56. The equation has been used to (1991, 1993). assess the impact of navigation on sediment resuspension in the Mississippi River basin (García et al. 1999; Admiraal et that are similar to Eqs. (2-155) and (2-156). Van Rijn’s rela- al. 2000) as well as the inception of channels in submarine tions are fans (Imran et al., 1998). Of all the formulations in the lit- erature, this equation is one of the few that has been gener- 2 1 ⎛ H ⎞ alized to handle sediment mixtures. This generalization of ln 12 (2-227) C f s κ ⎝⎜ ⎠⎟ the formula is shown in Eq. (3-143a) of Chapter 3. Because k s the equation was developed with data from small-sized to where for uniform material k s 3D . In this equation, the ref- medium-sized sand-bed streams, Wright and Parker (2004; / Δ Δ erence level b 1 2 , where bed form height or b k s 2005b) have found that it needs to be slightly modifi ed for when the bed form height is not known. Note that in Eq. (2- the case of large, low-slope sand-bed rivers. This modifi ca- 227), the total depth H is used, in contrast to Eq. (2-156) where tion of the formula is shown in Eq. (3-143e) of Chapter 3. Hs is used. Van Rijn’s formulation has been used extensively Another relation that has been found to perform well is in numerical models of suspended sediment transport and bed that of van Rijn (1984b). This relation takes the form, morphology (e.g., Duan et al. 2001; Zeng et al. 2006). Its appli- 1.5 cation for two-dimensional and three-dimensional sediment τ* / * D () s τc 1 c (2- 226) transport modeling is illustrated in Chapter 15. b 0.015 0.3 b D* A third relation that performs well is that of Smith and McLean (1977). Its origin can be traced back to the early / ν2 1/3 τ* Here D * D ( gR ) and s denotes the Shields stress work of Yalin (1963) on bed load transport and was fi rst pro- τ due to skin friction. Van Rijn computes bs from relations posed by Smith (1977). It can be expressed as

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* / * 1. Compute depth-discharge relations for fl ows up to γ () ττs c 1 c 0.65 o bank-full (lower regime only) using the Engelund- b γ * / * (2-228a) 11 o () ττs c Hansen method. Plot H versus Q . Use the results of τ* the Engelund-Hansen method to compute values of s where as well. τ * 2. Use the values of s to compute the bed load discharge γ Q q B using the Ashida-Michiue formulation (Eq. o 0.0024 (2-228b ) b b 2-94). For each value of H and U , back-calculate the _ composite roughness k . Then compute the suspended The value b at which c is to be evaluated is given by the c b load Q q B from the Einstein formulation and the relation below: s s relation for E s due to García and Parker. Plot Q b , Q s , and QT Q b Q s as functions of water discharge Q . * / * b26.3 () ττs c 1 Dk s (2-228c) For example, compute the fl ow depth, bed load discharge, and suspended load discharge as a function of fl ow discharge for a stream with the following properties: Here k s denotes the equivalent roughness height (i.e., Nikuradse’s roughness) for a fi xed bed. The Smith-Mclean S 0.0004 formulation is used extensively in benthic boundar layer -4 fl ows and oceanic sedimentation (e.g., Wiberg et al. 1994; Ds 0.35 mm 3.5 10 m and McCave 2001). After the comparative analysis of different entrainment R 1.65 formulations by García and Parker (1991), Zyserman and Fredsøe (1994) proposed an empirical relation using the Fort B 75 m at bankfull Collins experimental data. It reads H 2.9 m at bankfull 175. 0. 331()ττ** The calculations are performed for fl ows up to bank-full. sc cb (2-229) For fl ows below bank-full, the following relation is used to ττ**175. 1072. ()sc calculate the stream width:

In this formulation the near-bed equilibrium concentra- ⎛ ⎞ 0.1⎛ ⎞ 0.1 _ B Q U H B tion c is assigned to a reference level b 2 D . Even though (i) ⎜ ⎟ ⎜ ⎟ b 50 ⎝ Q ⎠ ⎝ Q ⎠ this relation was developed with laboratory data, it has been Bbf bf bf used to estimate near-bed sediment concentration in coastal sediment transport (e.g., Soulsby 1997). where the subscript bf indicates bankfull values. Solving for the stream width B yields:

1/ 0.9 2.9.8 Example of Depth-Discharge and Sediment ⎡ ⎛ ⎞ 0.1 ⎤ Load Computation (ii) ⎢ U H ⎥ B Bbf ⎜ ⎟ ⎢ ⎝ Q ⎠ ⎥ Consider the example stream of Section 2.8.2. For this ⎣ bf ⎦ stream, S 0.0004 and D 0.35 mm (uniform material). At bankfull fl ow, the stream width is 75 m. For fl ows below The methods used to determine Q , Qb , Qs , and Q bf bankfull, the following sample relation is assumed (Parker are described below. A computer program can be writ- and Peterson 1980): ten or a spreadsheet can be used to perform the necessary calculations. All computations and results are summarized

0.1 in Table 2-7. B ⎛ Q ⎞ 2.9.8.1 Depth-Discharge Calculations The depth- ⎜ ⎟ discharge relation is computed using the Engelund-Hansen Bbf ⎝ Qbf ⎠ method. The calculations are performed by assuming a value

for H s (the fl ow depth that would be expected in the absence A more precise relation can be worked out for any cross of bedforms), and then calculating the actual fl ow depth (H)

section using measured cross-sectional profi les in a stream. and the fl ow discharge ( Q ). Hs is varied between 0.22 m and Here the subscript bf denotes bank-full. Assume that the the bank-full value of 2.9 m. The fi rst step in calculating the

stream is wide enough to equate the hydraulic radius R h with depth-discharge relation is to compute the resistance coef- the cross-sectionally averaged depth H. fi cient due to skin drag (C fs ) from Hs :

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1

2 * 0 Hs U H Width B Discharge qb Qb kc u Rouse# qs Qs Qt τ* τ* 3 * 2 3 2 3 3 (m) Cfs (m/s) s Depth (m) (m) Q (m /s) q b (m /s) (m /s) (m) Zu Es (m/s) ZR J1 J2 (m /s) (m /s) (m /s)

0.10 0.003 0.353 0.069 0.152 0.220 45.282 3.516 0.01296 0.00000 0.00002 0.00192 2.51992 0.00001 0.02936 4.76463 0.01219 0.03363 0.00000 0.00000 0.00002

0.20 0.003 0.548 0.139 0.443 0.640 53.541 18.781 0.22362 0.00001 0.00032 0.00280 3.56370 0.00007 0.05010 2.79237 0.02417 0.06165 0.00000 0.00005 0.00036

0.30 0.002 0.706 0.208 0.608 0.878 57.038 35.358 0.62296 0.00002 0.00094 0.00256 4.36462 0.00021 0.05868 2.38406 0.03008 0.07401 0.00001 0.00031 0.00124

0.40 0.002 0.844 0.277 0.737 1.064 59.434 53.359 1.16862 0.00003 0.00183 0.00233 5.03983 0.00042 0.06460 2.16564 0.03447 0.08272 0.00002 0.00106 0.00289

0.50 0.002 0.969 0.346 0.846 1.221 61.284 72.500 1.83808 0.00005 0.00297 0.00214 5.63470 0.00074 0.06923 2.02077 0.03809 0.08962 0.00004 0.00269 0.00566

0.60 0.002 1.083 0.416 0.943 1.361 62.801 92.588 2.61681 0.00007 0.00433 0.00199 6.17251 0.00116 0.07309 1.91423 0.04122 0.09539 0.00009 0.00569 0.01002

0.70 0.002 1.190 0.485 1.031 1.488 64.093 113.487 3.49443 0.00009 0.00590 0.00186 6.66707 0.00170 0.07641 1.83093 0.04400 0.10040 0.00017 0.01065 0.01655

0.80 0.002 1.291 0.554 1.111 1.605 65.220 135.098 4.46304 0.00012 0.00767 0.00176 7.12740 0.00237 0.07935 1.76309 0.04652 0.10484 0.00028 0.01825 0.02592

0.90 0.002 1.387 0.623 1.187 1.713 66.221 157.342 5.51640 0.00015 0.00962 0.00167 7.55975 0.00318 0.08200 1.70620 0.04883 0.10884 0.00044 0.02925 0.03887

1.00 0.002 1.478 0.693 1.258 1.816 67.124 180.160 6.64937 0.00018 0.01176 0.00159 7.96868 0.00412 0.08441 1.65745 0.05098 0.11249 0.00066 0.04449 0.05625

1.10 0.002 1.566 0.762 1.325 1.912 67.947 203.503 7.85765 0.00021 0.01407 0.00152 8.35762 0.00521 0.08663 1.61496 0.05299 0.11585 0.00095 0.06488 0.07894

1.20 0.002 1.651 0.831 1.388 2.005 68.704 227.329 9.13758 0.00024 0.01654 0.00146 8.72925 0.00645 0.08869 1.57740 0.05488 0.11898 0.00133 0.09138 0.10792

1.30 0.002 1.732 0.900 1.450 2.093 69.404 251.603 10.48599 0.00028 0.01917 0.00141 9.08569 0.00784 0.09062 1.54385 0.05667 0.12189 0.00180 0.12503 0.14421

1.40 0.002 1.811 0.970 1.508 2.177 70.057 276.297 11.90008 0.00031 0.02196 0.00136 9.42866 0.00938 0.09243 1.51358 0.05837 0.12463 0.00238 0.16689 0.18885

1.50 0.002 1.888 1.039 1.564 2.259 70.669 301.385 13.37737 0.00035 0.02490 0.00132 9.75959 0.01109 0.09414 1.48607 0.05999 0.12722 0.00309 0.21805 0.24295

1.60 0.002 1.963 1.108 1.619 2.337 71.244 326.844 14.91565 0.00039 0.02799 0.00128 10.07967 0.01294 0.09577 1.46089 0.06155 0.12967 0.00392 0.27963 0.30762

1.70 0.002 2.036 1.177 1.671 2.413 71.788 352.654 16.51293 0.00044 0.03123 0.00124 10.38988 0.01496 0.09731 1.43770 0.06304 0.13200 0.00491 0.35277 0.38400

1.80 0.002 2.107 1.247 1.722 2.487 72.303 378.798 18.16739 0.00048 0.03460 0.00121 10.69110 0.01712 0.09878 1.41625 0.06447 0.13421 0.00607 0.43860 0.47320

1.90 0.002 2.176 1.316 1.772 2.558 72.793 405.260 19.87737 0.00052 0.03812 0.00118 10.98406 0.01944 0.10019 1.39631 0.06585 0.13634 0.00739 0.53825 0.57637

2.00 0.002 2.244 1.385 1.820 2.628 73.260 432.026 21.64136 0.00057 0.04177 0.00115 11.26941 0.02190 0.10155 1.37769 0.06719 0.13837 0.00891 0.65283 0.69460

2.10 0.002 2.311 1.455 1.867 2.696 73.706 459.083 23.45795 0.00062 0.04555 0.00112 11.54771 0.02451 0.10285 1.36026 0.06848 0.14032 0.01063 0.78342 0.82897

2.20 0.002 2.376 1.524 1.913 2.762 74.134 486.419 25.32585 0.00067 0.04946 0.00110 11.81946 0.02726 0.10410 1.34387 0.06973 0.14220 0.01256 0.93107 0.98053

2.30 0.002 2.440 1.593 1.958 2.826 74.544 514.024 27.24386 0.00072 0.05350 0.00108 12.08510 0.03014 0.10531 1.32843 0.07094 0.14401 0.01471 1.09677 1.15027

2.40 0.002 2.502 1.662 2.001 2.890 74.939 541.887 29.21083 0.00077 0.05767 0.00105 12.34502 0.03315 0.10648 1.31384 0.07212 0.14575 0.01710 1.28147 1.33914 11/21/2008 5:44:51 PM / 2 1 / 2 0 0 8

5 : 4 4 : 5 1

P M suspended load 121

2 2.9.8.3 Sediment Load Discharge Calculations The ⎡ 1 ⎛ H ⎞ ⎤ (iii) ⎢ ln 11 s ⎥ Einstein formulation is used to compute the suspended load C fs κ ⎝⎜ ⎠⎟ ⎣ k s ⎦ transport rate per unit width q s , ⎡ ⎛ ⎞ ⎤ where κ is the von Karman constant (0.4) and k is given by 1 H s (xiii) q cub * H ⎢ J12 ln ⎜30 ⎟ J⎥ s κ ⎣ ⎝ k ⎠ ⎦ 4 4 c (iv) k s 2.5 Ds 2.5 (3.5 10 ) 8.75 10 m The depth-averaged fl ow velocity (U ) can be found from where C and H fs s (xiv) u* gHS g H S (v) U s _ If the suspension is assumed to be at equilibrium, cb Cfs Es . The dimensionless rate of entrainment (E s ) is calculated τ* with the relation of García and Parker (1991), The Shields stress due to skin friction ( s ) is given by 5 τ HS A Z u * bs s E s (vi) τ s (xv) ⎛ ⎞ ρ A 5 g R DssR D ⎜1 Z ⎟ ⎝ 0.3 u⎠ According to Engelund-Hansen, the total Shields stress for where A is equal to 1.3 ×10 7 and the lower regime can be found from the following relation: u (xvi) *s 0.6 τ* 6 Z u Rep (vii) τ* s vs 0.4 (xvii) u*s g H s S The fl ow depth can be calculated from the total Shields stress as follows: R g DD (xviii) R ss ep ν τ* R D (viii) H s S Notice that for the entrainment formulation, the shear

velocity associated with skin friction u * s has to be used. The Finally, the discharge can be calculated from the results temperature is assumed to be about 20°C; therefore, the kine- of Eqs. (v) and (viii): matic viscosity v is about 10 6 m2 /s. An iterative method, or Eq. (2-47a), is used to calculate the terminal fall velocity (ix) Q U H B of the sediment particles vs, which is found to be 5.596 10 2 m /s. The composite roughness (k ) is calculated accord- where B must be adjusted according to Eq. (ii) for fl ows c ing to the following relation: less than bankfull. A plot of the depth-discharge relation is shown in Fig. 2-57(a). ⎧ κ ⎫ U 2.9.8.2 Bed Load Discharge Calculations The (xix) k c 11 H exp⎨ ⎬ dimensionless bed load transport rate (q * ) is found from the ⎩ u* ⎭ Ashida-Michiue formulation: The Einstein integral parameters J 1 and J 2 are found for δ 0.05 with the help of Equations (2-221a) and (2-221b). (x) q * 17( τ * τ * ) ͫ( τ * )0.5 ( τ * ) 0.5 ͬ b s c s c The suspended load transport rate per unit width calcu- lated according to Eq. (xiii) is used to compute the suspended where τ * is calculated in Eq. (vi) and τ * is taken to be 0.05. s c load transport rate suspended load per unit width q , so that The bed load transport rate per unit width q is given by s b the total suspended load can be obtained (in m3 /s) with * (xi) qqb g R Dss D (xx) Qs qs B

Therefore, the bedload transport rate (in m3 /s) is given by For fl ows less than bankfull fl ow, the channel width B must be adjusted according to Eq. (ii). (xii) Qb qb B 2.9.8.4 Determination of Bank-Full Flow Discharge ( Qbf ) The fl ow discharge at bankfull (Q bf ) is determined by Again, B must be adjusted according to Eq. (ii) for fl ows assuming that up to bank-full fl ow, lower regime conditions less than bank-full. exist. The bank-full fl ow depth for this stream is assumed to

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 112121 11/21/2008/21/2008 5:44:515:44:51 PMPM 122 sediment transport and morphodynamics be 2.9 m. Then for bank-full fl ow, the total shear stress τ * is g H S given by U s C (xxv) fs * HS 2.9 0.0004 9.81 2.42 0.0004 (xxi) τ 2.01 ms/ × 4 3 2.51 R Ds 1.65 3.5 10 1.5 10

From Engelund and Hansen, 3/ (xxvi) Qb UHB2.51 2.9 75 546.35 m s 2 (xxii) ττ**() ( )2 s 0.06 0.4 0.06 0.4 2.01 1.67 A plot of Q b , Qs , and Q T Q b Q s as functions of water discharge is shown in Fig. 2-57(b). For fl ows up to 100 m 3 /s, the bed load discharge is larger than the suspended τ* R 1.67 1.65 () 3.5 10 4 load discharge. As the fl ow discharge increases, the sus- (xxiii) s Ds H s 2.42 m pended load becomes much larger than the bed load all the S 0.0004 way up to bank-full fl ow conditions.

2 Notice also that the composite roughness k c (grain- friction ⎡ 1 ⎛ H ⎞ ⎤ plus form drag) fi rst increases with fl ow discharge for low ⎢ ln 11 s ⎥ C fs κ ⎝⎜ ⎠⎟ ⎣ k s ⎦ fl ows, but from then on it decreases monotonically as the bedforms begin to be washed out by the fl ow. For bank-full ⎡ ⎛ ⎞ ⎤ 2 (xxiv) 1 2.42 3 conditions, the bedforms have a small effect on fl ow resis- ⎢ ln ⎜ 11 ⎟ ⎥ 1.5 10 ⎣ 0.4 ⎝ 8.75 10 4 ⎠ ⎦ tance in this particular example.

Fig. 2-57. (a) fl ow discharge rating curve, and (b) sediment discharge rating curves for bed load, suspended load and total load.

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Fig. 2-57. (a) fl ow discharge rating curve, and (b) sediment discharge rating curves for bed load, suspended load and total load. (Continued)

2.10 DIMENSIONLESS RELATIONS Here six such relations are presented, those due to FOR TOTAL BED-MATERIAL LOAD IN Engelund and Hansen (1967); Ackers and White (1973); SAND-BED STREAMS Yang (1973); Brownlie (1981b); Karim and Kennedy (1981; 1990); and Molinas and Wu (2001). 2.10.1 Form of the Relations They apply only to sand-bed streams with relatively In the analysis presented in previous sections, the guid- uniform bed sediment. The Engelund and Hansen rela- ing principle has been the development of mechanistically tion together with the relations of Brownlie and Karim and accurate models of the bed load and suspended load Kennedy, are the most complete ones, as each is presented components of bed-material load. The total bed-material as a pair of relations for total load and hydraulic resistance. load is then computed as the sum of the two. That is, where Ackers and White (1973), Yang (1973), and Molinas and q denotes the volume bed load transport rate per unit width, Wu (2001) are presented as relations for total load only. In b most cases, it will also be necessary for the user to specify and qs denotes the volume suspended load transport rate per unit width (bed material only), the total volume transport a relation for hydraulic resistance in order to perform actual rate of bed material per unit width q is given by calculations. Computations for one-dimensional river mod- t eling using the Ackers-White, Engelund-Hansen, and Yang transport relations are presented in Chapter 14. qt qb qs (2-229) As stated earlier, the importance of using transport and Another, simpler approach is to ignore the details of the hydraulic resistance relations as pairs cannot be overem- physics of the problem, and instead use empirical techniques phasized (e.g., Parker and Anderson 1977). Consider, for such as regression analysis to correlate dimensionless param- example, the simplest generalization beyond the assumption

eters involving q t to dimensionless fl ow parameters inferred of normal fl ow, the case of quasi-steady, gradually varied to be of importance for sediment transport. This can be imple- one-dimensional fl ow. The governing equations for a wide mented in the strict sense only for equilibrium or quasi-equi- rectangular channel with fl ow in the streamwise x direction, librium fl ows, i.e., near-normal fl ow conditions. The resulting can be written as (Chow 1959) relations are no better than the choice of dimensionless parameters to be correlated. They are also less versatile than ⎛ 2 ⎞ physically based relations, because their application to non- d V (2-230a) ⎜ H ⎟ S S f steady, nonuniform fl ow fi elds is not obvious. On the other dx ⎝ 2 g ⎠ hand, they have the advantage of being relatively simple to use and of having been calibrated to sets of both laboratory and fi eld data often deemed to be trustworthy. qw UH (2-230b)

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Here the friction slope S f is given as transport calculations must be obtained from a predictor of hydraulic resistance. Chollet and Cunge (1979) demonstrate a useful approach to estimate friction slopes for unsteady fl ow τ U 2 b (2-231) computations (i.e., head losses) using two hydraulic resistance S f ρ C f g H g H predictors developed for steady, normal fl ow conditions (i.e., Einstein-Barbarossa and Engelund-Hansen). Chapter 14 also A slightly more general form for non-rectangular chan- presents relevant information on how the friction slope is esti- nels is mated for one-dimensional computational modeling. A few parameters are introduced here. Let Q denote the ⎛ 2 ⎞ d 1 Q Q ⎜ ξ ⎟ S S (2-232a) total water discharge, and st the total volume of bed-mate- d x ⎝ 2 g 2 b⎠ f A rial sediment discharge. Furthermore, let B a denote the active width of the river over which bed material is free to move as described in Fig. 2-58. In general, B is usually somewhat Q UA a (2-232b) less than water surface width B due to the common tendency for the banks to be cohesive, vegetated, or both. where It follows then that A channel cross-sectional area, And the friction slope S f is given as QBqw (2-235a)

2 τ Q Ba q b U st t (2-235b) S f ρ C f (2-233) g Rh g Rh One dimensionless form for dimensionless total bed material transport per unit width is q * where In the above equations, R h denotes the hydraulic radius t ξ and b denotes the water surface elevation above the deepest point in the channel, as shown in Fig. 2-58. q q* t (2-236) Note that in the case of normal fl ow, the momentum equa- t R g D D τ ρ tions reduce to S f S , or b gHS for the wide rectangular case and τ ρ gR S for the non-rectangular, natural case. In b h where the case of gradually varied fl ow, however, S ≠ S , in which f case the bed slope S cannot be used as a basis for calculating D grain size usually equated to D 50 . sediment transport. The appropriate choice is S , so that from f Another commonly used measure is concentration Eq. (2-233), for example, by weight in parts per million, here called C s . This can be τρ given as bhfg RS (2-234)

ρ Q It should be apparent for the case of gradually varying fl ow, C 6 s st (2-237) s 10 ρ ρ then, that the friction slope necessary to perform sediment Qs Qst

Fig. 2-58. Defi nition diagram for channel parameters.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 112424 11/21/2008/21/2008 5:44:545:44:54 PMPM dimensionless relations for total bed-material load in sand-bed streams 125 2.10.2 The Engelund-Hansen Relation 2.10.2.3 Computational Procedure for Normal Flow The water discharge Q , slope S , and grain size D must be 2.10.2.1 Sediment Transport This relation is one of 50 known. In addition, channel geometry must be known, so the simplest to use for sediment transport, and also one of that B , B , A , H , P, and R are all known functions of stage the most accurate. It was determined from a rather small set a h (water surface elevation) ξ . The procedure is best outlined of laboratory data (Guy et al. 1966), but performs quite well assuming that R is known and Q is to be calculated, rather as a fi eld predictor as well. It takes the form hs than vice versa. For any given value of R hs (or H s ), U can be * / computed from Eq. (2-239a). Noting that τ R S /( RD ) * * 5 2 s hs 50 * * C f qt 0.05 ()τ (2-238) τ τ and R h S /( RD50 ), , and thus, R h can be computed from ∼ Eqs. (2-239b e). The plot of Rh versus water surface eleva- where ξ ξ tion is used to determine b , which is then used to deter- mine B , B , H , A , P , etc. Discharge Q is then given by Q C total resistance coeffi cient (skin friction plus form a f UBH drag); and . In an actual implementation, this process is reversed ( Q is given and R , etc., is computed). This requires an iter- τ * total (skin friction plus form drag) Shields stress hs based on the size D . ative technique; Newton-Raphson is not diffi cult to imple- 50 ment (e.g., Parker 2005). 2.10.2.2 Hydraulic Resistance The hydraulic resis- Once the calculation of hydraulic resistance is com- tance relation of Engelund and Hansen has already been plete, it is possible to proceed with the computation of total introduced above. It must be written in several parts. The Q st . The friction coeffi cient C f is given key relation for skin friction is / 2 τ* by (gR h S ) U . Placing the known values of Cf and into * (2-238), q t, and thus, qt can be computed. It follows that Q st ⎛ ⎞ 1/2 U R hs q B . C 2.5ln ⎜ 11 ⎟ (2-239a) t a fs 2.10.2.4 Computational Procedure for Gradually g R h s S ⎝ k s ⎠ Varied Flow To implement the method for gradually var- ied fl ow, it is necessary to recast the above formulation into where an algorithm for friction slope S f , which replaces S every- where in the formulation of Eqs. (2-239a) to (2-239e). The ks 2 to 2.5 D50 . formulation is then solved in conjunction with Eqs. (2-230a) Here, R denotes the hydraulic radius due to skin fric- hs and (2-230b) or Eqs. (2-232a) and (2-232b) to determine tion, which often can be approximated by H s . The relation τ the appropriate backwater curve. Once C f and b are known for form drag can be written in the form everywhere, the sediment transport rate can be calculated from Eq. (2-238). Computations for gradually-varied ττ** f () s (2-239b) fl ow with the Engelund-Hansen relations are presented in Chapter 14. where for the lower regime

** 2 2.10.3 The Brownlie Relation ττs 0.06 0.4 () (2-239c) 2.10.3.1 Sediment Transport The Brownlie relations and for the upper regime are based on regressions of over 1000 experimental and fi eld data points. For normal or quasi-normal fl ow, the transport relation takes the form ⎧ τ * ;1 τ* ⎪ * ( 1 / 1.8 ) * τ s ⎨ ⎡ 1.8 ⎤ ; τ 1 0.3301 0.298 0.702 () τ* 1.978 ⎛ R ⎞ ⎪ ⎣⎢ ⎦⎥ 7115 () 0.6601 h ⎩ C sfc F ggoF S ⎝⎜ ⎠⎟ D50 (2-239d) (2-240a)

An approximate condition for the transition between the lower and upper regimes is given by where

* U τ s 0.55 (2-239e) F g (2-240b) R g D50 The Engelund-Hansen relation has been found to be a good predictor of both laboratory and fi eld data in spite of * 0.5293 0.1045 0.1606 (2-240c) its simplicity. Fgo 4.596 ()τc S σ g

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 112525 11/21/2008/21/2008 5:44:555:44:55 PMPM 126 sediment transport and morphodynamics

* 7.7 Y (2-240d) into Eq. (2-240a) in order to determine the concentration C τc 0.22Y 0.06 10 s in parts per million by weight. The transport rate Q st is then computed from Eq. (2-237). 0.6 Y Rep (2-240e) 2.10.3.4 Computation for Gradually Varied Flow The Brownlie relation is not presented in a form that obviously allows extension to gradually varied fl ow. The most unam- In Eq. (2-240a), cf 1 for laboratory fl umes and 1.268 τ* biguous procedure, however, is to replace S with S f in the for fi eld channels. The parameters c and R ep are the ones previously introduced in this chapter. resistance relation and to couple it with a backwater calcula- tion to determine S . The friction slope is then substituted 2.10.3.2 Hydraulic Resistance The Brownlie rela- f tions for hydraulic resistance were determined by regression into Eq. (2-240a) in place of the bed slope in order to deter- from the same set of data used to determine the relation for mine the sediment transport rate. sediment transport. The relation for lower regime fl ow is 2.10.4 The Ackers-White Relation

Rh 0.6539 0.09188 0.1050 Based on Bagnold’s stream power concept, Ackers and White S0.3724 (q) S σ g (2-241a) D50 (1973) applied dimensional analysis to express the mobil- ity and transport rate of sediment in terms of dimensionless The corresponding relation for upper regime fl ow is parameters. Several years later a corresponding relation for hydraulic resistance was also presented (White et al. 1980; 1982). Only the sediment load equation is presented here. It Rh 0.6248 0.08750 0.08013 takes the form S0.2836 (q) S σ g (2-241b) D50 ρ n m D ⎛ U ⎞ ⎛ F g r ⎞ 6 c s 50 1 (2-242a) In these relations, C s 10 ρ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ Rh u* Aa w

~ qw q (2-241c). where the so-called Ackers-White mobility number is given g D50 D 50 by

1n The distinction beween lower and upper regime is made uun ** as follows (see Fig. 2-44). For S 0.006, the fl ow is assumed Fgr (2-242b) R g always to be in the upper regime. For S 0.006, the largest D50

value of F g at which the lower regime can be maintained is taken to be U u * ⎛ ⎞ Rh (2-242c) 32 log ⎜ 10 ⎟ FFgg0.8 (2-241d) ⎝ ⎠ D50

And the smallest value of F for which upper regime can g The parameters n , m , c , and A aw were determined with be maintained is taken to be best-fi ts of laboratory data, as functions of a dimensionless

grain size D g r , where FFgg1.25 (2-241e) 23/ DRgr ep (2-242d) In the above relations in the following fashion. If D g r 60, then 1/3 Fg 1.74 S (2-241f) n 0 (2-242e)

2.10.3.3 Computational Procedure for Normal Flow σ m 1.5 (2-242f) It is necessary to know Q , S , D50 , g , and cross-sectional geometry as a function of stage. The computation is explicit, although trial and error may be required in order to deter- Aaw 0.17 (2-242g) mine the fl ow regime. Hydraulic radius is computed from Eq. (2-241a) or Eq. (2-241b), and the result can be substituted c 0.025 (2-242h)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 112626 11/21/2008/21/2008 5:44:575:44:57 PMPM dimensionless relations for total bed-material load in sand-bed streams 127 If 1 Dg r 60, then velocity and channel slope. Coeffi cients in Yang’s equation were determined by a multiple regression analysis of labora- tory fl ume data. The relation takes the form 9.66 n1 0.56 log ()D gr ; m 1.34 (2-242i) D g r ⎛ ⎞ ()U S U c S log C s a a2 log ⎜ ⎟ (2-243a) 1 ⎝ v v ⎠ 9.66 ss m 1.34 (2-242j) D g r where

0.23 ⎛ ⎞ ⎛ ⎞ Aaw 0.14 (2-242k) vs*D50 u a1 5.435 0.286 log ⎜ ⎟ 0.457 log ⎜ ⎟ D gr ν ⎝ ⎠ ⎝ vs ⎠

(2-243b) ⎡ ⎤ 2 log (c) 2.86 log ()D g r ⎣log ()D g r ⎦ 3.53 (2-242l) ⎛ ⎞ ⎛ v D50 ⎞ u 1.799 0.409 log s 0.314 log ⎜ * ⎟ a2 ⎜ ν ⎟ ⎝ ⎠ ⎝ vs ⎠ Note that all logarithms here are base 10; u * retains its previously introduced meaning as shear velocity. (2-243c) The procedure for the computation of sediment transport rate using Ackers and White’s approach can be summarized as follows: and Uc denotes a critical mean fl ow velocity for initiation of motion given by 1. Determine the value of D g r with Eq. (2-242d) from ρ ρ / ρ known values of D , R ( s ) , and kinematic υ viscosity . ⎧ u D 50 2.05 if * 70 2. Determine the value of n , A , m , and c corresponding ⎪ ν aw ⎪ to the value of Dgr from Eqs. (2-242e) to (2-242l). U c ⎨ 2.5 u D 50 if 1.2* 70 3. Determine the total sediment concentration v s ⎪ ⎛ u D 50 ⎞ ν C by weight in parts per million (ppm) with ⎪ log ⎜ * ⎟ 0.06 s ⎩ ⎝ ν ⎠ Eq. (2-242a). (2-243d) 4. Determine the total sediment transport rate Q st with Eq. (2-237) and a known value of fl ow discharge Q .

Note that the logarithms are all base 10, and that, v s retains Computations for gradually-varied fl ow in a river with its previous meaning as fall velocity. Computations using the Ackers-White relations are presented in Chapter 14. Yang’s approach for gradually-varied fl ow are presented in Brownlie (1981a) found that the Ackers-White sediment Chapter 14. load relation predicted laboratory observations quite well Yang and Molinas (1982) compared the Yang’s formula but fi eld observations were underestimated. In 1990, HR with 1093 laboratory data and 166 river data, yielding a Wallingford adjusted the coeffi cients in the Ackers-White value of 95% of the predicted transport rates within a fac- relation, because the original formula predicted transport tor of 2 of measured transport rates. Large-scale river data rates which were too large for fi ne sediments (D 50 0.2 mm). were not included in this comparison. Recently, Yang More recently, Niño (2002) has found that the Ackers-White and Simoes (2005) have used Yang’s unit stream power relation provides a good predictor for coarse sand and gravel approach (Yang 1979; Yang et al. 1996) to predict total transport in the rivers of Chile, and used this relation to pre- bed-material load as well as wash load in the Yellow River, dict longitudinal grain-size variation. The Ackers-White for- China. mulation is extended for application to sediment mixtures in Chapter 3.

2.10.6 The Karim-Kennedy Relation 2.10.5 The Yang Relation The Karim and Kennedy (1981, 1990) methodology for To determine total sediment concentration, Yang (1973) depth-discharge predictors described previously in Section also used dimensional analysis and the fundamental con- 2.8.3.5 also includes a total sediment discharge formula cept of unit stream power given by the product of mean fl ow obtained from nonlinear regression using a database of 339

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 112727 11/21/2008/21/2008 5:44:585:44:58 PMPM 128 sediment transport and morphodynamics river fl ows and 608 fl ume fl ows. The uncoupled sediment earlier (Eq. 2-72a) for the discussion of wash load and bed- load and resistance relations they obtained are given by material load. The formulation proposed by Celik and Rodi (1984) to estimate sediment transport capacity (see Table 2-6) is also based on Velikanov’s theory. Molinas and Wu (2001) q ⎡ U ⎤ logt 2.279 2.972 log ⎢ ⎥ argued that the predictors of Engelund-Hansen, Ackers and 3 ⎢ gRD ⎥ gRD50 ⎣ 50 ⎦ White, and Yang have been developed with fl ume experi- ments representative of shallow fl ows and cannot be applied ⎡ U ⎤ ⎡ uu ⎤ 1.060log ⎢ ⎥ log ⎢ **c ⎥ to large rivers having deep-fl ow conditions. Average depths ⎢⎣ gRD50 ⎦⎥ ⎣⎢ gRD50 ⎦⎥ vary between 12 and 68 m in the Amazon River, and between 3 and 22 m in the Missippi River (Posada 1995). These fl ow ⎛ ⎞ ⎡ ⎤ H uu**c depths are much larger than those commonly found in labo- 0.299log⎜ ⎟ log ⎢ ⎥ ⎝ ⎠ ratory experiments ( 1 m). At the same time, under labora- D50 ⎣⎢ gRRD50 ⎦⎥ tory conditions Reynolds numbers are much smaller, Froude (2-244a) numbers are larger, and water surface (energy) slopes are steeper than those observed in large natural rivers. Motivated and by the need for having a total bed-material load predictor for application to large sand-bed rivers, they used stream power and energy considerations together with data from large riv- ⎡ ⎤0.376 U qw 0.310 ers (e.g., Amazon, Atchafalaya, Mississippi, Red River), to 2.822 ⎢ ⎥ S (2-244b) obtain an empirical fi t for the total bed-material load concen- gRD50⎣⎢ gRD 50 ⎦⎥ tration (Eq. 2-237) in ppm,

in which q total volumetric sediment discharge per unit width. 1430( 0.86 ΨΨ) 1.5 t Cs (2-245a) The rest of the variables have been defi ned above. 0.016 ψ Equation (2-244b) can be used for fl ows well above incipient sediment motion. If it is necessary to take into account the where bed confi guration changes, Eq. (2-244b) should be replaced with Eq. (2-183d). ψ stream power, which is defi ned by More recently, Karim (1998) proposed a simpler power relation for the sediment transport equation using the same 3 Ψ U data sets employed in the Karim-Kennedy analysis, with the 2 ⎡ ⎛ H ⎞ ⎤ (2-245b) results given by gRHv ⎢log ⎥ s50 10 ⎝⎜ ⎠⎟ ⎣ D50 ⎦

2.97 1.47 q ⎡ U ⎤ ⎡u ⎤ t 0.00139 ⎢ ⎥ ⎢ * ⎥ (2-244c) Where 3 gRD ⎣⎢ gRD50 ⎦⎥ ⎣vs ⎦ 50 U mean fl ow velocity; g acceleration of gravity, H fl ow depth, Both relations yield approximately the same results but v fall velocity for sediment size D ; and Eq. (2-244c) would be easier to implement than the original s50 50 R submerged specifi c gravity of sediment. Eq. (2-244a). Karim (1998) applied his equation to labora- tory and fi eld data having non-uniform sediments by divid- Most parameters in this empirical relation can be measured ing the sediment into size fractions. More on this application and/or estimated in the fi eld, making it a useful formulation can be found on Chapter 3. for practical use in large sand-bed rivers. One advantage of this approximation is that the energy slope (S ) does not have to be measured directly, which is always a challenge in 2.10.7 The Molinas-Wu Relation large alluvial rivers. On the other hand, since Molinas and This empirical relation is based on Velikanov’s gravitational Wu (2001) do not mention how the wash load was separated power theory, which assumes that the power available in from the bed-material load and the same large river data fl owing water is equal to the sum of the power required to were used both to develop and to test their formulation, Eq. overcome fl ow resistance and the power required to keep (2-245) might overestimate bed-material load concentrations sediment in suspension against gravitational forces. In fact, when applied to other large rivers not included in the calibra- this is equivalent to the simplifi ed energy balance presented tion. More testing of this formulation with independent data

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 112828 11/21/2008/21/2008 5:44:585:44:58 PMPM morphodynamics of rivers and turbidity currents 129 sets is needed. Such data should be forthcoming given the approach, including their venting to prevent , was new technologies that are now available for fl ow and sedi- made by Fan (1986). A few years later, Sloff (1997) used the ment measurements in large rivers. method of characteristics to model turbidity currents in res- ervoirs, but did not address the morphodynamics of reservoir sedimentation. More recently, De Cesare et al. (2001) also 2.10.8 Other Relations for Total Bed-Material Load indicated that turbidity currents have played an important Several other total sand-discharge formulae can be found in the role in the siltation of many Swiss reservoirs. Thus, the main literature, including those of Shen and Hung (1972), Kikkawa reason for introducing turbidity currents in this section is to and Ishikawa (1978), Ranga-Raju et al. (1981), van Rijn (1984), motivate a succeeding section on the modeling of lake and and Pacheco-Ceballos (1989; 1992) among others. A more reservoir sedimentation. No attempt is made here to provide complete review and evaluation and ranking of various rela- a detailed discussion of either turbidity currents or reservoir tions can be found in Alonso (1980), Brownlie (1981), Yang sedimentation. The latter issue is addressed in more detail and Molinas (1982), Gomez and Church (1989), Yang and Wan in Chapter 12. Rather, the goal here is to guide the reader in (1991), Maza-Alvarez and García-Flores (1996), Yang (1996), understanding the processes involved in the morphodynam- and Bravo-Espinosa et al. (2003), and Yang (2005). ics of lake and reservoir sedimentation. In general, sediment transport predictors should be applied within the range of fl ow conditions and sediment character- 2.11.2 Equations Governing the Morphodynamics istics for which they were developed (Williams and Julien of Rivers 1989). Of paramount importance is to realize that sediment transport and fl ow resistance relations should be specifi ed Rivers evolve over time in accordance with the interaction in pairs (Parker and Anderson 1977). The most reliable pre- between the fl ow and sediment-transport fi elds over an erod- dictors for sediment load and fl ow resistance will continue ible bed (which changes the bed) and the changing morphol- to rely on direct observations in the fi eld. Some guidance ogy of the bed (which changes the fl ow and sediment-transport regarding the estimation of sediment discharge in the fi eld fi elds). This co-evolution is termed “morphodynamics” can be found in Appendix D. (Parker and García 2006). It is useful to introduce the rela- tions governing the morphodynamics of turbidity currents in analogy to the morphodynamics of one-dimensional fl ow in 2.11 MORPHODYNAMICS OF RIVERS AND a river of constant width. The parameters h , U and η denote TURBIDITY CURRENTS fl ow depth, depth-averaged fl ow velocity, and bed elevation, respectively. Depth-averaged fl ow velocity is defi ned as 2.11.1 Introduction

Turbidity currents are bottom currents of water laden with h Uh ∫ udz (2-246) suspended sediment that move downslope in otherwise still 0 bodies of water (García 1992). Turbidity currents are very similar to river fl ows, except for one important difference. where In the case of a river, gravity pulls the water downslope, and z upward normal coordinate with its origin at the bed; the water drags the sediment with it. In the case of turbid- and ity current in a body of still water, there would be no fl ow _ u local streamwise fl ow velocity averaged over turbu- if there were no suspended sediment. Gravity acts to pull lence at z . the suspended sediment in the bottom water downslope, and the sediment then drags the water with it. Turbidity currents Under appropriate simplifying assumptions, the one- occur in lakes, reservoirs, and the ocean. They provide major dimensional St. Venant shallow-water equations take the form mechanisms for moving sediment into deep water. Knowledge about sediment transport by turbidity cur- ∂h ∂Uh rents was limited when ASCE Manual 54 was fi rst pub- 0 (2-247a) ∂∂t x lished. Manual 54 cited frequent observations of turbidity currents in Lake Mead induced by the high sediment loads of the , suggesting that these fl ows could ∂Uh ∂Uh2 ∂h gh ghS C U 2 (2-247b) play an important role in lake and reservoir sedimentation ∂t ∂x ∂x f (Vanoni 2006, p.166). In China, the impact of turbidity cur- rents on reservoir sedimentation has long been recognized where and attempts have been made to use the transport capacity of these underfl ows to preserve storage capacity (Fan and t time; Morris 1992a, 1992b). One of the fi rst attempts at model- x boundary-attached (bed-attached) streamwise ing turbidity in reservoirs with a simple one-dimensional coordinate;

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 112929 11/21/2008/21/2008 5:44:595:44:59 PMPM 130 sediment transport and morphodynamics g gravitational acceleration; (2) dimensionless rate of entrainment of bed sediment into ρ water density; suspension Es , and (3) near-bed suspended-sediment con- _ S bed slope, given as centration c b. Numerous relations for q b and E s have been introduced in this chapter. In addition, if the fl ow and sus- ∂η pension do not deviate too strongly from equilibrium (i.e., S (2-248a) logarithmic and Rousean profi les), the following approxi- ∂x _ mate relation for cb is obtained from Eqs. (2-195), (2-204), and (2-249): and C f bed friction coeffi cient such that the bed shear stress τ crCbo (2-252a) b is given as

⎛ ⎞ τρ 2 h bfCU (2-248b) ln⎜11 ⎟ ⎝ k ⎠ c ro (2-252b) ZR 1 ⎡ ()1ζζ/ ⎤ ⎛ h ⎞ Here Eq. (2-247a) describes water-mass conservation and ∫ ⎢ ⎥ ln 30 ζζd ζ ζζ/ ⎝⎜ ⎠⎟ Eq. (2-247b) describes water-momentum conservation. b ⎣()1 bb⎦ kc The river is assumed to carry a dilute suspension of sedi- ment. Let the depth-fl ux averaged volume concentration C v s of suspended sediment be given by the relation ZR (2-252c) κu∗

h qUChucdz∫ (2-249) s 0 z ζ (2-252d) _ where c 1 is the local volume suspended-sediment con- h centration averaged over turbulence at elevation z , and q s where ζ 1 is a near-bed parameter. When ξ b /h denotes the volume transport rate per unit width of sus- b b 0.05, which is the commonly used value for this parameter, pended sediment. An approximate form for depth- averaged the shape factor r can be estimated with a simple fi t to the conservation of suspended sediment is as follows: o Rousean profi le (Parker et al. 1987)

∂Ch ∂UCh (2-250) 1.46 vEss() c b ⎛ u ⎞ ∂t ∂x r 1 31.5 * (2-252e) o ⎝⎜ ⎠⎟ vs where Thus, Eqs. (2-250) and (2-251) take the respective forms vs sediment fall velocity; Es dimensionless rate of entrainment of bed sediment into suspension; and ∂Ch ∂UCh _ _ vE() rC (2-253) ∂ ∂ ss o cb near-bed value of c , t x all of which were introduced earlier in this chapter. The Exner equation of conservation of bed sediment presented ∂η ∂q earlier takes the form (1λ )b vrC ( E ) (2-254) p ∂t ∂x so s

∂η ∂q λ b (2-251) Alternatively, between Eqs. (2-249), (2-250), and (2-251) (1p )vcsb ( E s ) ∂t ∂x it is found that

where q denotes the volume bedload transport rate per unit b ∂η ∂Ch ∂q ∂q width and λ denotes the bed porosity. (1λ ) bs (2-255a) p p ∂ ∂ ∂ ∂ Equations (2-247a), (2-247b), (2-250), and (2-251) are t t x x the basic equations describing the one-dimensional mor- phodynamics of rivers. The equations need to be closed by For the case of a dilute suspension (C 1), the amount of

specifying relations for (1) volume bed load transport q b , sediment stored in suspension can be neglected compared to

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 113030 11/21/2008/21/2008 5:44:595:44:59 PMPM morphodynamics of rivers and turbidity currents 131 the amount of sediment stored in the bed, so that the above equation accurately approximates to

∂η ∂q ∂q ∂q (1λ ) bs t (2-255b) p ∂t ∂x ∂x ∂x

Where qt denotes the total volume bed-material load per unit width. The formulation of Eq. (2-255b) can be imple- mented by assuming an appropriate relation for the volume

bed-load transport rate per unit width q b , and estimating the volume suspended-load transport rate per unit width q s for disequilibrium fl ow by the means of the local application of a formulation for equilibrium fl ow (based on, e.g., a logarith- mic velocity profi le and the Rousean profi le for suspended sediment). In point of fact the forms for Eqs. (2-247) and (2-251) are only approximate. The more general forms of Eq. (2-247) and (2-251) are, respectively, Fig. 2-59. Defi nition diagram for turbidity current.

∂Uh ∂α Uh2 ∂h 1 gh ghS C U 2 (2-256a) ∂t ∂x ∂x f

z and denote an upward vertical coordinate from the bed._ where u Just as in the case of rivers, the streamwise fl ow velocity_ and volume concentration of suspended sediment c , both 2 1⎛ u ⎞ averaged over turbulence, show notable variation in the αζ ∫ ⎜ ⎟ d (2-256b) 1 0 ⎝ ⎠ z U direction. Note, however, that because a turbidity_ cur- rent is a bottom fl ow, it can be expected that both u and _ → ∞ ∞ and c become vanishingly small as z , where here “ ” is shorthand for “high above the bottom current.” The turbidity current fl ows because it is laden with sedi- ∂α Ch ∂UCh 2 vE() rC (2-257a) ment, which renders it heavier than the ambient water above. ∂t ∂x ss o ρ ρ Let denote the density of water and s denote the density of ρ sediment. The density t (z) of the turbidity current at level where z is thus given as

1 1 ()()∫∫udζζ cd ρρ ρ ρ 1 c 0 0 ts(1cc ) (1 Rc ) (2-255) αζ∫ d (2-257b) 2 0 1 C ()∫ ucdζ 0 where where ζ is given by Eq. (2-252d). Here α and α are dimen- 1 2 ρ sionless shape factors governing_ _ the velocity and con- R s 1 (2-256) ρ centration profi les. Were u and c constant in the vertical, both shape factors would be equal to unity (i.e., top-hat approximation). They are often approximated to unity for denotes the “submerged specifi c gravity” of the sediment, simplicity. taking the value 1.65 for quartz. The turbidity current is driven downslope due to gravity acting on the density excess of the turbidity current relative to the ambient water. The 2.11.3 Equations Governing the Morphodynamics fractional density excess is given as of Turbidity Currents ρρ A turbidity current is illustrated in Fig. 2-59. Let x denote t Rc (2-257) a boundary-attached (bed-attached) streamwise coordinate ρ

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 113131 11/21/2008/21/2008 5:44:595:44:59 PMPM 132 sediment transport and morphodynamics The one-dimensional layer-averaged equation of conser- Here_ the case of dilute_ turbidity currents is considered, so that c 1 and thus R c 1 . vation of streamwise fl ow momentum is One-dimensional layer-averaged equations governing the fl ow of turbidity currents can be derived in analogy to ∂Uh ∂Uh221 ∂Ch the St. Venant equations of shallow water fl ow in rivers. g gRChS C U 2 (2-261) ∂t ∂x 2 ∂x f The suspended sediment that drives the turbidity current is

characterized here in terms of a single fall velocity v s . Layer thickness h and layer-averaged fl ow velocity U and volume The one-dimensional layer-averaged equations for con- concentration of suspended sediment C are defi ned in terms servation of suspended and bed sediment are given by the of the following three moments (e.g., Ellison and Turner same forms as for a river, that is, Eqs. (2-250) and (2-251), 1959): respectively. The governing equations of turbidity currents are thus seen ∞ to be very similar to those governing fl ow and sediment trans- Uh ∫ udz (2-258a) 0 port in a river. There is, however, one important difference. A river that carries no suspended sediment continues to fl ow, because the pull of gravity acts directly on the water. This effect ∞ Uh22 ∫ udz (2-258b) is embodied in the term ghS in Eq. (2-247). So gravity pulls the 0 water downhill, and the water pulls the sediment with it. In the case of a turbidity current, the corresponding downslope ∞ impelling term in Eq. (2-261) is gRChS . If the concentration of UCh ∫ u cdz (2-258c) 0 suspended sediment C drops to zero, the downslope impelling force also drops to zero, and the current must eventually die. In Note that Eqs. (2-258a) and (2-258c) for turbidity currents a turbidity current, then, gravity pulls the sediment downslope, are almost identical in form to the corresponding Eqs. (2-245) and the sediment drags the water with it. In a river, the dynam- and (2-249) used to defi ne the layer-averaged (more precisely ics of the fl ow is essentially decoupled from the dynamics of depth-averaged) quantities U and C in a river. The only dif- suspended sediment. In a turbidity current, the two are inti- ference is that the upper limit y h in a river is replaced mately linked (García 1992). with y ∞ in a turbidity current. That is, turbidity currents Equations (2-260) and (2-261), along with Eqs. (2-250) have no “layer thickness” in the precise sense of the word. and (2-251), can be closed with functional relations for

The extra relation of Eq. (2-258b) specifi es a momentum- Es , qb and r o that are similar to those used for rivers. In addi- based equivalent layer thickness h . tion, however, an equation for the entrainment coeffi cient of

A complete derivation of the one-dimensional fl ow equa- ambient water ew must be specifi ed. Parker et al. (1987) sug- tions for turbidity currents are given in Parker et al. (1986). gest the following functional form The one-dimensional equations contain a number of dimen- sionless shape factors analogous to α and α of Eqs. (2-256b) 1 2 0.075 and (2-257b), all of which take the value unity for the special ew (2-262) 2.4 0.5 case of the velocity and concentration profi les ()1 718 Ri

u c ⎧10; zh where Ri denotes the bulk Richardson number, given as ⎨ (2-259) U C ⎩0 ; zh RgCh Ri 2 (2-263) Parker et al. (1987) and García (1993, 1994) have evalu- U ated these shape factors for experimental turbidity currents. Because the values so determined do not deviate strongly The bulk Richardson number is related to the densimetric Fr from unity, values of unity are assumed below as well. Froude number d as The one-dimensional layer-averaged equation of conser- vation of water mass takes the form U 1/2 Frd Ri (2-264) gRCh ∂ ∂ h Uh eUw (2-260) ∂t ∂x A turbidity current is supercritical if Fr d 1 ( Ri 1) and subcritical if Fr d 1 ( Ri 1). (In fact the precise borderline where between subcritical and supercritical fl ow depends upon the ew coeffi cient of entrainment of ambient (sediment- shape factors discussed above, but a value of unity is generally free) water from above into the turbidity current. an acceptable approximation.) Supercritical turbidity currents

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 113232 11/21/2008/21/2008 5:45:015:45:01 PMPM morphodynamics of lake and reservoir sedimentation 133 tend to entrain ambient water from above, in analogy to the tendency of supercritical open channel fl ows to entrain air on spillways. In accordance with Eq. (2-262), highly subcritical turbidity currents tend to entrain little ambient water (García 1994). More material on water entrainment by underfl ows, including the role played by bottom slope and roughness, can be found in Fernandez and Imberger (2006).

2.12 MORPHODYNAMICS OF LAKE AND RESERVOIR SEDIMENTATION

The following material is intended to provide a brief sum- Fig. 2-60. Defi nition diagram for a fl uvial delta. mary of the morphodynamics of sedimentation in lakes and reservoirs. More detail on reservoir sedimentation is given in Chapter 12. These characteristic features are seen for the case of Lake Mead in Fig. 2-61 (Grover and Howard 1938; Smith et al. 2.12.1 Introduction: Topset, Foreset, and Bottomset 1954). The fi gure shows the history of reservoir sedimenta- of Delta tion from the closing of the in 1936 to 1948. The top- As a river enters the slack water of a lake or reservoir, the set, foreset, and bottomset deposits are clearly visible in the fl ow decelerates and its sediment drops out. The result is fi gure. It is of interest to note that the maximum slope of the the formation of a delta (Fig. 2-60). The coarser sediment foreset is less than 1°. An inset in Fig. 2-61 is expanded in (typically sand and/or gravel) deposits fl uvially to form an Fig. 2-62. This inset shows that the topset and foreset are aggrading topset and deposits by avalanching to form a pro- predominantly composed of sand, and the bottomset is pre- grading foreset. The fi ner sediment (typically mud, i.e., silt dominantly composed of mud (silt and clay). The foreset- and clay) deposits in deeper water to form a bottomset (e.g., bottomset interface clearly defi nes a moving boundary. This Morris and Fan 1997). feature is exploited in the modeling described here.

Fig. 2-61. Formation of fl uvial delta in Lake Mead, USA. Adapted from Smith et al. (1954).

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(a) Fig. 2-62. Inset of Fig. 2-61 showing the moving boundary between the muddy bottomset and the sandy topset-foreset.

The bottomset can be emplaced in three ways. If the sedi- ment-laden river water is lighter than the lake water, a muddy surface plume extends out into the lake from the delta front. Mud gradually rains out of this plume and deposits to form the bottomset, as shown in Fig. 2-63a. Lakes are often strati- fi ed, however, so that the is warmer and lighter than the water at a depth. If the sediment-laden river water has a density that is intermediate between that of the sur- (b) face and bottom water in the lake, the muddy plume partially plunges to form an interfl ow, as shown in Fig. 2-63b. During fl oods, when rivers are likely to carry their highest concen- trations of mud as wash load, the river water is often suf- fi ciently dense to plunge to the bottom of the reservoir and form a turbidity current, as shown in Fig. 2-63c (Bell 1942; Fan and Morris 1992a).

2.12.2 Fluvial Deposition of Topset and Foreset A simple one-dimensional morphodynamic model of delta topset and foreset evolution is fi rst considered. A river of constant width fl ows into a lake of the same width and infi - (c) ξ nite streamwise extent. The water surface elevation o in the lake is held constant. The river transports sand as bed-mate- Fig. 2-63. Illustration of the formation of a muddy surface plume rial load; this sand is approximated with a single grain size (a), a muddy interfl ow (b), and a muddy bottom turbidity current (c).

Ds . The of the river is approximated in the fol- lowing way; for a constant fraction I of the year the river is q2 f Fr2 w in fl ood, carrying a constant fl oodwater discharge per unit 3 (2-265b) ghf width q w. Otherwise, the river is assumed to be morphody- namically inactive. During fl oods sand enters the river at x 0 at volume rate per unit width q , where the subscript t In this backwater equation, tf h and η fl ow depth and bed elevation in a fl uvial zone denotes total bed material load and f denotes fl ood. The f f λ that includes the topset and foreset regions porosity of the sandy bed deposit is denoted as ps , where s denotes sand. Since the fl ood fl ow is assumed to be steady, but excludes the bottomset; Eqs. (2-246) and (2-247) can be reduced to the backwater Fr Froude number of the fl uvial fl ow; and C bed friction coeffi cient in the fl uvial region. equation ff The boundary condition on this equation is expressed in terms of the fl ow depth at a point x s where standing ∂η stand f 2 water elevation ξ is maintained. Let η (x , t ) denote the bed ∂h CFrff o f f ∂x (2-265a) elevation at any point x and time t. The boundary condition ∂x 1Fr2 is then

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 113434 11/21/2008/21/2008 5:45:035:45:03 PMPM morphodynamics of lake and reservoir sedimentation 135 The bed evolution or morphodynamics driven by the back- hxt(),,ξη () xt (2-265c) f of xssdtanxs sd tan water of the lake is computed from the Exner equation of sedi- ment continuity. Modifying Eq. (2-255b) for the fl uvial region The fl ow is assumed to be everywhere subcritical (Fr 1), so as to account for the fact that the river is morphologically so that Eq. (2-265a) is integrated in the upstream (negative x ) active only I fraction of the time, it is seen that f direction from x s stand . The solution of Eq. (2-265a) subject to Eq. (2-265c) constitutes the standard backwater formula- ∂η ∂q λ f t tion; in a consideration of reservoir sedimentation the back- (1ps ) I f (2-268) ∂t ∂x water curve of interest is the M1 curve (Chow 1959). The Shields number τ * of the fl uvial fl ow is given as The boundary condition on Eq. (2-268) is a specifi ed feed rate of sand, here taken to be constant; τ Cq2 τ∗ b fw (2-266) ρ 2 RgDss RgDhssf qq t x0 tf (2-269) where R denotes the submerged specifi c gravity for the sand. s The initial condition on the problem is here simplifi ed to In all following calculations, the total volume bed material a bed with constant slope S and a bed elevation η 0 at load per unit width of the river q is estimated using the rela- base f t x s . tion of Engelund and Hansen (1967) introduced earlier in stand The problem defi ned by Eqs. (2-265) to (2-269) has been the chapter, solved numerically by many authors, e.g., Hotchkiss and Parker (1991). Here techniques outlined in Parker (2005) are q ∗∗t 0.05 τ 52/ used to solve the problem. A predictor-corrector method is qt () (2-267) RgDDsss C ff used to solve the backwater equation over the bed at any given time, and the Euler step method is used to compute the

Here the friction coeffi cient C ff is assumed to be a specifi ed bed elevation at a later time from the above Exner formula- constant for simplicity. At any given time t , then, the total vol- tion. The input parameters are given as those of Case A of ume bed material transport rate per unit width of sand q (x ,t ) can t Table 2-8. The size D s of the sand is seen there to be 0.4 mm, be computed by (1) solving Eq. (2-265a) subject to Eq. (2-265c) or 400 μm. η to determine hf (x ,t ) over a known bed f (x ,t ), and then The results of a computer simulation of Case A for a simu- (2) using Eqs. (2-266) and (2-267) to determine q (x ,t ). t lated run time trun of 30 years are shown in Fig. 2-64. The

Fig. 2-64. Simulation of formation of sandy topset and foreset in a lake using a shock-capturing formulation and the input of Case A of Table 2.7.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 113535 11/21/2008/21/2008 5:45:055:45:05 PMPM 136 sediment transport and morphodynamics Table 2-8 Parameters Used in Numerical and the corresponding distance to the foreset-bottomset break

Modeling of Lake/Reservoir Sedimentation is denoted as s b (t). Both these parameters defi ne boundaries that move as the delta evolves. Likewise, the bed elevation at Parameter Units Case A Case B Case C Case D Case E η the topset-foreset break is defi ned as s (t ), and the bed eleva- q m2/s 6 6 6 6 2 η w tion at the foreset-basement break is defi ned as b (t ). 2 qtf m /s 0.001 0.001 0.001 0.001 0.000625 In a shock-fi tting formulation, delta is D μm 400 400 400 400 400 described in terms of a shock condition determined by inte- s grating the Exner equation (2-268) from the topset-foreset C 1 0.0025 0.0025 0.0025 0.0025 0.00694 ff break to the foreset-bottomset break, under the condition of I 1 0.1 0.1 0.1 0.1 1 f constant foreset slope S a . Kostic and Parker (2003a) gen-

Sbase 1 0.000025 0.000025 0.005 0.005 0.016 eralized the work of Swenson et al. (2000) to obtain the condition Rs 1 1.65 1.65 1.65 1.65 1.65 λ ps 1 0.4 0.4 0.4 0.4 0.4 ξ ⎡ Ι q ∂η ⎤ o m 10 10 53 53 203 1 f t xs s ⎢ s f ⎥ (2-270) s m 50,000 s ⎢ λ ∂ ⎥ stand (SSas ) (1ps )(ss b s ) t ⎣ xss ⎦ trun yrs 30 600 45 3000 0.246 η m 6.304 50 50 200 . sI In this relation s denotes the speed of migration of the η s bI m 0.931 0 0 100 topset-foreset break, and S 1 0.05 0.05 0.05 0.2 a ∂η f SfI 1 0.00023 0.00023 0.00028 0.00073 S (2-271) s ∂x xs ssI m 13,000 10,000 10,000 1000 s 2 qmf m s 0.003 0.003 0.00625 i.e., the slope of the topset at the topset-foreset breaks. μ Dm m 10, 15, 25 10 50 Eq. (2-270) indicates that the foreset progrades at a speed that C 1 0.00111 0.00111 0.00111 is commensurate with the rate of delivery of bed- material ft load (sand in the simulations reported here) to the topset- R 1 1.65 1.65 1.65 m foreset break. λ m 1 0.6 0.6 0.6 The shock condition of Eq. (2-270) is augmented with an elevation continuity condition at the foreset-basement break, smax m 30,000 30,000 6000 . S B m 80 which reduces to the following condition; where b denotes c the speed of migration of the foreset-basement break, θ f ˚ 90 θ ˚ 10 ∂η t ()()SSsSSsf (2-272) abbass∂ t method is fundamentally shock-capturing (the delta front xss being the shock); a delta forms of its own accord, steepens, In the above relation Sb denotes the slope of the subaque- and progrades downstream. The topset and foreset are clearly ous basement at the foreset-basement break. visible in Fig. 2-64. There is no bottomset because mud has not been included in the formulation. Although this formu- lation clearly captures the mechanism of formation and pro- gradation of the delta topset and foreset, is does not correctly describe foreset slope. In a shock-capturing method, the slope of the foreset is entirely dependent on the spatial step length. Once the delta has formed, it becomes possible to change the formulation to a shock-fi tting formulation, in which the

foreset has a prescribed slope of avalanching S a that can be selected based on physical considerations. Swenson et al. (2000) have developed such a formulation. The delta is defi ned in terms of a topset-foreset break and a foreset-basement break, as shown in Fig. 2-65. The foreset progrades over a subaque- ous basement of prescribed morphology. The distance from the sediment feed point at x 0 to the topset-foreset break is Fig. 2-65. Defi nition diagram for a moving-boundary shock- fi tting formlation of delta topset-foreset evolution. denoted as s s (t ) (the subscript s being shorthand for shoreline),

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 113636 11/21/2008/21/2008 5:45:065:45:06 PMPM morphodynamics of lake and reservoir sedimentation 137 ⎡ ⎤ The problem is conveniently solved numerically in terms 1 Ι qt(1, ) ∂η s ⎢ f t f f ⎥ (2-276) of moving boundary coordinates, according to which s λ  S ⎢(ss )(1 ) ∂t f ⎥ a ⎣ bs ps xf 1 ⎦ x xˆ f (2-273a) and the continuity condition at the foreset-basement break. sts () ∂η 1 f ˆtt (2-273b) ss f bsSt∂ˆ (2-277) a f xˆ f 1 That is, Eqs. (2-265) and (2-268) are transformed accord- ing to Eqs. (2-273a) and (2-273b) and solved in conjunction Details are given in Kostic and Parker (2003a; 2003b). with Eqs. (2-270) and (2-272). The transformed problem is The calculation is implemented here as Case B, the input stated here in terms of the backwater formulation, parameters for which are given in Table 2-8. Case B is chosen to be very close to Case A. Case B requires ini- ∂η 2 1 f qw η η η η C tial values ssI , sI and bI for s s , s and, b , respectively, as ∂ sx∂ˆ ff 3 1 hf s f ghf well as an initial fl uvial bed slope S fI . These values have (2-274a) s ∂xˆ q2 been estimated from the output of Case A at t 15 years. s f w 1 3 Basement slope S base is the same in Case B as that of Case ghf A; because it is constant, S b in Eq. (2-272) is always equal to S . The foreset avalanche slope S has been set equal htξη(1,ˆ ) (2-274b) base a f of to 0.05, i.e., 2.86°. xˆ f 1 The results of 600 years of simulation of Case B are the Exner equation of sediment continuity, shown in Fig. 2-66. Progradation forces the evolution of a topset with an upward-concave long profi le, i.e., one for ⎡⎛ ∂ηη∂ ⎞ ⎤ which slope declines in the streamwise direction. After 600 f s f 1 ∂q (1λ ) ⎢⎜ s xˆ ⎟ ⎥ Ι t (2-275) years the delta front has prograded well over 150 km. It ps ⎢⎝ ∂ˆt s ∂xsˆˆ⎠ ⎥ f ∂x ⎣ f s fs⎦ f should be noted that the progradation predicted by a one- dimensional model is considerably exaggerated as compared the shock condition for foreset progradation, to nature. This is because in a one-dimensional model the

Fig. 2.66. Simulation of progradation of sandy topset and foreset in a lake using a shock-fi tting formulation and the input of Case B of Table 2-7.

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only place for the topset sediment to deposit is in the chan- submerged specifi c gravity of the mud is denoted as R m . As nel, whereas in nature sediment deposits on the surface of a the river fl ow plunges, it invariably draws into it some ambi- much wider fan-delta over which the river channel avulses. ent water from the lake. The velocity at which this ambient

The two-dimensional case is considered below. water enters the muddy fl ow is denoted as U a . The coeffi cient of mixing of ambient water into the muddy fl ow, γ , is defi ned as 2.12.3 Plunging of a Muddy Turbidity Current Up to now the muddy bottomset has been excluded from the Uhap() h d formulation. Sand- and gravel-bed rivers often carry copious γ (2-278) Uh amounts of mud (sand and silt) as wash load, i.e., material pp that is carried in suspension but constitutes only a negligible fraction of the bed sediment. An example is the Colorado A value γ 0 is required for the muddy fl ow to plunge. River, USA (Smith et al. 1954). Because the mud does not The fl ow discharge per unit width just downstream of plung- deposit in the bed of the river, it may be neglected in a fi rst ing is related to that just upstream of plunging as model of the evolution of the topset and foreset. As seen in Figs. 2-63, however, when river meets the γγ standing water of a lake or reservoir, the sand is left behind Uhdd Uh p p()11 q w () (2-279) on the topset-foreset and the remaining muddy water con- tinues as a surface plume, interfl ow, or bottom turbidity The depth ratio ϕ and the densimetric Froude numbers current. The case considered here is the one for which the Frdp and Fr dd just upstream and downstream of the plunge muddy water is suffi ciently dense to plunge and form a bot- point are defi ned as tom turbidity current. When muddy river water is denser than lake water at every h level of the lake, the river water plunges somewhere above ϕ d (2-280a) the foreset to create the bottom turbidity current schematized hp in Fig. 2-67. A number of relations are available to predict plunging of muddy water in a lake; these are reviewed in U 2 q2 Parker and Toniolo (2007). Here the treatment of Parker and Fr2 p w dp 3 (2-280b) Toniolo (2007) is offered as a sample. RCmmpp gh RCmmpp gh The analysis is applied to muddy water fl owing into a lake with no ambient stratifi cation. The fl ow near the plunge U 2 point is illustrated in Fig. 2-68. Except for the mud, the river 2 d Frdd (2-280c) water is assumed to have the same density as the sediment- RCmmdd gh free lake water. The fl ow velocity, depth, and volume mud concentration in the river water just upstream of the plunge Parker and Toniolo (2007) obtain the following relations

point are denoted as U p , hp and C mp, respectively; the cor- for these parameters: responding fl ow velocity, layer thickness and volume mud ()1γ 3 concentration in the turbidity current just downstream of the Fr22 Fr dd dp ϕ3 (2-281) plunge point are denoted as U d , hd , and C md , respectively. The

Fig. 2-68. Detailed illustration of fl ow near the plunge point with Fig. 2-67. Defi nition diagram illustrating plunging. defi nitions.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 113838 11/21/2008/21/2008 5:45:095:45:09 PMPM morphodynamics of lake and reservoir sedimentation 139 Under these assumptions, Eqs. (2-253), (2-260) and 2 1 ϕ 3 Frdp ()1 (2-282) (2-261) can be reduced to the forms 2γ 2

ϕ 3 ϕ2 1 3 11() 22 ∂η ⎛ 1 ⎞ 1 v ()1ϕ ()()111γϕ 0 Ri t eRiCRir⎜2 ⎟ sm γ 2 γ 2 ϕ ()1γ ∂h ∂x wfto⎝ 2 ⎠ 2 U (2-285a) t t (2-283) dx 1 Ri

Once the mixing coeffi cient γ is specifi ed, these equa- ∂η ⎛ 1 ⎞ 1 v tions allow the determination of the plunge point and the Ri t eRiCRir⎜1 ⎟ sm ∂ ∂ wfto⎝ ⎠ Ut x 2 2 UttU fl ow below it. Let q mf denote the volume feed rate of mud per unit width. Because the mud does not settle out upstream ∂x 1Ri h t of the plunge point, the volume concentration of mud, C , mp (2-285b) just upstream of the plunge point (and, indeed, everywhere upstream of the plunge point) is given from the relation ∂q v q m r sm m (2-285c) ∂x o U h qqCmf w mp (2-284) t t

ϕ / Where The ratio h d hp is obtained from Eq. (2-283). Once η elevation of the bed below the turbidity current; this value is known, h p is computed from Eqs. (2-282) and t h layer thickness of the turbidity current; (2-280b) and h d is further computed from this value and the t ϕ U layer-averaged velocity of the turbidity current; previously computed value of . The value of U d is then t and computed from Eq. (2-279), and the value of C md is com- q volume discharge of mud per unit width of the tur- puted from Ud , hd , and Eqs. (2-280c) and (2-281). m Plunging occurs at the point of the foreset where the fl u- bidity current, related to the layer-fl ux averaged mud concentration C as vial depth hf becomes equal to the value hp predicted from t the above relations. It can be verifi ed from these relations that a necessary condition for plunging is a mixing coeffi - qUhCmttt (2-286) cient γ 0.14. Under such conditions, the river fl ow just upstream of the plunge point is subcritical in the sense of the In addition, v denotes the fall velocity of the mud and C densimetric Froude number, i.e., Fr 1, and the turbidity sm ft dp denotes the coeffi cient of bed resistance of the turbidity cur- current just downstream of the plunge point is supercritical, rent, here approximated as constant. Finally, the Richardson i.e., Fr 1. dd number Ri is given from Eqs. (2-263) and (2-286) as

RgCh Rgq 2.12.4 Linked One-Dimensional Model of Topset, Ri mtt mm 23 (2-287) Foreset and Bottomset Evolution Ut Ut As noted above, Eqs. (2-260), (2-261), (2-253), and (2-254) describe, respectively, fl ow mass balance, momentum bal- The assumption of supercritical fl ows implies that Ri ance, suspended-sediment mass balance, and conservation 1 everywhere. This assumption allows Eqs. (2-285a), of bed sediment for the case of a turbidity current. Here the (2-285b), and (2-285c) to be integrated downstream from the following assumptions are made to simplify the problem. plunge point. Let x sp denote the position of the plunge point. The boundary conditions on (2-285a), (2-285b), and • The turbidity current is assumed to be steady and driven (2-285c) are then by the steady infl ow from the river. • The turbidity current is assumed to be fl owing over a hht d (2-288a) bed that is suffi ciently steep to render it everywhere su- xsp percritical in the densimetric sense. • The turbidity current is purely depositional, so the term E UUt d s in Eqs. (2-253) and (2-254) can be neglected. xsp (2-288b) • The turbidity current is allowed to run out infi nitely in the streamwise direction. • Bed load transport is neglected. qUhCm dd md (2-288c) xsp

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where h d , Ud and Cmd are obtained from the calculation of Sample calculations of the coevolution of the sandy top- plunging. set-foreset and muddy bottomset of a one-dimensional delta Once the fl ow over the bed is solved, the morphodynamic are presented as Case C, which has three subcases Ca, Cb, μ evolution of the lake bed is solved from Eq. (2-251) subject and Cc, corresponding to mud size D m 10 , 15, and 25 m, to these assumptions, which reduce it to the form respectively. The relevant modeling parameters are given

in Table 2-8; for these cases S base corresponds to the slope of the antecedent subaqueous bed over which the turbidity ∂η v q ()1λ t Ir sm m (2-289) current deposits mud. Fall velocities v have been com- pm ∂ fo sm t Ut ht puted using the values of R m and D m listed in Table 2-8, the kinematic viscosity ν of water at 20°C and the relation of λ where pm denotes the porosity of the mud deposit. The solu- Dietrich (1982) presented in Eq. (2-47a). Water entrainment tion of Eqs. (2-285a), (2-285b), and (2-285c) subject to Eqs. has been computed using Eq. (2-262). The coeffi cient r o has (2-288a), (2-288b), and (2-288c) for the fl ow over the bed, been set equal to 2 (García 1994). The plunging formulation and Eq. (2-289) for the bed evolution describes the evolu- used in the computation is a highly simplifi ed version that tion of a bottomset emplaced by a purely depositional one- is less rigorous than the one presented in the previous sec- dimensional muddy turbidity current. tion. It serves, however, to illustrate the morphodynamics of Kostic and Parker (2003a, 2003b) have devised a moving- coevolution. In the calculation the turbidity current runs a boundary formulation that dynamically links the moving- short distance down the foreset from the plunge point before boundary model of the topset-foreset evolution described debouching onto the bottomset. It is assumed that the foreset in Section 2.12.2 with the model of the bottomset evolution is too steep to allow for the deposition of mud. Mud deposi- described in this section. All the equations describing top- tion commences at the foreset-bottomset break. set-foreset evolution remain the same except for Eq. (2-272) The results of the calculations for Cases Ca, Cb and Cc are describing the evolution of the foreset-basement break. In presented in Fig. (2-69). In Case Ca with 10-μm material, the a linked model the foreset-basement break is replaced with bottomset is quite thin due to the small fall velocity of the mud. a foreset-bottomset break located at x s b , as described in (The bottomset extends far beyond the point smax 30,000 m Fig. 2-66. The modifi ed relation is in Cases Ca, Cb, and Cc.) Progradation of the foreset simply pushes the bottomset lakeward. In Case Cb with 15-μm mate- ∂η ∂η rial the foreset and bottomset are seen to interact: progradation ()()SSsSSsf t abbass∂ ∂ (2-290) of the foreset pushes the bottomset lakeward, and tts ss s of the bottomset reduces the elevation drop of the foreset. The interaction is seen to be even stronger for the 25-μm mate-

where S b is now given as the slope of the bottomset at the rial of Case Cc. Note that as the mud becomes coarser it falls foreset-bottomset break: out more rapidly from the turbidity current, so that deposition on the bottomset becomes more proximal to the foreset. ∂η Figure 2-69(c) in particular bears a remarkable resem- t Sb (2-291) blance to the delta of Lake Mead shown in Fig. 2-61. In both ∂x xsb images the interface between the sand and mud (foreset- bottomset break) is fi rst seen to move upward and lakeward, The moving boundary formulation for topset/foreset evolu- and then to move downward and lakeward. tion remains the same as that of Section 2.12.2, except for The restriction to supercritical fl ow means that the back- Eq. (2-277), which takes the amended form water formulation of Eqs. (2-285a), (2-285b), and (2-285c) for the turbidity (actually a frontwater formulation, because ⎛ ⎞ supercritical fl ow dictates integration in the downstream direc- 1 ∂η ∂η ss⎜ f t ⎟ (2-292) tion) can only be implemented on relatively high subaqueous bsSt⎜ ∂ˆˆ∂ t ⎟ a f t xˆ 0 bed slopes. On lower slopes the turbidity current undergoes a ⎝ xˆ f 1 t ⎠ hydraulic jump to subcritical fl ow (García 1993). The prob- where the moving boundary coordinates for the bottomset lem can still be solved, but steady forms of Eqs. (2-285a), region are given as (2-285b), and (2-285c) must be replaced by their correspond- ing unsteady forms, even when the incoming fl ow is constant. The unsteady forms of the equations allow automatic shock xs ˆtt sˆ b (2-293) capturing of any hydraulic jump. Kostic and Parker (2003a, tt ssmax b 2003b) have implemented such calculations using a moving- boundary formulation that adds an extra moving boundary. In addition to the moving boundaries of the topset-foreset and x s max denotes the downstream limit of the range on which the calculation is to be performed. and foreset-bottomset breaks, the model also captures the

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 114040 11/21/2008/21/2008 5:45:115:45:11 PMPM morphodynamics of lake and reservoir sedimentation 141 migration of the front of the turbidity current until such time deposited across the entire fan by gradual channel shift and θ as it exits the calculational domain. Tracking the front prop- . Here an axially symmetric fan with angle f is con- agation constitutes a challenging problem (e.g., Choi and sidered, as illustrated in Fig. 2-71. A radial coordinate r is García 1995, 2001). Finally, a quasi-steady state turbidity defi ned from the apex of the fan. The equations governing current is achieved in a relatively short amount of time when fl ow in the channel remain unchanged from the one-dimen- the infl ow conditions are steady. sional formulation (except for the transformation from x to radial coordinate r ). The radial width B is given as 2.12.5 Linked Quasi-Two-Dimensional Model f of Topset, Foreset and Bottomset Evolution θ Brff (2-294) Most real deltas at the upstream end of lakes are fan- deltas. That is, they spread out in the transverse direction. Fig. 2-70 shows an example of such a fan-delta: the Eau Claire River The appropriate form of the Exner equation of sediment fl owing into Lake Altoona, a reservoir in Wisconsin. As the conservation for the topset and foreset is river channel on a fan-delta aggrades, it migrates and avulses to fi ll the available surface of the fan-delta as it progrades. That is, it fi lls the area of the fan rather than the area of the channel ∂η ∂q ()1 λ B f IB t (2-295a) in Fig. 2-71. Since the river fi lls a much wider area than just ps f ∂t fc∂r the bed of the channel itself, the rate of delta progradation is greatly slowed compared to the one-dimensional case. The two-dimensional morphodynamics of a fan-delta or rearranging, can be approximated using a one-dimensional model in the following way. The river on the fan is assumed to have an ∂η ∂ B f Bc qt effective constant width c for transporting sediment. In the ()1λ I (2-295b) ps ∂ f ∂ long-term average, however, the sediment is assumed to be t Bf r

Fig. 2-69. Simulation of coevolution of a sandy topset-foreset and a muddy bottomset foreset in a lake using a shock-fi tting formulation and the input of Cases Ca(a),Cb(b), and Cc(c) of Table 2-8. The mud has a size of (a)10 μm, (b) 15 μm, and (c) 25 μm.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 114141 11/21/2008/21/2008 5:45:115:45:11 PMPM 142 sediment transport and morphodynamics

(b)

(c)

Fig. 2-69. Simulation of coevolution of a sandy topset-foreset and a muddy bottomset foreset in a lake using a shock-fi tting formulation and the input of Cases Ca(a),Cb(b), and Cc(c) of Table 2-8. The mud has a size of (a)10 μm, (b) 15 μm, and (c) 25 μm. (Continued)

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 114242 11/21/2008/21/2008 5:45:125:45:12 PMPM morphodynamics of lake and reservoir sedimentation 143 / As the fan progrades the ratio B c Bf must become small where in accordance with Eq. (2-294). The result is a greatly sup- pressed rate of delta progradation and aggradation as com- ∂η pared to a purely one-dimensional formulation in which all S f (2-297) the topset sand deposits within the channel. s ∂r rs As illustrated in Fig. 2-71, the topset-foreset break is s r s t located at s ( ) and the foreset-bottomset break is located The continuity condition at the foreset-bottomset break r s t at b ( ); both defi ne moving boundaries. The shock con- remains unaltered from the one-dimensional case (except for dition at the foreset now takes the form the transformation x → r ). The turbidity current is assumed to spread out axially at θ angle t . It is assumed to have an upstream width equal to ⎡∂η ⎤ IBq that of the effective river channel, as described in Fig. 2-71. 1 f fctrs ()ss22θ ⎢ ()SSs ⎥ s Let r denote the position of the virtual origin of the wedge 21bsf⎢ ∂t ass⎥ ()λ t ⎣ s ⎦ ps θ s of the turbidity current with angle t in Fig. 2-71. The gov- (2-296) erning equations of a steady, axially symmetric, purely dep- ositional turbidity current can be reduced to the forms

∂η ⎛ 1 ⎞ h ⎛ 1 ⎞ 1 v Ri t Ri e⎜2 Ri⎟ C t ⎜1 Ri⎟ Rir sm dh ∂rwft⎝ 2 ⎠ r ⎝ 2 ⎠ 2 o U t t dr 1Ri

(2-298a)

∂η ⎛ 1 ⎞ 1 h 1 v Ri t eRiCRi⎜1 ⎟ t Rir sm dU ∂rwft⎝ 2 ⎠ 2 r 2 o U U t t t dr 1 Ri ht (2-298b)

dq q v q mm sm m ro (2-298c) drr Ut ht

where Fig. 2-70. Fan-delta of the Eau Claire River as it fl owed into Lake Altoona, a reservoir in Wisconsin, in 1988. rrr t (2-299)

The turbidity current is assumed to migrate across the subaqueous fan, fi lling it in a manner analogous to that of a subaerial fan-delta. That is, the turbidity current deposits over an area that is much larger than that of the turbidity current itself at any given time. The appropriate form of the Exner equation of sediment continuity becomes

∂η v q ()1 λθr t r sm m θ ()rr (2-300) pm f ∂ o tt t Ut ht

This quasi-two-dimensional formulation can be cast into a moving-boundary framework and solved in a manner that is completely analogous to the one-dimensional case. The input parameters for the calculation are shown as Case D in

Table 2-8. As in all previous cases, the sand size Ds is 0.4 μ μ θ mm (400 m); the mud size is 10 m. The fan angle f is taken to be 90° and the angle of spread of the turbidity cur- Fig. 2-71. Defi nition diagram for a two-dimensional fan-delta. θ ° rent t is taken to be 10 .

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 114343 11/21/2008/21/2008 5:45:145:45:14 PMPM 144 sediment transport and morphodynamics

Fig. 2-72. Simulation of coevolution of a sandy topset-foreset and a muddy bottomset foreset in a lake using a quasi-2D shock-fi tting formulation and the input of Case D of Table 2-8. The mud has a size of 10 μm.

The results of the quasi-two-dimensional calculation of Case D are shown in Fig. 2-72. The analogous one-dimen- sional case is that of Case Cc. It is seen that after 3000 years the quasi-two-dimensional fan-delta has prograded out much less than in the corresponding one-dimensional case over 45 years. The effect of the two-dimensional geometry in limit- ing the rate of progradation is clearly apparent.

2.12.6 Formation of a Muddy Pond in a Reservoir: Detrainment and Sediment Trap Effi ciency Fig. 2-74. Quasi-steady fl ow in a reservoir after the establishment of an internal hydraulic jump and a muddy pond. In the calculations presented in the previous two sections the turbidity current has been allowed to run out infi nitely in the streamwise direction. In a reservoir, however, a sustained turbidity current eventually hits the dam and refl ects off it.

Fig. 2-75. Diagram illustrating the gradual decline in trap effi - Fig. 2-73. Process by which a muddy pond is set up in a ciency of a reservoir as the level of the muddy interface gradually reservoir. rises above an outlet.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 114444 11/21/2008/21/2008 5:45:165:45:16 PMPM morphodynamics of lake and reservoir sedimentation 145 This creates an upstream-migrating bore that eventually sta- can stabilize well below the water surface of the reservoir, bilizes upstream as an internal hydraulic jump (García and while water continues to detrain at the discharge v sm A . Parker 1989; García 1993). The result is the formation of a Initially the interface of the muddy pond may stabilize at a deep, slow-moving muddy pond downstream of the hydrau- level that is below any outlet, as shown in Fig. 2-75. In such a lic jump (Lamb et al. 2006). The setup process is illustrated case the trap effi ciency of the reservoir is 100 %. As the foreset in Fig. 2-73, and the resulting quasi-steady fl ow after setup progrades and the bottomset aggrades, however, the interface of the muddy pond is illustrated in Fig. 2-74. should rise in time, so that mud is washed out of a reservoir The muddy pond is a zone containing a nearly stagnant outlet. This process is illustrated in Fig. 2-75. The result is a turbidity current, for which the densimetric Froude number trap effi ciency that gradually drops below 100 percent. Toniolo et al. (2007) have developed a moving boundary F rd satisfi es the condition numerical model to describe the coevolution of a sandy top- set-foreset and a muddy bottomset, in which a dam forces the Frd 1 (2-301) formation of a muddy pond. The model is one- dimensional, but it could easily be adapted to a quasi-twodimensional confi guration. The model requires a fully unsteady treatment It is subject to a phenomenon which Toniolo (2002) of the fl ow in order to capture the hydraulic jump (e.g., Choi and Lamb et al. (2004) have called “water detrainment.” In and García1995). Once its position is stabilized, however, a the absence of incoming fl ow from upstream, the interface quasi-steady fl ow is maintained in the presence of slow delta between the muddy pond and the clear water above would progradation and bottomset. Again, the turbidity current is gradually migrate downward at a rate equal to the fall velocity treated as purely depositional.

vsm of the mud. Thus muddy water is converted to clear water The equations governing fl ow mass and momentum con- at a detrainment discharge equal to v sm A , where A is equal to servation and conservation of suspended sediment must the surface area of the interface. When muddy water is con- be adapted to include water detrainment from a stagnant, stantly being replaced from upstream as it passes through the muddy pond for which F rd 1. The forms of these relations hydraulic jump, however, the elevation of the muddy pond used in the analysis are

Fig. 2-76. Simulation of coevolution of a sandy topset-foreset and a muddy bottomset foreset in a reservoir using a one-dimensional shock-fi tting formulation and the input of Case E of Fig. 2-77. Prediction of reservoir trap effi ciency for Case E of Table 2-8. The mud has a size of 50 μm. Note the presence of the internal hydraulic jump and the muddy pond.

6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 114545 11/21/2008/21/2008 5:45:185:45:18 PMPM 146 sediment transport and morphodynamics ∂h ∂Uh effi ciency. It should be pointed out that all the sediment escap- ttt()1 δδeU v (2-302a) ∂t ∂x wt sm ing the model reservoir was mud, not sand. Venting of turbidity currents through bottom outlets to prevent reservoir sedimentation has been considered around ∂∂Uh Uh2 tt t t δUv the world, particularly in China where the sediment loads ∂t ∂x tsm (2-302b) of rivers are extremely large (e.g., Bruk 1985; Morris and 1 ∂Ch2 Fan 1992b). For instance, the effectiveness of bottom outlets Rg tt RgChS CU2 2 m ∂x mtt ftt to discharge sediment-laden water will be greatly infl uenced by the characteristics of turbidity currents reaching the dam where they are placed (Toniolo and Schultz 2005). The loca- ∂Ch ∂UCh tt t tt δrv C (2-302c) tion and frequency of operation of bottom outlets will also ∂ ∂ osmt t x be infl uenced by sediment-laden underfl ows. The approach presented here, which is based on the dynamics of plung- Here the parameter δ is set equal to 1 in the muddy pond, ing turbidity currents, provides a way to assess how effective but is set equal to 0 elsewhere. The details of the moving- and useful different technologies might be for extending the boundary analysis can be found in Toniolo et al. (2007). life of reservoirs. The treatment uses the plunging formulation of Parker and γ Toniolo (2007) with a value of of 1.3. 2.12.7 Remarks in Closing Toniolo et al. (2007) have applied the model at fi eld scale. The analyses of lake and reservoir sedimentation presented The parameters are given as Case E of Table 2-8. Note that r o is set equal to 1 in the analysis, because the hydraulic jump here are highly simplifi ed. They should, however, serve to should render the ponded zone well-mixed up, to the interface illustrate the main processes by which sedimentation occurs. They also should provide guidelines for the numerical pre- with the clear water above. The distance s max from the origin diction of lake and reservoir sedimentation as well as the to the wall of the dam is 6000 m. The fl ow intermittency I f has been set equal to unity. This means that the 90 days of operation of reservoirs to reduce sediment deposition by calculation time corresponds to 90 days of continuous fl ood, open-channel fl ows and turbidity currents. A more practi- which might translate to, e.g., 2 to 10 years of real time. cal analysis of reservoir sedimentation is given in Chapter The results of the calculation are shown in Fig. 2-76. Note 12. One-dimensional modeling of sedimentation processes that the bottomset deposit in the ponded zone is of nearly uni- in rivers is addressed in Chapter 14, while two-dimensional form thickness. This feature is driven by the ponding itself. In and three-dimensional sediment transport models are cov- performing the calculation, water was vented from the reser- ered in Chapter15. Turbulence models for sediment-laden voir at an elevation that was 38 m below the level of the water fl ows are considered in Chapter 16, while numerical mod- surface. As shown in Fig. 2-77, the trap effi ciency was initially eling of sediment transport associated with dam removal is 100 percent because the settling interface was below the out- presented in Chapter 22. let. Reservoir sedimentation caused the level of the interface to rise in time, however, leading to a gradual reduction in trap REFERENCES

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