CHAPTER 2
Sediment Transport and Morphodynamics Marcelo H. García
ASCE Manual 54 , Sedimentation Engineering, prepared suspended load, fl uid turbulence comes into play, carrying under the leadership of Professor Vito A. Vanoni, has pro- the particles well up into the water column. In both cases, the vided guidance to theoreticians and practitioners’ world driving force for sediment 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 drag. Whether the mode of transport is saltation 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 rivers. 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 bed load 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. Sediment transport is accomplished in the near- past 30 years as the most complete reference on sedimenta- shore of lakes and oceans 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 sea. Landslides, debris fl ows and mud 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 clay particles to silt, sand, gravel, cobbles, and boulders. decades. A concerted effort is made to relate the mechanics Rock types can include quartz, feldspar, limestone, granite, of sediment transport in rivers and by turbidity currents to basalt, and other less common types such as magnetite. The the morphodynamics of lake 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 river. 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 snow avalanche, 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 suspension. 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 antidunes, 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 avalanches, 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 Desert sand dunes 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 erosion and deposition. If a stream is suffi ciently wide, it will water, quartz sand particles saltate in long, high trajectories, braid rather than meander, 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 canyons, 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 beach 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-current-sediment problem, as well, can be treated with the techniques of fl uid interaction. The boulder levees 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, wind, and glaciers 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. Rills, gullies, streams, 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 discharge, 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 channel, 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 mountain 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 geomorphology 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 shear stress, 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 ττ 1 (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 wetted perimeter), 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 viscosity 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 sands 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 / Reynolds number, 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 Missouri river, 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, dune-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 sediments. 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 mountains, 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, muds and fl at fi ne sands are
6690-003-2pass-002-r03.indd90-003-2pass-002-r03.indd 2299 11/21/2008/21/2008 5:42:405:42:40 PMPM 30 sediment transport and morphodynamics
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 gravels 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 Table 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 grain size 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