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CHAPTER 8: FLUVIAL BEDFORMS. The interaction of flow and sediment transport often creates bedforms such as ripples, dunes, antidunes, and bars. These bedforms in turn can interact with the flow to modify the rate of sediment transport.

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## CHAPTER 8: FLUVIAL BEDFORMS

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**CHAPTER 8:**FLUVIAL BEDFORMS The interaction of flow and sediment transport often creates bedforms such as ripples, dunes, antidunes, and bars. These bedforms in turn can interact with the flow to modify the rate of sediment transport. Dunes in the North Loup River, Nebraska, USA; image courtesy D. Mohrig**TOUR OF BEDFORMS IN RIVERS: RIPPLES**Ripples are characteristic of a) very low transport rates in b) rivers with sediment size D less than about 0.6 mm. Typical wavelengths are on the order of 10’s of cm and and wave heights are on the order of cm. Ripples migrate downstream and are asymmetric with a gentle stoss (upstream) side and a steep lee (downstream side). Ripples do not interact with the water surface. View of the Rum River, Minnesota USA Ripples in the Rum River, Minnesota USA at very low flow; ~ 10 - 20 cm.**TOUR OF BEDFORMS IN RIVERS: DUNES**Dunes are the most common bedforms in sand-bed rivers; they can also occur in gravel-bed rivers. Wavelength can range up to 100’s of m, and wave height can range up to 5 m or more in large rivers. Dunes are usually asymmetric, with a gentle stoss (upstream) side and a steep lee (downstream) side. They are characteristic of subcritical flow (Fr sufficiently below 1). Dunes migrate downstream. They interact weakly with the water surface, such that the flow accelerates over the crests, where water surface elevation is slightly reduced. (That is, the water surface is out of phase with the bed.) Dunes in the North Loup River, Nebraska USA. Two people are circled for scale. Image courtesy D. Mohrig.**TOUR OF BEDFORMS IN RIVERS: DUNE MIGRATION**Double-click on the image to see the video; video courtesy D. Mohrig. rte-bookmohrigloup.mpg:to run without relinking, download to same folder as PowerPoint presentations.**TOUR OF BEDFORMS IN RIVERS: ANTIDUNES**Antidunes occur in rivers with sufficiently high (but not necessarily supercritical) Froude numbers. They can occur in sand-bed and gravel-bed rivers. The most common type of antidune migrates upstream, and shows little asymmetry. The water surface is strongly in phase with the bed. A train of symmetrical surface waves is usually indicative of the presence of antidunes. Trains of surface waves indicating the presence of antidunes in braided channels of the tailings basin of the Hibbing Taconite Mine, Minnesota, USA. Flow is from top to bottom.**TOUR OF BEDFORMS IN RIVERS: CYCLIC STEPS (CHUTE AND POOL**TRAINS) Trains of cyclic steps occur in very steep flows with supercritical Froude numbers. They are long-wave relatives of antidunes. The steps are delineated by hydraulic jumps (immediately downstream of which the flow is locally subcritical). The steps migrate upstream. These features are also called chute-and-pool topography. Train of cyclic steps in a small laboratory channel at St. Anthony Falls Laboratory. The water has been dyed to aid visualization; two hydraulic jumps can be seen in the figure.**TOUR OF BEDFORMS IN RIVERS: CYCLIC STEPS (contd.)**Cyclic steps form in the field when slopes are steep, the flow is supercritical and there is a plethora of sediment. jumps flow Trains of cyclic steps in a coastal outflow channel on a beach in Calais, France. Image courtesy H. Capart.**TOUR OF BEDFORMS IN RIVERS: ALTERNATE BARS**Alternate bars occur in rivers with sufficiently large (> ~ 12), but not too large width-depth ratio B/H. Alternate bars migrate downstream, and often have relatively sharp fronts. They are often precursors to meandering. Alternate bars may coexist with dunes and/or antidunes. Alternate bars in the Naka River, an artificially straightened river in Japan. Image courtesy S. Ikeda.**TOUR OF BEDFORMS IN RIVERS: MULTIPLE-ROW LINGUOID BARS**Multiple-row bars (linguoid bars) occur when the width-depth ratio B/H is even larger than that for alternate bars. These bars migrate downstream. They may co-exist with dunes or antidunes. Plan view of superimposed linguoid bars and dunes in the North Loup River, Nebraska USA. Image courtesy D. Mohrig. Flow is from left to right.**BEDFORMS IN THE LABORATORY AND FIELD: DUNES**Dunes in a flume in Tsukuba University, Japan: flow turned off. Image courtesy H. Ikeda. Dunes on an exposed point bar in the meandering Fly River, Papua New Guinea**Rhine River, Switzerland**BEDFORMS IN THE LABORATORY AND FIELD: ALTERNATE BARS Alternate bars in a flume in Tsukuba University, Japan: flow turned low. Image courtesy H. Ikeda. Alternate bars in the Rhine River between Switzerland and Lichtenstein. Image courtesy M. Jaeggi.**BEDFORMS IN THE LABORATORY AND FIELD: MULTIPLE-ROW**(LINGUOID) BARS Linguoid bars in a flume in Tsukuba University, Japan: flow turned off. Image courtesy H. Ikeda. Linguoid bars in the Fuefuki River, Japan. Image courtesy S. Ikeda.**Ohau River, New Zealand**WHEN THE FLOW IS INSUFFICIENT TO COVER THE BED, THE RIVER MAY DISPLAY A BRAIDED PLANFORM Braiding in a flume in Tsukuba University, Japan: flow turned low. Image courtesy H. Ikeda. Braiding in the Ohau River, New Zealand. Image courtesy P. Mosley.**RIPPLES**Ripples are small-scale bedforms that migrate downstream and show a characteristic asymmetry, with a gentle stoss face and a steep lee face. Ripples require the existence of a reasonably well-defined viscous sublayer in order to form. In rivers, a viscous sublayer can exist only when the flow is very slow and well below flood conditions. Because of the viscous sublayer, ripples do not interact with the water surface. Engelund and Hansen (1967) have suggested the following condition for ripple formation: D v, where v = 11.6 /u* denotes the thickness of the viscous sublayer (Chapter 6). This relation can be rearranged to yield the threshold condition where The above relation can be solved with the modified Brownlie relation of Chapter 6 to yield a maximum value of Rep for ripple formation. The value so obtained is 91, corresponding to a grain size of 0.8 mm with = 0.01 cm2/s and R = 1.65. In practice, ripples are observed only for D < 0.6 mm. Ripples can coexist with dunes.**DEFINITION OF DUNES AND ANTIDUNES**Dunes are 1D (or quasi-1D) bedforms for which the water surface fluctuations are approximately out of phase with the bed fluctuations. That is, the water surface is high where the bed is low and vice versa. As is shown below dunes migrate downstream. Antidunes are 1D (or quasi-1D) bedforms for which the water surface fluctuations are approximately in phase with the bed fluctuations. That is, the water surface is high where the bed is high and vice versa. As shown below, most antidunes migrate upstream, but there is a regime within which they can migrate downstream.**RESPONSE OF FLOW TO BED UNDULATIONS:**INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS Steady, uniform flow over a flat erodible bed (base flow; no bedforms) has flow depth Ho and flow velocity Uo = qw/Ho. Unperturbed bed elevation is at = 0. The bed is then given a slight wavy perturbation of the form where ’ << Ho denotes the amplitude of the perturbation and denotes the wavelength of the perturbation. How does the flow and water surface respond to such a perturbation?**RESPONSE OF FLOW TO BED UNDULATIONS:**INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS contd. Consider inviscid (frictionless) steady 1D shallow water flow over an undulating bed. The St. Venant shallow water equations simplify as follows: The equation in the box can be made dimensionless using the depth Ho of the base flow:**RESPONSE OF FLOW TO BED UNDULATIONS:**LINEAR INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS Solving for the variation in flow depth, The variation in water surface elevation is given as Now the bed perturbation can be represented in dimensionless form as follows: Here denotes the dimensionless amplitude of the bed perturbation and k denotes the dimensionless wavenumber of the bed perturbation. We further write the response of the depth and water surface elevation to the perturbation as where denotes the dimensionless amplitude of the response of depth to the bed perturbation, and denotes the corresponding dimensionless response in water surface elevation.**RESPONSE OF FLOW TO BED UNDULATIONS:**LINEAR INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS contd. Now as long as << 1, With this approximation, substituting into gives the results and**SHALLOW-WATER RESPONSE OF WATER SURFACE TO BED PERTURBATION**When Fro < 1, and the water surface perturbation is out of phase with the bed perturbation: the water surface is low where the bed is high and the water surface is high where the bed is low. According to long wave theory, then, dunes can occur in subcritical flow (Fro < 1) When Fr0 > 1, and the depth perturbation is in phase with the bed perturbation: the water surface is high where the bed is high and the water surface is low where the bed is low. According to long wave theory, then, antidunes can occur in supercritical flow (Fro > 1).**PREDICTIONS OF LINEAR INVISCID SHALLOW-WATER THEORY FOR**DUNES AND ANTIDUNES**BEYOND THE SHALLOW-WATER APPROXIMATION:**POTENTIAL FLOW FORMULATION The shallow-water theory of bedforms is not entirely accurate. This is because the wavelength of dunes and antidunes usually scales as a multiple of the flow depth H, and so the condition H/ << 1 is usually not satisfied. In more precise terms, the wavenumber of the bedforms k = 2Ho/ does not usually satisfy the condition k << 1. A better view of bedforms is obtained by solving for the linearized potential flow over a wavy bed. This formulation includes the vertical coordinate z as well as the horizontal coordinate x, and describes the vertical as well as the horizontal structure of the response of the flow to bed. Such a solution was first implemented by Anderson (1953) and extended by Kennedy (1963). Let = velocity potential function:**POTENTIAL FLOW FORMULATION contd.**In general, subcritical flow is a flow for which the water surface perturbation is approximately out of phase with a bed perturbation, and supercritical flow is a flow for which the water surface is approximately in phase with a bed perturbation. Potential flow theory indicates that the border between subcritical and supercritical flow is a function of both Froude number Fro and wavenumber k = 2Ho/ as follows: Now as k 0, Fro 1, indicating that for long bedforms the division between subcritical and supercritical flow is given by the long wave (shallow-water) limit of 1. If e.g. the bedform has a wavelength equal to 5 Ho (a reasonable guess for many dunes and antidunes), k = 1.26 and the borderline between subcritical and supercritical flow is Fro = 0.82. That is, the zone of supercritical response extends somewhat into the range Fro < 1, and antidunes can occur in flows for which Fro < 1. In general, lower-regime flow refers to truly subcritical flow in the sense Fro < [tanh(k)/k]1/2, and upper-regime flow refers to truly supercritical flow in the sense Fro > [tanh(k)/k]1/2. It is important to realize that part of the zone of upper-regime flow is subcritical in the long-wave sense.**POTENTIAL FLOW FORMULATION contd.**In addition to the criterion dividing subcritical from supercritical response, potential flow reveals another criterion further dividing the regime of supercritical flow. When Fro < [tanh(k)/k]1/2 both the water surface and depth are out of phase with the bed, and the flow accelerates over bed crests and decelerates over bed troughs. This gives rise to downstream-migrating dunes. When Fro > [tanh(k)/k]1/2 and Fro < [k tanh(k)]-1/2, both the water surface and the depth are in phase with the bed, and the flow decelerates over crests and accelerates over troughs. This gives rise to upstream-migrating antidunes. When Fro > [tanh(k)/k]1/2 and Fro > [k tanh(k)]-1/2 the water surface is in phase with the bed, so the bedforms are antidunes, but the depth is out of phase with the bed, and the flow accelerates over the crests and decelerates over the troughs. These antidunes (which cannot be obtained from the St. Venant formulation) thus migrate downstream.**FLOW IN THE DUNE REGIME**Fro < [tanh(k)/k]1/2 Water surface is out of phase with the bed. Depth variation is out of phase with the bed Flow accelerates from trough to crest. Sediment transport increases from trough to crest. Bedform migrates downstream. Bedform becomes asymmetric.**FLOW IN THE UPSTREAM-MIGRATING ANTIDUNE REGIME**[tanh(k)/k]1/2 < Fro < [k tanh(k)]-1/2 Water surface is in phase with the bed. Depth variation is in phase with the bed Flow decelerates from trough to crest. Sediment transport decreases from trough to crest. Bedform migrates upstream (or hardly at all). Bedform stays symmetric.**FLOW IN THE DOWNSTREAM-MIGRATING ANTIDUNE REGIME**[k tanh(k)]-1/2 < Fro Water surface is in phase with the bed. Depth variation is out of phase with the bed. Flow accelerates from trough to crest. Sediment transport increases from trough to crest. Bedform migrates downstream. Bedform becomes asymmetric. These are antidunes that look like dunes: not too common, but they are observed.**PHASE DIAGRAM FOR DUNES AND ANTIDUNES BASED ON LINEAR**POTENTIAL THEORY OVER A WAVY BED**LOWER- AND UPPER-REGIME PLANE BED**Dunes usually do not form in gravel-bed streams. This is because such streams usually fall into a regime known as lower-regime plane bed, for which the flow is subcritical and neither dunes nor ripples form. Chabert and Chauvin (1963) have described this regime experimentally, and Engelund (1970) and Fredsoe (1974) have developed stability analyses for bedforms which describe this regime. In sand-bed streams, there is a second regime in the vicinity of the line Fro = [tanh(k)/k]1/2 within which neither dunes nor antidunes form. This regime is known as upper-regimeplane bed. Engelund (1970) and Fredsoe (1974) have explained this region as one of competition between the effects of bedload and suspended load. The former favors the formation of dunes, and the latter favors the formation of antidunes. Within the regime of upper-regime plane bed, the two effects cancel each other, and a plane bed prevails. A rough sketch of the zones for lower-regime plane bed, dunes, upper regime plane bed, upstream-migrating antidunes and downstream-migrating antidunes is given in the following diagram based on potential flow. It should be pointed out, however, that the analyses of Engelund (1970) and Fredsoe (1974) result in somewhat modified criteria for the divisions between supcritical and supercritical flow, and upstream and downstream migrating antidunes. (See Engelund and Fredsoe, 1982).**EXPERIMENTAL RESULTS OF CHABERT AND CHAUVIN (1963)**Chabert and Chauvin (1963) report on experiments which yield a threshold conditions for ripples that is very similar to that proposed by Engelund and Hansen (1967). In addition, they obtain a criterion for the threshold between lower-regime plane bed and dunes that can be approximated as where c* is given by the modified Brownlie relation, Thus in the limit of coarse material (Rep >> 1, gravel-bed streams) dunes should not form until * exceeds 0.0816. It was seen from Slide 21 of Chapter 3, however, that this condition is not common for gravel-bed streams at bankfull flow. Dunes can form in gravel-bed streams if the conditions are right; e.g. see Dinehart (1992).**SHIELDS DIAGRAM INCLUDING RESULTS OF CHABERT AND CHAUVIN**(1963)**CYCLIC STEPS (CHUTE-AND-POOL TOPOGRAPHY)**Trains of cyclic steps occur in very steep flows with supercritical Froude numbers. They are long-wave relatives of antidunes (Winterwerp et al., 1992; Taki and Parker, in press). The steps are delineated by hydraulic jumps (immediately downstream of which the flow is locally subcritical). The steps migrate upstream. These features are also called chute-and-pool topography (Simons et al., 1965). Their regime of formation is schematized in the previous Slide 30. Train of cyclic steps in a small laboratory channel at St. Anthony Falls Laboratory. The water has been dyed to aid visualization; two hydraulic jumps can be seen in the figure.**1D BEDFORM REGIME DIAGRAMS**A number of diagrams have been proposed to characterize bedform regime. The most useful of these are dimensionless. Following the analysis of Vanoni (1974) and Parker and Anderson (1978), the following general relation can be posited: Here Rep, R and g have their standard meanings of explicit particle Reynolds number, sediment submerged specific gravity and geometric standard deviation of bed sediment. In addition, X1 and X2 are two dimensionless parameters describing the flow which must be independent from each other. Suitable choices include Shields number * = u*2/(RgDs50), Froude number Fr = qw/[(gH)1/2H], bed slope S, dimensionless depth = H/Ds50, dimensionless unit stream power US/vs etc. Vanoni (1974) has provided a relatively complete set of bedform diagrams for sand, including dunes, antidunes, ripples, flat (by which he means lower-regime flat bed), transition (by which he means upper-regime flat bed) and chute-and-pool topography (in which cyclic steps should be included. Vanoni chooses X1 = Fr and X2 = , In addition, he assumes that R is constant at 1.65, and he neglects the effect of g (i.e. assumes uniform material), so that**BEDFORM REGIME DIAGRAM 1 OF VANONI (1974)**Fr Bedform Chart for D50 = 0.011 mm and 0.088 – 0.15 mm (Rep = 0.11 and 2.4 – 5.4)**BEDFORM REGIME DIAGRAM 2 OF VANONI (1974)**Fr Bedform Chart for D50 = 0.12 – 0.20 mm (Rep = 3.9 – 8.3)**BEDFORM REGIME DIAGRAM 3 OF VANONI (1974)**Fr Bedform Chart for D50 = 0.15 – 0.32 mm (Rep = 5.4 – 16.8)**BEDFORM REGIME DIAGRAM 4 OF VANONI (1974)**Fr Bedform Chart for D50 = 0.23 – 0.45 mm (Rep = 10.2 – 28.0)**BEDFORM REGIME DIAGRAM 5 OF VANONI (1974)**Fr Bedform Chart for D50 = 0.4 – 0.6 mm (Rep = 23.5 – 43.1)**BEDFORM REGIME DIAGRAM 6 OF VANONI (1974)**1.0 Fr 0.1 102 103 Bedform Chart for D50 = 0.93, 1.20 and 1.35 mm (Rep = 83.3, 122, 146)**BEDFORM REGIME DIAGRAM OF ENGELUND AND HANSEN (1966)**This diagram uses the hydraulic parameters X1 = Fr and X2 = U/u*. The parameter Rep is not included, and the diagram is valid only for sand. The diagram clearly shows an extensive range of flow for which Fr < 1 but antidunes form. The “plane bed” regime on the left-hand side of the diagram is upper-regime plane bed. Lower-regime plane bed is not shown in the diagram. U/u* Fr**REFERENCES FOR CHAPTER 8**Anderson, A. G., 1953, The characteristics of sediment waves formed by flow in open channels, Proceedings, 3rd Midwest Conference on Fluid Mechanics, University of Minnesota. Chabert, J. and Chauvin, J. L., 1963, Formation des dunes et de rides dans les modeles fluviaux, BulletinC.R.E.C., No. 4. Dinehart, R. L., 1992, Evolution of coarse gravel bed forms; field measurements at flood stage, Water Res. Res., 28(10), 2667-2689. Engelund, F., 1970, Instability of erodible beds, J. Fluid Mech., 42(2). Engelund, F. and Fredsoe, J., 1982, Sediment ripples and dunes, Annual Review of Fluid Mechanics, 14, 13-37. Engelund, F. and Hansen, E., 1966, Hydraulic resistance in alluvial streams, Acta Polytechnica Scandanavica, V. Ci-35. Engelund, F. and Hansen, E., 1967, A Monograph on Sediment Transport, Technisk Forlag, Copenhagen, Denmark. Fredsoe, J., 1974, On the development of dunes in erodible channels, J. Fluid Mech., 64(1), 1-16. Kennedy, J. F., 1963, The mechanics of dunes and antidunes in erodible bed channels, J, Fluid Mech., 16(4). Parker, G. and Anderson, A., 1977, Basic principles of river hydraulics, J. Hydr. Engrg., 103(9), 1077-1087.**REFERENCES FOR CHAPTER 8 contd.**Simons, D. B., Richardson, E. V. and Nordin, C. F., 1965, Sedimentary structures generated by flow in alluvial channels, Special Pub. No. 12, Am. Assoc. Petrol. Geologists. Taki, K.. And Parker, G., 2005, Transportational cyclic steps created by flow over an erodible bed. Part 1. Experiments, J. Hydr. Res., in press, downloadable from http://cee.uiuc.edu/people/parkerg/preprints.htm . Vanoni, V., 1974, Factors determining bed forms of alluvial streams, Journal of Hydraulic Engineering, 100(3), 363-377. Winterwerp, J. C., Bakker, W. T., Mastbergen, D. R., and Van Rossum, H, 1992, Hyperconcentrated sand-water mixture flows over erodible bed, J. Hydr. Engrg., 119(11), 1508-1525.

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