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1 Incipient Motion Particle movement will occur when the instantaneous fluid force on a particle is just larger than the instantaneous resisting force related to the submerged particle weight and the friction coefficient. The driving forces are strongly related to the local near-bed velocities. In turbulent flow conditions the velocities are fluctuating in space and time, which make together with the randomness of particle size, shape and position that initiation of motion is not merely a deterministic phenomenon but a stochastic process as well. Incipient motion is important in the study of sediment transport, channel degradation, and stable channel design. Due to the stochastic nature of sediment movement along an alluvial bed, it is difficult to define precisely at what flow condition a sediment particle will begin to move. Let us consider the steady flow over the bed composed of cohesionless grains. The forces acting on the grain is shown in Fig Figure 1.10 Forces acting on a grain resting on the bed. The driving force is the flow drag force on the grain, Hydraulic Structures II - Lecture Note Page 28

2 where the friction velocity u * is the flow velocity close to the bed. α is a coefficient, used to modify u * so that αu * forms the characteristic flow velocity past the grain. The stabilizing force can be modeled as the friction force acting on the grain. If u *, c, critical friction velocity, denotes the situation where the grain is about to move, then the drag force is equal to the friction force, i.e. F D = f (W F L ), which can be re-arranged into Shields parameter is then defined as u 2 * * (1.15) s 1 g d We say that sediment starts to move if where or where u u * *,c critical friction velocity u*,c u = τ b ρ τ b = the mean bed shear stress and r = fluid density τ b > τ b,c critical bottom shear stress τ b = ρghs S = bed slope, h = water depth critical Shields parameter or * *, c *, c s 1 g d Fig.1.11 shows Shields experimental results, which relate τ *,c to the grain Reynolds number defined as u 2 *, c Hydraulic Structures II - Lecture Note Page 29

3 The figure has 3 distinct zones corresponding to 3 flow situations u* d n 1) Hydraulically smooth flow; for R e 2. (1.16) d n is much smaller than the thickness of viscous sublayer. Grains are embedded in the viscous sublayer and hence, τ *,c is independent of the grain diameter. By experiments it is found that τ *,c = 0.1/Re. 2) Hydraulically rough flow; for Re 500. The viscous sublayer does not exist and hence, τ *,c is independent of the fluid viscosity. τ *,c has a constant value of ) Hydraulically transitional flow; for 2 Re 500. Grain size is the same order as the thickness of the viscous sublayer. There is a minimum value of τ *,c of corresponding to Re = 10. Note that the flow classification is similar to that of the Nikurase pipe flow where the bed roughness k s is applied instead of d n. Figure The Shields diagram giving τ *,c as a function of Re (uniform and cohesionless grain). Hydraulic Structures II - Lecture Note Page 30

4 Figure 1.12 Turbulent velocity distribution Bed Load, Suspended Load, Wash Load and Total Load Transport When the values of the bed shear velocity just exceeds the critical value for initiation of motion, the bed material particles will be rolling and/or sliding in continuous contact with the bed. For increasing values of the bed shear velocity the particles will be moving along the bed by more or less regular jumps, which are called saltations. When the value of the bed shear velocity begins to exceed fall velocity of the particles, the sediment particles can be lifted to a level at which the upward turbulent forces will be of comparable or higher order than the submerged weight of the particles and as a result the particles may go into suspension. Usually, the transport of particles by rolling, sliding and saltating is called bed load transport, while the suspended particles are transported as suspended load transport. The suspended load may also include the fine silt particles brought into suspension Hydraulic Structures II - Lecture Note Page 31

5 from the catchment area rather than from the streambed material (bed material load) and is called the wash load. A grain size of 63 μm (dividing line between silt and sand) is frequently used to separate between bed material and wash load. Bed load and suspended load may occur simultaneously, but the transition zone between both modes of transport is not well defined. The following classification and definitions are used for the total sediment transported in rivers. Hydraulic Structures II - Lecture Note Page 32

6 Bed Load Transport Usually, the transport of particles by rolling, sliding and saltating is called the bed load transport. Saltation refers to the transport of sediment particles in a series of irregular jumps and bounces along the bed (see Figure 1.13). Bed load transport occurs when the bed shear stress, τ 0 exceeds a critical value (τ 0 ) c. In dimensionless terms, the condition for bed-load motion is: Bed load transport * * c where τ * is the Shields parameter (i.e. * 0 s parameter for initiation of bed load transport. and (τ * ) c is the critical Shields The sediment transport rate may be measured by weight (units: N/s), by mass (units: kg/s) or by volume (units: m 3 /s). In practice the sediment transport rate is often expressed per unit width and is measured either by mass or by volume. Bed load, as one part of the bed material load, is often quantitatively small and hence does not represent a severe problem of sedimentation. On the other hand, as the main factor of the bed formation process, it is always of major importance. Roughness of alluvial channels is to a great extent determined by the movement of the bed load. 1 gd Hydraulic Structures II - Lecture Note Page 33

7 Figure 1.13 Bed load motion. (a) Sketch of saltation motion. (b) Definition sketch of the bed-load layer. Bed Load Formulae Various formulas are developed in the past for estimation of bed load discharge. Estimates of bed load transport using different formula for the same set of given data are also found to give widely different results. The most common formulae and approaches are: a) The discharge approach (bed load expressed in terms of discharge) b) Shear stress approach c) The probabilistic approach Out of these methods the shear stress approach shall be discussed next. Hydraulic Structures II - Lecture Note Page 34

8 Shear Stress Approach This approach is much more favored today, because of the importance accorded to the shear stress in all aspects of the sediment movement in alluvial channels. Formulae of this type are those of Straub-Du Boys, Shield, Kalinske, Meyer-Peter and Mueller, etc. The best known of these, and probably the most widely used, is the Meyer-Peter and Mueller formula; it also gives the best agreement with measured data. Meyer-Peter and Mueller Formula The original Meyer-Peter formula was the discharge type. The new type of formula was arrived at in collaboration with R. Mueller. The formula is given as follows: n.25 G q 2 3 b h S s d s nb where q b = dry weight of transported sediment (N/s/m width of channel) n G = Manning s grain roughness coef. n B = Manning s bed roughness coef. h = depth of flow (m), S = energy slope, d = d m = representative grain size of the bed material (m) (1.17) The roughness coefficients n B, which comprises of bottom roughness due to the sediment and to form resistance should be estimated. The grain roughness coef. n G is defined by n G =d 90 1/6 /26. Equation (1.17) is valid for fully developed turbulence. The representative grain diameter, d, should best be determined by dividing the grain size distribution curve into several fractions and then computing the grain size by d m d p (1.18) 100 where d = average size of grains in a size fraction p = percentage of a given fraction in respect to the total Hydraulic Structures II - Lecture Note Page 35

9 Equation (1.17) gives fair agreement with measured quantities for coarse sediments, i.e. for gravel or cobble-bed streams. Shields Formula The semi-empirical formula derived by Shields for a level bed is q q S c 10 d b (1.19) s s where d = d 50, and S = bed slope. In this formula τ and τ c can be calculated from c s gd50 and g R S Equation (1.19) is dimensionally homogeneous, and can be used for any system of units. The critical shear stress can also be obtained from Shields diagram. Suspended Load Transport Suspended load refers to sediment that is supported by the upward components of turbulent currents and stays in suspension for an appreciable length of time. In most natural rivers, sediments are mainly transported as suspended load. The suspended load transport can be defined mathematically as q sv h u c dz a h (1.20a) q sw s u c dz (1.20b) a where q sv and q sw are suspended load transport rates in terms of volume and weight, respectively; u and c are time averaged velocity and sediment concentration, by volume at a distance z above the bed, respectively; a is thickness of the bed load transport; and h is the water depth. Before eq. (1.20) is integrated, u and c must be expressed mathematically as a function of z. Under steady equilibrium conditions, the downward movement of sediment due to Hydraulic Structures II - Lecture Note Page 36

10 the fall velocity must be balanced by the net upward movement of sediment due to turbulent fluctuations, i.e., dc C s 0 (1.21) dz where ε s is the momentum diffusion coefficient for sediment, which is a function of z; ω is fall velocity of sediment particles; and C is sediment concentration. For turbulent flow, the turbulent shear stress can be expressed as du z m (1.22) dz where ε m is kinematic eddy viscosity of fluid or momentum diffusion coefficient for fluid. It is generally assumed that s m (1.23) where β is a factor of proportionality. For fine sediments in suspension, it can be assumed that β = 1 without causing significant error. Eq.(1.21) can also be written as dc dz 0 (1.24) C s and integration of eq.(1.24) yields z dz C C a exp (1.25) a s where C and C a are sediment concentrations by weight at distance z and a above the bed, respectively. The shear stress at a distance z above the bed is z z S h z 1 (1.26) h where τ and τ z are shear stresses at channel bottom and a distance z above the bed, respectively. Assume the Prandtl von Karman velocity distribution is valid, i.e., Hydraulic Structures II - Lecture Note Page 37

11 du U* (1.27) dz k z From equations (1.22), (1.26) and (1.27), z m k U* (h z) (1.28) h z and s k U* (h z) (1.29) h Equation (1.28) indicates that ε m =0 at z = 0 and z =h. The maximum value of ε m occurs at z = ½ h. On substituting eq. (1.29) into eq. (1.24) and integration from a to z, assuming β = 1, yields where C C a Z h z a z h a (1.30) Z is known as the Rouse constant and equation (1.30) is called the Rouse k U * equation. This equation gives the distribution of the suspended sediment concentration over the vertical for various values of Z (see Fig. 1.14). Figure 1.14 Suspended sediment distribution according to equation (1.30) Hydraulic Structures II - Lecture Note Page 38

12 Total Load Transport Based on the mode of transportation, total load is the sum of bed load and suspended load. Based on the source of material being transported, total load can also be defined as the sum of bed material load and wash load. Wash load consists of fine materials that are finer than those found in the bed. The amount of wash load depends mainly on the supply from the watershed, not on the hydraulics of the river. Consequently, it is difficult to predict the wash load based on the hydraulic characteristics of a river. Most total load equations are, therefore, total bed material load equations. General Approaches There are two general approaches to the determination of total load: (1) computation of bed load and suspended load separately and then adding them together to obtain total load indirect method, and (2) determination of total load function directly without dividing it into bed load and suspended load direct method. Out of the number of methods available for total load computation, only the Engelund and Hansen method (which is based on the second approach) is presented here. Engelund and Hansen Method The basic expression for this method is given by 5 2 f 0. 1 (1.31) where f = total friction factor, computed from Darcy- Weisbach equation for friction losses, Φ = dimensionless sediment discharge, given by q T 3 s g 1 d 50 (1.32) where q T = bed material discharge per unit width and time, s = specific gravity of sediments, and Hydraulic Structures II - Lecture Note Page 39

13 h S s 1 d 50 s 1d 50 where θ = dimensionless form of the bed shear stress, τ h = mean flow depth, S = hydraulic gradient. (1.33) By analogy with Darcy-Weisbach equation for friction losses, the total friction factor can be expressed as 2 S f (1.34) 2 F r where F r = Froude number of the stream flow. 1.4 Cross-sectional Index and Meandering Index Alluvial stream channels, due to the continuous process of erosion and deposition, have ever-changing cross-sections, now being aggraded (deposition), now being degraded (erosion). In order to express these changes, a characteristic ratio, called cross-section index, is often used. in which d =A/B - hydraulic depth; B - water-surface width,. (1.35) Alluvial streams rarely flow for any appreciable length along a straight line, but rather in a series of curves, called meanders. Even those streams which at first sight appear not to meander for longer stretches, will invariably prove to be also subject to natural oscillations. The geometric aspect of a meandering stream is expressed by a characteristic index which denotes to what extent a given alluvial stream deviates from following the centerline of the valley, (see Fig. 1.15). The ratio of the actual stream channel alignment to the corresponding length of the valley line is called the meandering index, denoted M, Hydraulic Structures II - Lecture Note Page 40

14 Figure 1.15 Meandering stream (1.36) According to the very nature of alluvial streams, meandering index is never a constant for a given alluvial channel, but is rather continuously varying around a characteristic value. Lack of constancy for both the cross-section index and the meandering index is another expression of the fact that alluvial streams are generally in temporary and precarious equilibrium only. 1.5 Development Process of a Stream Stream channel formation Streams exhibit a wide range of physical characteristics at different phases of their formation and will react differently to management or restoration efforts by resource managers. Stream managers or users must understand stream channel formation to adequately address stream problems and restoration Dynamic equilibrium All streams try to move towards a state of dynamic equilibrium. One way to describe this equilibrium is the amount of sediment delivered to the channel from the watershed is in long-term balance with the capacity of the stream to transport and discharge that sediment. Sediment suspended in water eventually equals sediment settling out of the water column or being deposited. Sediment load is the total amount of sediment, Hydraulic Structures II - Lecture Note Page 41

15 including that in the bed of the stream, being moved by flowing water. The streams dynamic equilibrium can be expressed with the stream power proportionality equation developed by Lane (see Figure 1.16). Figure 1.16 Lane s stream power proportionality equation that expresses the stream dynamic equilibrium According to Lane s equation, the products of Q S and Q s D 50 are proportional to each other although not equal to each other. The equilibrium occurs when all four variables are in balance. The bottom line is that a given amount of water with a certain velocity can only move so much sediment of a given size. For example, if slope is increased and streamflow remains the same, either the sediment load or the size of the particles must also increase. Likewise, if flow is increased (e.g., by an interbasin transfer) and the slope stays the same, sediment load or sediment particle size has to increase to maintain channel equilibrium. A stream seeking a new equilibrium tends to erode more sediment and of larger particle size. Stable streams are in dynamic equilibrium and called graded (poised). The slope of a graded stream, over a period of years, has delicately adjusted to provide, with the Hydraulic Structures II - Lecture Note Page 42

16 available discharge and prevailing channel characteristics, the velocity required for the transportation of the sediment load supplied from the drainage basin. A graded stream can have depositional and erosional events but overall the sediment transported and supplied to the stream is balanced over long periods. Disturbance of the equilibrium leads to unstable streams that are degrading (eroding) or aggrading (depositing). Degrading streams have a deficit of sediment supply, while aggrading streams have an excess of sediment supply. In both degrading and aggrading streams, the stream is trying to adjust its slope based on the sediment supply. A stream can typically exhibit all three equilibrium states in various reaches along the same stream The channel evolution model In addition to channel stability (dynamic equilibrium), sediment transport and channel dimensions (width and depth of the channel) are very important characteristics for describing streams. These characteristics are incorporated in a conceptual model called the incised channel evolution model (CEM). This model builds upon the dynamic equilibrium theory and describes the stages a stream goes through to reach a new dynamic equilibrium following a disturbance. It also describes the stream bank erosion processes (downcutting, headcutting, or lateral erosion) that are dominant during the different stages. There are five different stages (Figure 1.17): Stage I (Stable): The stream flow discharge of Q 2 will spill in the floodplain and deposit sediment and organic matter. Q 2 is a discharge that has a probability of occurring every two years and is associated with bankfull discharge of undisturbed streams in normal to wet environments. The stream bank height h is below the critical height h c. Critical height h c is that height above which the banks have high potential of collapsing by gravitational forces. Stage II (Incision): This stage starts after disequilibrium conditions occur. These conditions occur as a result of higher Q (stream flow discharge) or S (slope), which lead to an increase in Q s (sediment discharge) capacity in order to maintain the Hydraulic Structures II - Lecture Note Page 43

17 dynamic equilibrium. The increased Q s capacity causes downcutting of the stream bed. The height of the stream banks increases to higher than critical. As a result the banks now can hold stream flow discharge of Q 10. Q 10 is the discharge that has the probability to occur every ten years. A knick point can indicate the movement of incision upstream and in the tributaries. A knick point is a point in the stream profile where the slope abruptly changes. Stage III (Widening): The extensive increase in bank heights (higher than the critical height) of the channel leads to excessive stream bank instability. The banks start collapsing and the stream starts widening. These streams are extremely deep and wide. Most of the sediment is still moving downstream. Stage IV (Stabilizing): Excessive sediment deposition from the stream banks in the channel makes it impossible for the stream flow discharge to remove all of it. The stream bank height starts decreasing (typically equals the critical height). Vegetation starts growing on the sloughing material that is not removed. A new lower capacity stream channel is formed. Stage V (Stable with terraces): A new channel develops and the new banks have heights shorter than the critical bank height. The new floodplain is connected with the stream. Terraces are the remnants of the original floodplain. Within each of the five stages of channel development described by the Channel Evolution Model, channel adjustment is dominated by one of the several processes. For example, in Stage II, downcutting yields the majority of the stream sediment while in Stage III lateral (stream bank) erosion is the primary mechanism of channel adjustment. The difference between these types of erosion has implications for determining the type of restoration efforts. In Stage II success of restoration depends upon stopping stream downcutting by what is called bed stabilization. Bed stabilization is usually done by installing grade control structures such as gabions or check dams. In Stage III, where lateral erosion dominates, restoration efforts should focus on the stream banks. Hydraulic Structures II - Lecture Note Page 44

18 Figure 1.17 The five stages of channel evolution model 1.6 Bed forms and alluvial roughness In alluvial channels the movable bed will take on different and changing forms, depending on the interaction between the sediment and the flow of water. A general picture of bed forms and their relationship with flow regimes is essential for engineering purposes, as the resistance to flow in alluvial channels is largely determined by bed configuration. Hydraulic Structures II - Lecture Note Page 45

19 Flowing over an alluvial bed, water exerts a shear stress on individual sediment particles, given as τ = g RS. If Manning equation for uniform flow is used, this shear stress can be expressed as (1.37) in which n is the Manning roughness coefficient, R the hydraulic radius and V the mean velocity. Assuming further that n and R are constant; this is a simple quadratic relation between τ and V. If, on the other hand, roughness coefficient n changes as a result of the shear stresses on the loose bed, the above simple relationship generally assumes a form similar to the one shown in Figure Figure 1.17 Schematic relationship τ = f (V) in alluvial channels Bed forms Many types of bedforms can be observed in nature. When the bed form crest is perpendicular (transverse) to the main flow direction, the bedforms are called transverse bedforms, such as ripples, dunes and anti-dunes (see Fig. 1.18). Ripples have a length scale smaller than the water depth, whereas dunes have a length scale much larger than the water depth. Hydraulic Structures II - Lecture Note Page 46

20 Ripples and dunes travel downstream by erosion at the upstream face (stoss side) and deposition at the downstream face (lee side). Antidunes travel upstream by lee side erosion and stoss side deposition (see Figure 1.19). Bedforms with their crest parallel to the flow are called longitudinal bedforms such as ribbons and ridges. In laboratory flumes the sequence of bedforms with increasing flow intensity is Flat bed Ripples Dunes High stage flat bed Antidunes Plane (flat) bed: is a plane bed surface without elevations or depressions larger than the largest grain of the bed material. Ripples: Ripples are formed at relatively weak flow intensity and are linked with fine materials, with d 50 less than 0.7 mm. The size of ripples is mainly controlled by grain size. By observations the typical height and length of ripples are At low flow intensity the ripples have a fairly regular form with an upstream slope 6 and downstream slope 32. Ripple profiles are approximately triangular, with long gentle upstream slopes and short, steep downstream slopes. Dunes: The shape of dunes is very similar to that of ripples, but it is much larger. The size of dunes is mainly controlled by flow depth. Dunes are linked with coarse grains, with d 50 bigger than 0.6 mm. With the increase of flow intensity, dunes grow up, and the water depth at the crest of dunes becomes smaller. It means a fairly high velocity at the crest, dunes will be washed-out and the high stage flat (plane) bed is formed. Transition: This bed configuration is generated by flow conditions intermediate between those producing dunes and plane bed. In many cases, part of the bed is covered with dunes while a plane bed covers the remainder. Antidunes: These are also called standing waves. When Froude number exceeds unity antidunes occur. The wave height on the water surface is the same order as the antidune height. The surface wave is unstable and can grow and break in an upstream direction, which moves the antidunes upstream. Hydraulic Structures II - Lecture Note Page 47

21 Chutes and Pools: These occur at relatively large slopes with high velocities and sediment concentrations. Figure 1.19 Bed form types in rivers Figure 1.20 Bed form migration in lower and upper regimes Hydraulic Structures II - Lecture Note Page 48

22 Bars: These are bed forms having lengths of the same order as the channel width, or greater, and heights comparable to the mean depth of the generating flow. Bars have generally elongated shapes, usually reaching lengths equal to channel width or more. These are point bars, alternate bars, middle bars, and tributary bars. Point bars are formed on the convex side of channel bends or meandering alluvial streams. Alternate bars are generally a characteristic feature of crossings, i.e. straight stretches between successive meanders. They appear alternately along both banks of the stream, and as a rule occupy much less than the width of the channel. Tributary bars are formed at confluence of tributaries with the main stream, and they extend downstream. Tributary bars, developed during high flows may appear as detached small islands during low water. Figure 1.21 Types of bars Alluvial Cones and Fans At locations in which an alluvial stream suddenly changes its slope from relatively steep to mild, as for instance when leaving mountainous area and entering alluvial plain, or where a steep tributary meets a flat stream, an alluvial fan may develop. Hydraulic Structures II - Lecture Note Page 49

23 Several chronological stages of the alluvial-fan development are schematically shown on Fig Stage (a) - A relatively unobstructed or recently regulated stream channel carries the water safely within its banks; Stage (b) - Beginning of sediment deposition, part of the available cross section is blocked; Stage (c) - Sediment deposition continues, stream channel fills up and the water starts to overflow the banks, flooding part of the adjacent area, the flood-plain; Stage (d) - The water which has overflown the banks moves with a very low velocity over the flood plain and fine sediment settles down on both sides of the stream-bed. This is already a highly undesirable situation, since the flood-waters cannot be drained back into the main stream channel; Stage (e) - At this stage the channel bed is higher than the surrounding area, and this situation is justly known as an "elevated stream-bed". It consists mainly of recently deposited fine sediment. The area of sediment deposition outside the stream channel proper grows continuously from this stage on. This elevated area is called the alluvial cone. When the water flowing over the cone fans out in the form of branching gullies, it is known by the name of alluvial fan (see Figure 1.23). As mentioned before, the main cause for the formation of an alluvial cone is probably the abrupt changes of the channel slope, but recent studies and field observations seem to indicate that the deposition is also the result of change in channel width and the corresponding reduction of flow volume as the water fans out over an ever larger area. Hydraulic Structures II - Lecture Note Page 50

24 Figure Alluvial cone and fan formation stages. Figure 1.23 Alluvial fan. Hydraulic Structures II - Lecture Note Page 51

25 The apex of the cone is located at the head of the mountain front, and from this point deposits spread out fan-like into the plain and finally merge with it. At the apex the sediment is generally a mixture of gravel and sand, becoming progressively finer toward the margins. Slope of the cone is the steepest at the apex, diminishing toward the periphery. Stream Delta When a stream finally reaches the sea, or any other expanse of water, it loses most of its tractive power and deposits all of its sediments, including the finest silt and clay fractions. The deposited sediment generally takes the form similar to an alluvial fan, but its formation is much more complex, because of additional parameters that are of considerable influence, such as sea waves and breakers, offshore currents and tidal motion. From the engineering point of view, stream mouths may generally be of three main types: 1) estuaries, 2) lagoons, and 3) deltas. In the following only delta-type stream mouths will be reviewed, not only because they are by large the most widespread, but also because they have many features in common with all the other forms. Delta is a highly dynamic natural phenomenon, since it is actually the result of a continuous contest and interaction between the stream and the sea or other water expanse. The stream manages to deposit its sediments more quickly and efficiently than the dispersive action of sea waves and currents to carry them off into the sea. Climatic conditions of the drainage basin determine to a great extent the shape of the delta. In temperate climates, where the flow and the sediment load generally are more or less evenly distributed throughout the year, stream channels of the delta tend to be stable and well adapted to the whole range of discharges conveyed by the stream. In arid climates, on the other hand, flood conditions tend to be erratic, and large quantities of water and sediment are carried by the stream during relatively short periods of time. As a consequence, distributary channels never fully adjust themselves to such large influx of water and sediment. Many new channels are rapidly formed during the flood wave, to be filled up and abandoned as floodwaters recede. This Hydraulic Structures II - Lecture Note Page 52

26 process, often repeated, causes instability and migratory tendency of the whole deltaic system. Within the delta proper, sediment transportation is in form of bed load and suspended load, but mainly in the latter form. Stability of distributary channels depends to some degree on the form of sediment transportation: Channels carrying larger volumes of bed load generally tend to be wide and shallow, and are subject to rapid lateral migration; High concentrations of suspended load tend to cause narrow and deep channels that are relatively much more stable. A typical cross-section through a stream delta generally shows several sets of bed layers superimposed one on top of the other. In the subaerial part (see Fig. 1.24) mostly channel sands and natural silts are found. Deposits at the delta front are laid down in a subaqueous environment immediately seaward of the delta coastline, and are of much finer gradation. In general, it can be said that all deposits tend to grade from coarser to finer in the offshore direction, and from finer to coarser in a vertical section, starting upward from the bottom. Figure 1.24 Schematic drawing of a river delta. Hydraulic Structures II - Lecture Note Page 53

27 In the case of a radical reduction of sediment discharge, due to large-scale engineering interference in the upstream course of the stream channel (such as the erection of a large dam), the existing temporary equilibrium between the sediment transport and the sea currents may be grossly disturbed. As a consequence, sea currents are likely to attack and erode existing sediment deposits, thus actually shortening the delta. A situation of this kind has been evolving at the Nile river delta after the construction of the High Aswan Dam. Stream Confluence and Bifurcation Confluences are mainly present in the upper reach of a river whereas bifurcations are usually present in the lower reach. A few typical cases will be qualitatively analyzed in the following. - It is assumed that the bottom elevation of the tributary at the confluence is roughly the same as that of the main stream (case 1, 2, & 3). - Liquid discharge, sediment discharge, mean grain size and the hydraulic gradient in the main stream are Q 1, Q s1, d s1, and S 1, respectively; in the tributary they are Q 2, Q s2, d s2, and S 2 For a confluence, the continuity of equation for water (Q) and sediment (Q S ) hold. Q 0 = Q 1 + Q 2 Q S0 = Q S1 + Q S2 - Both streams carry the maximum sediment discharge according to their respective sediment transport capacity (STC) under the given flow conditions. Case 1: The flood wave in both streams occurs roughly at the same time (Fig. 1.25a). - Water stage in the main stream during the passage of the flood wave is usually higher than in the tributary, and hence back-water curve will develop in the tributary (Fig. 1.25a). As a result of this, hydraulic gradient in the tributary, in the reach Hydraulic Structures II - Lecture Note Page 54

28 upstream of the confluence, will decrease, causing a part of the sediment to be deposited close to the stream-mouth. - Downstream of the confluence liquid discharge in the main stream will increase from Q 1 to (Q 1 + Q 2 ), and the sediment discharge from Q S1 to somewhat less than (Q s1 + Q S2 ), because a relatively small part of Q S2 will already have been deposited. - It can be assumed that there will be deposition, because STC will not be sufficiently high. Sand bars downstream of the confluence will mainly consist of coarser sediment fractions from the tributary, since the finer ones will probably be carried by the increased water volume in the main stream. Case 2: During the flood wave in the tributary, there is low water in the main stream (Fig. 1.25b); - The situation now will be the reverse: water level in the tributary will be higher than in the main stream, and hence a drawdown curve will have to form in the tributary upstream of the confluence. - Due to flow velocities higher than for normal flow, STC of the tributary will be high enough in the vicinity of the meeting point to carry the entire sediment load, Q S2. So the tributary is likely to stay clean. - In the downstream main channel the combined discharge (Q 1 + Q 2 ) may well be too low to carry the aggregate sediment load (Q s1 + Q S2 ), and hence a considerable part of it is likely to be deposited downstream of the confluence, causing large sand bars. Figure 1.25 Flow situation at confluence. Hydraulic Structures II - Lecture Note Page 55

29 Case 3: Low water in the tributary during the passage of a flood wave in the main stream; - Back-water curve extend much farther upstream into the tributary. - In spite of the reduced flow velocities caused by the backed-up water, sediment load in the tributary will be relatively low, and the tributary is likely to be capable of handling it. Hence little deposition is expected to take place in the tributary. - There is relatively modest addition of sediment from the tributary. - High flow velocities in the main stream are likely to raise its STC just enough to carry the additional load without much difficulty, and hence probably little or no deposition in the main channel either (see Fig. 1.25a). Case 4: water level in the tributary is higher than in the main stream and the bed elevation of the tributary at the confluence is higher than in the main stream (Fig. 1.26); - There will be a drawdown curve in the tributary upstream of the confluence, accompanied by high velocities, - Severe erosion is to be expected along the bed of the tributary. - After some time, a part of channel bed may collapse, shifting the drop from 1 to 2; this process, generally known as back-erosion, may repeat itself several times (points 3, 4, etc.), and thus endanger the stability of the channel. - The eroded material will ultimately be carried into the main stream, settling downstream of the confluence until entrained by high water during flood waves Figure 1.26 Back-erosion at confluence Although in both cases of confluence and bifurcation the main watercourse meets two streams, there are some important differences. The geometry of the branching channels Hydraulic Structures II - Lecture Note Page 56

30 at the bifurcation and the available head determine the magnitude of discharges Q 1 and Q 2. The sorting of the sediment at the bifurcation also depends mainly on the geometry. There is generally no backwater effect at the bifurcation. Meandering and Braided Stream Channels Alluvial streams generally flow in a succession of clockwise and anti-clockwise bends, interconnected by relatively short straight reaches called crossings. Such geometrical alignment is generally known as a meandering river (Fig. 1.27). A watercourse is generally called a meandering stream when the ratio between its actual length and the length of the valley is 1.5 or more (the ratio is rarely more than about 2.5). The actual shape of bends in a meandering stream is rarely symmetrical and geometrically well-defined. Radius of curvature varies over a wide range, depending upon the type of bend. Free bends in plain alluvial material, easily erodible and mobile, generally have the ratio of the radii of curvature to the width of the stream in the range of 4-5, while in case of more consolidated bank materials; the ratio may be as high as 7-8. On the other hand, in forced bends, formed by a stream being deflected by a practically non-eroding bank, the ratio may be as low as 2-3. Figure Schematic layout of a meandering stream. Hydraulic Structures II - Lecture Note Page 57

31 A general characteristic of all meandering watercourses is the migration of the bends downstream and under certain circumstances even laterally. The migration velocity changes from stream to stream, and there are slow-moving and fast-moving streams. Short straight reaches connecting consecutive bends are known under the name of "crossings", and they generally are relatively shallow compared to deep parts of the bends that precede and follow them. A considerable part of the bed material eroded from the concave bank of the bend is deposited in the crossings by the spiral cross currents which do not decay as soon as they leave the bend, but extend downstream. At lower discharges, sand bars also may be formed in the crossings. The main erosion process is to be expected at the concave side of the flow channel. Figure 1.27a An alternative alignment of an oscillating alluvial watercourse is known as a braided stream. The characteristic features of such a configuration are a wide channel, unstable and poorly defined banks and shallow water. The watercourse consists of a number of entwined channels divided by islands, which meet, cross and separate again. The main causes which bring about the braiding of a stream seem to be Hydraulic Structures II - Lecture Note Page 58

32 1) Supply of more sediment than warranted by its STC, hence part of the load is deposited, 2) Steep longitudinal slopes that tend to produce a wide and shallow channel in which bars readily form, become stabilized by armouring and vegetation and form islands, and 3) Easily erodible banks, allowing the widening of the stream channel at high flows. It is generally assumed that a braided channel has a steep slope, a large bed load in comparison with the suspended load, and usually small amounts of silt and clay particles in both bed and banks. A decrease in longitudinal slope may often change a channel from braided into meandering. Figure A braided stream Bed Roughness In open channel hydraulics with rigid-boundary, the roughness coefficient can be treated as a constant. After the roughness coefficient has been determined, a resistance formula can be applied directly for the computation of velocity, slope, or depth. In fluvial hydraulics, the boundary is movable and the resistance to flow or the roughness coefficient is variable. In this case, a resistance formula cannot be applied directly without knowledge of how the resistance coefficient will change under different flow and sediment conditions. Resistance to flow with a movable boundary consists of two parts. The roughness that is directly related to grain size is called GRAIN ROUGHNESS. The roughness that is due to the existence of bedforms and that changes with changes of bedforms is called FORM ROUGHNESS. Hydraulic Structures II - Lecture Note Page 59

33 If Manning s roughness coefficient is used, the total coefficient n can be expressed as n n n (1.38) where n = Manning s coefficient due to grain roughness, and n = Manning s roughness due to form roughness. The value of n is proportional to the sediment particle size to the sixth power. There is no reliable method for the computation of n, which poses a major problem in the study of alluvial hydraulics. Manning s Formula One of the most commonly used resistance equations for open channel flows is Manning s equation, namely, V n R S (1.39) Strickler defined Manning s n as a function of sediment particle size as: n 1 6 d 21.1 (1.40a) where d = sediment size of uniform sand in m. Meyer-Peter and Mueller, considering a sand mixture, transformed Strickler s formula to n d (1.40b) 26 where d 90 = sediment size (in m) for which 90% of the mixture is finer. Similar to the division of total roughness into grain roughness and form roughness, the shear stress or drag force acting along an alluvial bed can be divided into two parts, i.e., S R R (1.41) where τ = total drag force acting along alluvial bed, τ and τ = drag force due to grain roughness and form roughness, respectively, γ= specific weight of water, S = energy Hydraulic Structures II - Lecture Note Page 60

34 slope, and R and R = hydraulic radii due to grain roughness and form roughness, respectively. Hydraulic Structures II - Lecture Note Page 61