Abstract
The mode of sediment transport where the sediment particles slide, roll, or travel in succession of low jumps close to the bed is known as bed-load transport. In this chapter, theories of bed-load and formulations to predict the bed-load transport rate are presented. The pioneering attempt to predict the bed-load transport rate was due to MP du Boys in 1879, who expressed bed-load transport rate as a function of excess bed shear stress, that is the bed shear stress exceeding the threshold bed shear stress. Thereafter, number of researchers suggested du Boys type equations making use of the excess bed shear stress in different forms and coefficients. Other concepts to predict the bed-load transport rate are the discharge concept (Schoklitsch type), the velocity concept, the bedform concept, the probabilistic concept (Einstein type), the deterministic concept (Bagnold type), and the equal mobility concept. The additional features of this chapter are the discussion on particle saltation, sediment sorting, streambed armoring, and sediment entrainment probability to bed load. The effects of bed load on velocity distribution, length scales of turbulence, and von Kármán constant are also discussed in details. The method of computation of bed-load transport is illustrated through worked out examples.
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Keywords
5.1 General
The transport of sediment in rivers, by which the river morphological changes are closely related, is an important aspect in fluvial processes. The term load, as often used to define the sediment transport, refers to the quantity of sediment that is transported in a stream. More specifically, it is used to define the rate (volume or weight per unit time and width) at which the sediment is transported.
When the bed shear stress τ 0 induced by the flow exceeds the threshold bed shear stress τ 0c for the initiation of sediment motion, the sediment particles forming the bed are set in motion. The bed-load transport is the mode of sediment transport where the sediment particles slide, roll, or travel in succession of low jumps, termed saltation, but belong close to the bed, from where they may leave temporarily. The dislodgment of the sediment particles is rather intermittent, as turbulence (velocity fluctuations) interacts with the bed particles randomly to play an important role in transporting them. It is, however, convenient to distinguish the modes of sediment transport as bed load (slide, roll, and saltation) and suspended load. Figure 5.1 presents a schematic of different modes of sediment transport.
Schematic of different modes of sediment transport
At relatively small excess bed shear stress (τ 0 − τ 0c), the bed-load transport takes place in a sliding and/or rolling mode. It therefore describes a sediment motion generally in contact with the bed; while individual sediment particles have intermittent motion, but substantially continuous. The bed-load transport in this mode is known as contact load. With an increase in excess bed shear stress, increasingly sediment particles are driven streamwise in a short succession of jumping or bouncing mode of motion, as the particles lose contact with the bed for a short while to attain a mean height in water of a number of particle diameters. The bed-load transport in this mode is called saltation. According to Einstein (1942, 1950), the bed-load transport is defined as the transport of sediment particles within a thin layer having a thickness of two particle diameters above the bed by sliding, rolling, or traveling in succession of jumps with a streamwise distance of a few particle diameters. On the other hand, Bagnold (1956) defined the bed-load transport that takes place by successive contacts of the particles with the bed being limited by the gravity effect.
With a further increase in excess bed shear stress, the production of turbulence near the bed and its diffusion in upward direction lift up relatively finer sediment particles from the bed keeping them in suspension, as they are transported by the flow. The upward diffusion of turbulence retains the particles in the fluid domain against the gravity; while relatively coarser particles are still transported as bed load. In reality, the particles stay occasionally in contact with the bed and are displaced by making more or less large jumps to remain often surrounded by the fluid. The sediment transport in suspension mode is termed suspended load. Bagnold (1956) defined the suspended-load transport that takes place by balancing submerged weight of the particles with upward diffusion of turbulent eddies. In both bed-load and suspended-load transports, the sediment transport is established by the action of gravity on the fluid phase driving the sediment particles by the induced drag.
It is useful to provide approximate limiting values to separate different modes of sediment transport:
where u * is the shear velocity and w s is the settling or terminal velocity of particles. Generally, the amount of bed load transported through a large deep river is approximately 5–25 % of the suspended load.
In natural stream, wash load is the portion of sediment that is carried by the flow such that it always remains close to the free surface. It is in near-permanent suspension and transported without deposition, essentially passing straight through the stream. It consists of very fine sediment particles, such as silt and clay. The composition of wash load is distinct because it is almost entirely made up of particles that are only found in small quantities in the bed. Nevertheless, wash-load particles are also brought in by the overland flow or from the cohesive stream banks. As the wash-load particles tend to be very fine, they have a small settling velocity, being easily kept in suspension by the turbulence in flow. A physical characterization of the wash load is a difficult proposition, as the wash load, by definition, cannot be determined by the given flow characteristics of a river.
5.2 Definition of Bed-Load Transport
The term bed-load transport is defined as the sediment particles, such as silt, sand, gravel, etc., carried by the stream flow in the streamwise direction immediately above the bed as sliding, rolling, and/or saltating at a velocity less than that of the stream flow. The bed-load transport rate q b is generally expressed as the solid volume of sediment transported per unit time and width. It is also expressed as the weight of sediment transported per unit time and width, denoted by g b, or the submerged weight of sediment transported per unit time and width, denoted by g bs. However, in nondimensional form, the bed-load transport rate is designated as bed-load transport intensity and denoted by Φb. The bed-load transport intensity Φb is related with q b, g b, and g bs as follows:
where Δ is the submerged relative density (= s − 1), s is the relative density of sediment (= ρ s/ρ), ρ s is the mass density of sediment, ρ is the mass density of water, g is the acceleration due to gravity, and d is the representative sediment size, that is the median or weighted mean diameter.
The bed-load transport rate q b can be defined as the product of the particle velocity u b in streamwise direction, the volumetric concentration C of particles transported as bed-load, and the thickness δ b of bed-load transport layer. It is therefore given by
The bed-load transport rate q b can also be defined as the product of the particle velocity u b in streamwise direction, the number of particles in motion N b per unit area, and the volume of particles V b. It is thus
Further, by defining the particle velocity u b as the ratio of saltation or step length λ s to saltation or step period t e, that is u b = λ s/t e, Eq. (5.4) can be rewritten as
where E b and D b are the degraded or aggraded volume of particles per unit time and area.
Another way of defining bed-load transport rate is the pickup rate. It is, in fact basically, defined as the number of particles picked up per unit time and area. Later, the definition of pickup rate E p has been modified to the mass of particles picked up per unit time and area. The nondimensional pickup rate, known as the sediment pickup function Φp, is defined according to Einstein (1950) as
Although different researchers studied pickup rate (Einstein 1950; Fernandez Luque 1974; Yalin 1977; Nakagawa and Tsujimoto 1980; de Ruiter 1982, 1983; van Rijn 1984b; Dey and Debnath 2001), it, however, remains almost unclear whether contact load or saltation contributes to pickup rate.
5.3 Bed Shear Stress Concept for Bed-Load Transport
5.3.1 du Boys’ Approach
The pioneering attempt to predict the bed-load transport rate was due to MP du Boys in 1879, who was a French engineer. His analysis was based on the force balance between the force applied to the top layer of sediment bed by the flowing fluid and the frictional resistance between the top layer of sediment particles and the layers beneath it.
du Boys (1879) assumed that the sediment particles move in series of superimposed layers of individual thickness Δε by the tractive force offered by the uniform flow as given by the bed shear stress τ 0 = ρghS 0 applied to the surface of the top layer; where h is the flow depth and S 0 is the streamwise bed slope. The mean velocity of the successive layers that are sliding over each other increases linearly toward the bed surface. It implies that the velocity is highest at the top layer forming the bed surface and zero (minimum) at the lowest layer at a depth of Δε·m; where m is the number of layers. Figure 5.2 illustrates the definition sketch of du Boys model. Under the equilibrium condition, the top layer is one where the tractive force balances the frictional resistance force between these layers. The coefficient of frictional resistance μ f between successive layers is assumed to be constant, such that the force balance is
Definition sketch of du Boys’ bed-load model
The fastest moving layer being the top layer moves with a velocity of (m − 1)u s, where u s is the velocity of the second lowest layer. As the layers between the first and the m-th move according to a linear velocity distribution, the sediment transport rate (in volume per unit time and width, that is, m3 s−1 m−1) is given by
The threshold condition at which sediment motion is just about to begin can be obtained by setting m = 1. Then, from Eq. (5.7), threshold bed shear stress τ 0c can be determined, and thus, m is obtained as the ratio of applied bed shear stress to threshold bed shear stress as follows:
It is introduced into Eq. (5.8) and then
du Boys referred the first term within the parenthesis in right-hand side of Eq. (5.10) as a characteristic of sediment coefficient and denoted by χ. Thus, the equation becomes
The sediment coefficient χ was determined from the experimental data obtained by Schoklitsch (1914). According to Graf (1971), it is
Straub (1935) related χ with the particle size d (in mm) (0.125 < d < 4 mm) as
5.3.2 du Boys Type Equations
du Boys equation that is characterized by the excess bed shear stress is one of the classical equations of bed-load transport. Later, investigators have tried to put forward improved version of bed-load transport equations, known as du Boys type equations, based on excess bed shear stress. They are discussed below:
Shields (1936) obtained the threshold bed shear stress that had a value for which the extrapolated sediment flux (bed-load transport) became zero. Therefore, he basically studied the flow conditions corresponding to the bed-load transport rate greater than zero. He obtained an empirical equation of bed load as
where Θ and Θc are the Shields and threshold Shields parameters, respectively, q is the flow rate per unit width (= Uh), and U is the depth-averaged flow velocity. The Shields parameter is given by Θ = τ 0/(Δρgd) and Θc corresponds to τ 0c.
Meyer-Peter and Müller (1948) gave the following equation of bed load including the effects of particle roughness:
where C R is the total Chézy coefficient due to effective bed roughness k s, that is 18log(12 h/k s) or U/(R b S 0)0.5, R b is the hydraulic radius, and \( C^{\prime}_{\text{R}} \) is the Chézy coefficient due to particle roughness d 90, that is 18log(12 h/d 90). In Eq. (5.15), Meyer-Peter and Müller recommended the value of Θc = 0.047. Their formula corresponded well with the experimental data for coarse sands and gravels. The η C was reported to vary from 0.5 to 1 that corresponds to coarse sand and a small form drag. Considering k s ≈ d 90, the η C becomes unity; and the Meyer-Peter and Müller formula can be simplified to
Subsequently, Frijlink (1952) proposed a formula that can approximate Meyer-Peter and Müller formula, but it is not a du Boys type equation. It is
However, Chien (1954) showed that the Meyer-Peter and Müller formula can be replaced by
Further, Wong and Parker (2006) reanalyzed the experimental data used by Meyer-Peter and Müller and found a better fitting for the Meyer-Peter and Müller formula with the following equation:
For the high bed-load transport rate, Wilson (1966) put forward an empirical equation as
Chang et al. (1967) suggested that the bed-load transport can be determined from the following relationship:
where K b is a constant and ϕ is the angle of repose of the sediment. In the above, K t represents a constant defining the bed-load transport and can be determined using Fig. 5.3.
Variation of K t with (U/u *)ΘS 0 for different sediment sizes (Chang et al. 1967)
Ashida and Michiue (1972) analyzed micro-mechanical particle collision with the bed, but not considered the saltation. They obtained the following equation of bed-load transport intensity for the range of particle size 0.3 ≤ d ≤ 7 mm:
In the above, Ashida and Michiue recommended the value Θc = 0.05.
Fernandez Luque and van Beek (1976) used laboratory experimental data to suggest bed-load transport intensity as
They considered a range of Θc within 0.05 ≤ Θc ≤ 0.058 for 0.9 ≤ d ≤ 3.3 mm.
For gravel-bed rivers, Parker (1979) proposed
Smart (1984) measured bed-load transport rate in steep channels (0.03 ≤ S 0 ≤ 0.2) for the gravel sizes 2 ≤ d ≤ 10.5 mm. Based on his data and the data of Meyer-Peter and Müller, he proposed
The bed-load transport intensity equation derived by van Rijn (1984a) for 0.2 ≤ d ≤ 2 mm is
where D * is the particle parameter, that is d(Δg/υ 2)1/3, and υ is the kinematic viscosity of water.
Graf and Suszka (1987) gave a bed-load transport intensity formula for steep bed slopes as
Madsen (1991) recommended
where K b = 8/tanϕ for sliding and rolling sand particles, and K b = 9.5 for saltating sand particles in water. However, Niño and García (1998) proposed a similar equation with K b = 12/μ d for saltating particles. They determined a dynamic coefficient of friction μ d = 0.23.
Nielsen’s (1992) equation for sand and gravel (0.69 ≤ d ≤ 28.7 mm) transport is
Damgaard et al. (1997) conducted experiments for the wide variation of streamwise bed slope (–32° ≤ θ ≤ 32°; where θ is the streamwise bed angle with the horizontal). They introduced a correction factor f θ to Meyer-Peter and Müller formula as
where
where \(\Theta _{\text{c}\theta}\) is the threshold Shields parameter on streamwise bed slope.
Lajeunesse et al. (2010) suggested
5.3.3 Other Empirical Relationships Involving Bed Shear Stress
Kalinske (1947) emphasized on the near-bed turbulence that plays an important role in analyzing bed particle motion. The time-averaged bed-load transport rate q b was expressed as a product of three quantities: volume of a particle, number of particles in motion per unit area, and time-averaged particle velocity \( \bar{u}_{\text{b}} \). It is
where p n is the fraction of moving particles. The time-averaged particle velocity \( \bar{u}_{\text{b}} \) can be obtained as
where c 0 is the constant of proportionality, u d is the instantaneous flow velocity at the particle level, u cr is the threshold velocity (at the particle level) for the particle motion, and f(u d) is the frequency distribution of u d. The f(u d) is given by
where σ u is the standard deviation of u d. Assuming \( \tau_{{0{\text{c}}}} /\tau_{0} = \, (u_{\text{cr}} /\bar{u}_{\text{d}} )^{2} \), where \( \bar{u}_{\text{d}} \) is the time-averaged value of u d, the following functional relationship is obtained:
Using Eq. (5.36), Eq. (5.33) can be expressed a functional relationship as
Figure 5.4 shows this relationship.
Variation of τ 0c/τ 0 with q b/(u * d) (Kalinske 1947)
Frijlink (1952) formula, as already given by Eq. (5.17) that can approximate Meyer-Peter and Müller formula, was one that falls under the category to involve bed shear stress. Further, the bed-load transport formula that was widely used by Engelund and Hansen (1967) for sand transport in terms of bed shear stress is
In case of weak bed-load transport rate, Paintal (1971) obtained a bed-load transport formula for 1 ≤ d ≤ 25 mm as
The relationships proposed by Misri et al. (1984) to involve bed shear stress due to particle roughness are as follows:
where Θ′ is the Shields parameter due to particle roughness, that is \( \tau^{\prime}_{0} /(\Delta \rho gd) \), and \( \tau^{\prime}_{0} \) is the bed shear stress due to particle roughness.
On the other hand, Cheng (2002) gave a relationship for moderate bed-load transport rate as
The above equation yields results similar to those obtained from Meyer-Peter and Müller formula for moderate transport rate and Paintal formula for weak transport rate.
For high bed-load transport rate, Rickenmann (1991) reported that the particles transport like a sheet flow, when Θ > 0.4. Hanes (1986) suggested that under a sheet flow type transport, the intense bed-load transport can be approximated as
5.4 Discharge Concept for Bed-Load Transport
Schoklitsch (1934) was the pioneer to use discharge for the estimation of bed load. He used the data of Gilbert (1914) with his own to propose a bed-load transport rate formula for particle size 0.305 ≤ d ≤ 7.02 mm as
where g b is the bed-load transport rate in mass per unit time and width (kg s−1 m−1), d is in mm, and q c is the discharge per unit width corresponding to sediment threshold. Schoklitsch determined \( q_{\text{c}} = 1. 9 4 4\times 10^{{{-} 5}} /S_{0}^{4/3} \left( {{\text{m}}^{ 2} \,{\text{s}}^{{{-} 1}} } \right) \) by plotting a curve of bed-load transport rate versus bed slope. He then extrapolated the curve to zero transport rate (g b = 0) to determine the corresponding value of q as q c. Schoklitsch later modified the equation for d ≥ 6 mm as
He redefined the threshold discharge as \( q_{\text{c}} = h_{\text{c}}^{5/3} S_{0}^{0.5} /n = 0. 2 6\Delta ^{ 5/ 3} d^{ 1. 5} /S_{0}^{7/6} \left( {{\text{m}}^{ 3} \,{\text{s}}^{{{-} 1}} \,{\text{m}}^{{{-} 1}} } \right) \); where d is in m, n is the Manning coefficient, and h c is the flow depth corresponding to sediment threshold.
5.5 Velocity Concept for Bed-Load Transport
Donat (1929) used the Chézy equation in Eq. (5.11) and obtained the following equation of bed-load transport using average flow velocity:
where U cr is the average threshold velocity.
Barekyan (1962) proposed bed-load transport equation using average flow velocity as
Based on the stream power concept, Dou (1964) established an empirical equation of bed-load transport for sand as
5.6 Bedform Concept for Bed-Load Transport
Bedforms are discussed comprehensively in Chap. 8. Note that the bed load is the mode of sediment transport in lower flow regime when the bed is covered by ripples and/or dunes. The particles transport up the face of the mild slope of the ridge of a bedform and then drop down the steep slope being deposited on the downstream face and in the trough. As a result of sediment removal from the upstream and deposition on the downstream slope, the bedforms move downstream (Fig. 5.5). The bed-load transport can therefore be calculated directly from the movement of the bedforms. The continuity equation of sediment transport resulting in a change of bed level was given by Exner (1925) as
where η is the elevation of the sand-bed with respect to a horizontal reference, t is the time, x is the horizontal distance from a reference point, and ρ 0 is the porosity of sediment.
Bed-load transport with migration of bedforms
Assuming that the bedforms migrate with a velocity of U b being independent of time, the following transformation can be used:
By using Eq. (5.49), Eq. (5.48) yields
Integrating Eq. (5.50) yields
Assuming that the simplified bedforms are triangular shaped with an average height or pick-to-pick amplitude of a m and noting that the constant of integration A = 0 for the initial boundary condition, Eq. (5.51) becomes
The above equation can be used to determine the bed-load transport rate from the information of the bedform migration velocity and its height.
5.7 Probabilistic Concept for Bed-Load Transport
5.7.1 Einstein’s Approach
Einstein (1942, 1950) was the pioneer to develop a bed-load transport model based on the probabilistic concept. Primarily, he had two fundamental considerations that departed from the then earlier concepts. Firstly, the threshold criterion was avoided, as it is always a difficult proposition to define, if not impossible. Secondly, the transport of sediment particles was related to the velocity fluctuations instead of the time-averaged velocity. As a result of which, the beginning and the ceasing of sediment motion are expressed with probabilistic concept that relates to the ratio of submerged weight of the particle to instantaneous hydrodynamic lift induced to the particle. Some of the key issues toward the bed-load transport of sediment particles, as experimentally observed by Einstein, are as follows:
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A rigorous, but steady, exchange of sediment particles is prevalent between the bed surface and mobile bed-load layer.
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The particles travel along the bed in a series of quick steps. A particle does not, however, remain in motion continuously, but temporarily deposited on the bed after some steps with comparatively long intermediate resting periods.
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The average step, which is always the same and about 100 times the particle diameter, is simply proportional to the particle diameter and independent of the hydraulic condition and the transport rate.
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The transport rate is dependent on the average time period between two steps and the thickness of the mobile bed-load layer.
Einstein’s (1942, 1950) bed-load transport model was based on the aforementioned aspects. He first presented an empirical relationship in 1942, which was then replaced by a semitheoretical approach in 1950.
Dynamic equilibrium during the bed-load transport is established by exchanging the particles from the bed within the bed-load transport layer. Thus, the conservation of sediment mass is maintained balancing the number of particles removal (washed out by the flow) per unit time and area by those deposited (put down by the flow) per unit time and area.
Rate of deposition: The average traveling distance L x of a particle is defined by the distance that a particle travels from its starting point until it is deposited on the bed. The single step length of a particle having diameter d can be expressed as λ s d and for spherical particles, λ s = 100. As a particle travels a step by a brief jump (Fig. 5.6), it goes down on the bed at a location where a local lift force exceeds the submerged weight of the particle. Thus, the particle does not stop moving but travels for a second step and so on until it is temporarily deposited on the bed with comparatively long intermediate resting periods. In this way, the sediment particles passing a section (across the flow) per unit time deposit within a length of the channel that is equal to L x , regardless from where they have started to move. If g b represents the bed-load transport rate in dry weight and i bs is the fraction of bed load to be deposited of a given sediment size d, then the rate at which the particles of a size d are deposited per unit time and width is g b i bs. Therefore, the number of particles N d deposited per unit time and area is given by
where k 1 is the factor related to particle volume. The term within the parenthesis in the denominator defines the weight of a particle.
Sketch of a particle traveling along the bed in a series of steps
If p is the probability of lift force to exceed the submerged weight of the particles, then n(1 − p) particles deposit on the bed after traveling a step length, where n is the number of particles in motion. Thus, only np particles continue to move. Subsequently, the np(1 − p) more particles deposit and only np 2 particles remain in motion after traveling the second step length, and so on. In this way, all n particles deposit on the bed after elapsing some time. The average traveling distanceFootnote 1 can therefore be determined as
Using Eq. (5.54) into Eq. (5.53), the number of particles deposited per unit time and area becomes
Rate of removal: Depending on the availability of the particles and the flow conditions, a particle of a given size d is removed. If the fraction of sediment of a given size d to be removed is i br, then the number of such particles per unit area can be given by i br/(k 2 d 2); where k 2 is the factor related to the projected area of the particle. If p is the probability of a particle to begin to move at any location, then p/t e is the probability of removal per unit time. Here, t e is the time consumed by each exchange. Therefore, the number of particles removed N r per unit time and area is given by
The exchange time t e or the time for a particle to remove is assumed to be proportional to the time for a particle to fall a height of one diameter with a terminal velocity w s in a still water. Thus, it is
where k 3 is a constant for time scale. Using Eq. (5.57) into Eq. (5.56), the number of particles removed per unit time and area is
Equilibrium of bed-load transport: Sediment transport is in equilibrium if the rate of sediment deposition on the bed is balanced by the rate of sediment removal from the bed. Thus, equating Eqs. (5.55) and (5.58), the equation of dynamic equilibrium is obtained as
The bed-load transport equation is therefore obtained from Eq. (5.59) as
The parameter Φb* is called bed-load transport intensity, and the probability p of rate of sediment removal is given by
Probability determination: The probability p of a sediment particle removal is a function of the ratio of submerged weight F G of the particle to instantaneous hydrodynamic lift F L induced to the particle. The condition of removal is therefore p(F G/F L) < 1. It can therefore be expressed as
where C L is the lift coefficient and u δ′ is the effective instantaneous flow velocity at the edge of the viscous sublayer. Einstein and El-Samni (1949) observed that for uniform sediment particles, if the flow velocity at an elevation z = 0.35X is taken as the effective flow velocity u δ′ , the distribution of lift force fluctuations follows the Gaussian distribution with a standard deviation equaling half of the mean value and the lift coefficient as C L = 0.178 (a constant value). Here, X is the characteristic size of the bed sediment particles. The random function parameter η t (t) represents the lift force fluctuations with time t being distributed according to the normal error law, where the standard deviation η 0 is a universal constant having a value η 0 = 0.5. Using a nondimensional number η * that represents the lift force fluctuations, it can be written as η t = η 0 η * .
The effective instantaneous flow velocity u δ′ is expressed as
where κ is the von Kármán constant, Δ k is the apparent roughness (= k s/x k), x k is a correction factor, \( u^{\prime}_{*} \) is the shear velocity due to particle roughness, that is \( (gR^{\prime}_{\text{b}} S_{0} )^{0. 5} \hbox{,}\,R^{\prime}_{\text{b}} \) is the hydraulic radius due to particle roughness, and δ′ is the viscous sublayer thickness \( ( = { 11}. 6\upsilon/u^{\prime}_{*}) \). Einstein (1950) considered Nikuradse's equivalent sand roughness as k s = d 65. The correction factor x k can be obtained from the curve given by Einstein (1950) (Fig. 5.7), and thus, apparent roughness Δ k (= k s/x k) can be determined.
Variation of correction factor x k with k s/δ′, where Nikuradse's equivalent sand roughness k s = d 65
Hence, the lift force can be expressed as
The probability p of sediment removal is expressed as the probability of the ratio of the submerged weight F G to the instantaneous lift F L. The ratio has to be smaller than unity, that is
Using symbols, Eq. (5.65) can be reduced to
In the above, \( \Psi _{\text{b}}^{{\prime }} \) is known as flow intensity parameter due to particle roughness, which is reciprocal of the Shields parameter.
Einstein (1950) proposed two correction factors ξ and Y termed hiding factor and lift correction factor, respectively, which were determined experimentally (Figs. 5.8 and 5.9). Particles in the sediment mass smaller than X likely to hide between larger ones or within the viscous sublayer, as such the lift experienced by the smaller particles is to be corrected by a factor ξ −1. Einstein gave a curve for the hiding factor ξ as a function of d/X (see Fig. 5.8). The lift correction factor Y takes care of the change of lift coefficient in the sediment mass due to different roughness and is expressed as a function of k s/δ′ (see Fig. 5.9).
Variation of hiding factor ξ with d/X (Einstein 1950)
Variation of lift correction factor Y with k s/δ′ (Einstein 1950)
The fluctuations of lift force are caused by the velocity fluctuations. The lift force is always positive regardless of the velocity fluctuations to be positive or negative. Thus, the inequality for the lift force can be modified as
Therefore, the threshold condition for the bed particle motion is as follows:Footnote 2
It implies that between these two values of η *, no sediment transport takes place. Therefore, the probability p of sediment motion, as the lift force fluctuations follow Gaussian distribution, is
Using Eq. (5.69) into Eq. (5.60), Einstein’s bed-load transport equation is
Einstein experimentally obtained the values of the constants that are η 0 = 0.5, A * = 43.5 and B * = 1/7. The variation of Ψb* with Φb* from Eq. (5.70) is shown in Fig. 5.10.Footnote 3 The Ψb*(Φb*) curve matches well with the experimental data of Gilbert (1914), Meyer-Peter et al. (1934) and Chien and Wan (1999).
5.7.2 Empirical Refinement of Einstein Formula
Brown (1950) refined the Einstein formula by curve fitting and showed that the majority of flume data of Gilbert and Meyer-Peter et al. could be expressed by the following relationships:
For the sediment transport at higher Shields parameter (Ψb ≤ 1.92), Julien (1998) suggested
In the above equations, the parameter K f that appears in Rubey (1933) formula for terminal fall velocity was introduced by Brown to account for the effects of fall velocity of the sediment particles.
5.7.3 Modified Einstein’s Approach
The derivation of Einstein’s bed-load formula involves some oversimplified assumptions concerning the step length of a particle, exchange time, and probability of particle removal. Later, Wang et al. (2008) proposed a modification of the Einstein formula.
They argued that conceptually, the step length of a particle increases with the magnitude of the lift force exerted by the flow, but decreases with the submerged weight of the particle. The step length can thus be given by λ s d/Ψb. The rate of particle deposition g dep per unit area is obtained as
The number of particles per unit area can be estimated as 1/(k 2 d 2), and their total weight is k 1 ρ s gd 3/(k 2 d 2). If p is the probability of a particle to begin to move, sediment with a total weight of (k 1/k 2)ρ s gdp is removed from the bed per unit time and area.
Based on the finding by Hu and Hui (1996) that the upward velocity of a particle is approximated by a linear relationship of shear velocity u *, the time for a particle to be removed from the bed is inversely proportional to u *. Wang et al., therefore, suggested that the exchange time t e can be expressed as
The rate of particle removal g rem per unit area is obtained as
Equilibrium is reached when the rate of sediment removal from the bed equals the rate of deposition on the bed. Equating Eqs. (5.73) and (5.75) yields
Wang et al. assumed that a particle is removed only if the lift force exceeds the submerged weight of the particle, that is
where B′ is the coefficient. The probability p of particle removal is given by
Combining Eqs. (5.76) and (5.78) and introducing nonuniformity of sediments, the following relationship is obtained
Based on the measured data used by Einstein (1950), the values of the constants were determined as B′/η 0 = 0.07, η 0 = 0.5 and A * = 20. The variation of Ψb (read Ψb* as Ψb) with Φb* obtained from Eq. (5.79) is shown in Fig. 5.10. The Ψb(Φb*) curve departs to some extent from the experimental data plots of Gilbert (1914), Meyer-Peter et al. (1934) and Chien and Wan (1999), and the curve of Einstein (1950).
5.7.4 Engelund and Fredsøe’s Approach
Engelund and Fredsøe (1976) developed a bed-load transport model for the flow conditions close to the threshold of sediment motion. In this type of flow, the superficial bed particles are only transported. The model is based on the concept of Fernandez Luque and van Beek (1976), who hypothesized that the transported bed particles are to reduce the maximum fluid bed shear stress to its threshold value for the bed particle motion by exerting an average reaction force on the ambient fluid.
If the particles are transported with a mean velocity \( \bar{u}_{\text{b}} \), when they are in motion, the hydrodynamic drag force F D acting on a transported particle is given by
where C D is the drag coefficient and αu * is the flow velocity at the bed particle level. If the particle is at a distance of one to two particle diameters above the mean bed level, then α = 6–10.
The stabilizing resistance F R on the moving particle is
where μ d is the dynamic coefficient of friction for the bed particles.
At dynamic equilibrium, the hydrodynamic drag force is balanced by the stabilizing resistance (F D = F R). Thus, equating Eqs. (5.80) and (5.81) and then simplifying yield
where Θ0c is the threshold Shields parameter for a particle protruding from the bed surface. In fact, Θ0c differs from Θc, which is the conventional threshold Shields parameter for the initiation of particle motion in a compactly arranged bed. As a particle lying on the bed is easier to move than a particle within the bed, it implies that Θc > Θ0c. From the experimental data, Fernandez Luque and van Beek (1976) found Θ0c = 0.5Θc. Thus, Eq. (5.82) becomes
For a sandy bed, α ≈ 9.3. Engelund and Fredsøe treated sediment particles as spheres of diameter d, so that the number of spherical particles per unit area of bed surface is approximately 1/d 2. For a given flow intensity, the probability of the particles on the bed surface to move is p. Hence, the bed-load transport rate g b is
Using Eq. (5.83) into Eq. (5.84) yields
According to Bagnold, the applied bed shear stress τ 0 by the flow is composed of dispersive particle bed shear stress τ 0b and interfacial (intergranular) fluid bed shear stress τ 0f. Furthermore, he suggested that during bed-load transport, the interfacial fluid bed shear stress τ 0f equals the threshold bed shear stress τ 0c for the initiation of particle motion. This phenomenon is further discussed in the following section using a shear stress diagram. The estimation of probability p of surface bed particle removal is based on the assumption that only τ 0c of the applied bed shear stress τ 0 by the flow is transmitted directly to the immobile-bed particles as a skin frictional stress; whereas the residual fluid bed shear stress (τ 0 − τ 0c) is directly transmitted to the mobile particles as a drag induced bed shear τ 0b (= nF D) and indirectly transmitted to the bed by intermittent surface creep. Hence,
where n is the number of particles moving per unit area of bed surface. As F D = F R, inserting Eq. (5.81) into Eq. (5.86) leads to an estimation of p as
Using Eq. (5.87) into Eq. (5.85), the bed-load transport rate, expressed as bed-load transport intensity Φb, is obtained as follows:
For an intense bed-load transport rate Θ ≫ Θc, Eq. (5.88) can be approximated as Φb = 9.3Θ1.5/μ d.
5.8 Deterministic Concept for Bed-Load Transport
5.8.1 Bagnold’s Approach
Bagnold (1954) identified the limitation in Einstein’s approach by revealing an inconsistency toward the stability criterion of the bed during bed-load transport. Let it be discussed with an ideal example of the flow over a plane bed formed by uniform spherical sediment particles. This situation of a streambed leads to an equal exposure of all the bed particles to the flow; and hence, the stochastic variations due to turbulence can be ignored. When the applied bed shear stress exceeds its threshold value for the particle motion, all particles in the uppermost layer are in motion simultaneously and removed by the flow. As a result, the next layer of particles comes in contact with the flow and is subsequently also removed and so on. In this way, all the subsequent underlying layers of particles are removed and equilibrium toward a stable bed never exits as long as the bed shear stress exceeds the threshold value. Bagnold, however, argued this inconsistency by decomposing the applied shear stress τ by the flow into the dispersive particle shear stress τ b that is the shear stress transmitted due to exchange of momentum for the collision of moving particles and the interfacial fluid shear stress τ f that is the shear stress transmitted by the interfacial fluid (Fig. 5.11). The background of the idea was that the sediment-laden flows are a result of shear that includes shear between the layers of the particles and that between the sediment and the surrounding fluid. An applied bed shear stress τ 0 induced by the fluid tractive force that acts in the streamwise direction to sustain such a shear is developed by the gravity in the streamwise direction (Fig. 5.11).
Decomposition of applied shear stress into dispersive particle shear stress and interfacial fluid shear stress
The bed shear stress decomposition is therefore
Bagnold further argued that with the removal of a layer of particles, a dispersive pressure on the subsequent layer of particles is developed as a stabilizing force. The number of layers to be removed is governed by the interfacial fluid bed shear stress τ 0f until it equals the threshold bed shear stress τ 0c that acts on the first immobile layer. The applied bed shear stress τ 0 induced by the fluid tractive force is therefore greater than the threshold bed shear stress τ 0c. Hence, the τ 0 is partially transmitted to the moving particle as τ 0b and rest to the immobile bed as τ 0c.
Bagnold (1956) assumed that the saltation is the primary mode of bed-load transport. The momentum component in the streamwise direction when a saltating particle drops down to the bed is m G u 1. Here, m G is the submerged mass of the particle, and u 1 is the velocity component of the particle in the streamwise direction when it collides with the bed. The particle at the same time is acted on by a force from the bed particles producing a momentum component m G(–u 0) opposite to the streamwise direction. Here, –u 0 is the reduction of particle velocity component in the streamwise direction due to collision with the bed particles. To maintain the saltation of a particle, the flowing fluid therefore must act on the particle to provide a momentum component m G u 0 in the time interval Δ t between successive collisions of the saltating particle with the bed particles.
Therefore, the fluid flow exerts a force on the particle with a component in the streamwise direction as
If \( \bar{u}_{\text{b}} \) is the average velocity of the particle, then the work done per unit time by the flowing fluid on the particle is \( F_{x} \bar{u}_{\text{b}} \). Also, the energy consumed per unit time by the flow is \( F_{\text{G}} \bar{u}_{\text{b}} \;{ \tan}\phi_{\text{d}} \); where ϕ d is the dynamic frictional angle. Equating them and using Eq. (5.90) yield
The vertical distance z n is the location at which the particle is acted upon by a force F x to accelerate the particle from u 1 − u 0 to u 0. If the flow velocity at z n is \( \bar{u}_{\text{n}} \), then the \( u_{\text{r}} ( = \bar{u}_{\text{n}} - \bar{u}_{\text{b}} ) \) exists at an elevation z = z n. As a number of particles are in motion along the bed during bed-load transport, then
where τ bn is the shear stress for maintaining sediment motion at z = z n. So, the bed-load transport rate g bs (in submerged weight per unit time and width) is
Using a coefficient a, the shear stress τ bn is given by
The flow velocity is considered to follow the logarithmic law in the flow region z ≥ z n, and the velocity at z = 0.4h is considered to be equal to the depth-averaged flow velocity U. Then,
Using Eqs. (5.94) and (5.95) into Eq. (5.93) yields
Determination of a: Bagnold argued a = 0 at the threshold condition and a → 1 for the high flow velocity corresponding to intense bed-load transport. It is thus given by
Determination of u r: The hydrodynamic drag force exerted by the flow on a particle is balanced by the bed resistance. It can be expressed as
where C Dx is the drag coefficient for the drag force acting in the streamwise direction.
When a particle falls with a terminal fall velocity w s in a still fluid, the drag force F Dz acting on the particle is balanced by the submerged weight F G of the particle. Then,
where C Dz is the drag coefficient for a settling particle. From Eqs. (5.98) and (5.99), the following relationship is obtained:
It was found from the measured data that C Dx ≈ C Dz and tan0.5 ϕ d ≈ 1. Therefore, Eq. (5.100) becomes
Determination of z n: In the absence of any bedforms, the average elevation of the saltating particles is proportional to their diameter. Thus,
where K 1 is a coefficient. In laboratory experiments, K 1 = 0.4 was found by Francis (1973); but in rivers, it becomes 2.8 for sands and 7.3–9.1 for gravels (Bagnold 1977).
Equation of bed-load transport rate: Using Eqs. (5.97), (5.101) and (5.102) into Eq. (5.96), the equation of bed-load transport rate obtained by Bagnold in terms of submerged weight is given by
Later, Bagnold (1966) simplified the analysis introducing an efficiency factor e b for the bed-load transport. He balanced the available fraction of flow energy per unit time and area (that is the stream power) τ 0 Ue b with the work done required to move the bed-load particles \( F_{\text{G}} \bar{u}_{\text{b}} { \tan }\phi_{\text{d}} ( = g_{\text{bs}} { \tan }\phi_{\text{d}} ) \). Thus, equation of bed-load transport rate is
The variation of bed-load transport efficiency e b with U for different particle sizes d given by Bagnold is shown in Fig. 5.12. The prediction of e b is possible for d = 0.01–1 mm.
Variation of bed-load transport efficiency e b with U for different particle sizes d
5.8.2 Yalin’s Approach
Yalin (1977) proposed a bed-load transport model based on the analysis of forces acting on a sediment particle. The equations of force acting on a moving sediment particle in the streamwise and normal directions are
where F x and F z are the force components of flow acting on a particle in the streamwise and normal directions, respectively, and u b and w b are the velocity components of a sediment particle in the streamwise and normal directions, respectively. The force components F x and F z are given by
where u d is the instantaneous streamwise flow velocity at the particle level.
A particle detaches from the bed by the action of hydrodynamic lift force F L. The difference F L − F G > 0 near the bed reduces with distance from the bed and becomes F L − F G = 0 at an elevation where the particle reaches its maximum vertical velocity component [w b]max. The [w b]max can be determined from the following equation:
Equation (5.107) represents the initial condition of Eq. (5.105b). To solve these equations, Yalin made the assumptions: (1) The F L/F G ratio decreases with z/d according to the exponential law, that is F L/F G ~ exp(–z/d), (2) the drag coefficients C Dx and C Dz are constants, and (3) the nondimensional flow velocity u/u * in the vicinity of the bed is constant.
As a result, he obtained an expression for u b and then its average value \( \bar{u}_{\text{b}} \) over the time when the particle is in motion. It is given by
where \( a_{ 1} = { 2}. 4 5{\Theta}_{\text{c}}^{ 0. 5} /s^{0. 4} \) and C 1 is a constant to be determined. He determined the submerged weight of the bed-load transport per unit area W s from the dimensional analysis. It follows
where Θ = R b S 0/(Δd), R b is the hydraulic radius, and R *d = u * d/υ. Equation (5.109) can be rewritten as
At the threshold of particle motion, Θ(W s = 0) = Θc, and thus
Equations (5.110) and (5.111) are combined to
Yalin assumed that the left-hand side of Eq. (5.112) is linearly proportional to nondimensional excess bed shear stress. Hence,
where C 2 is a constant to be determined.
Substituting Eqs. (5.108) and (5.113) into Eqs. (5.105a, b) and determining the constants from the measured data, the bed-load transport rate g b in weight per unit time and width is given by \( g_{\text{b}} = (s/\Delta )g_{{{\text{bs}}}} = (s/\Delta )W_{\text{s}} \bar{u}_{\text{b}} \). Thus, the bed-load equation of Yalin is
Equation (5.114) can be expressed in nondimensional form as
For initiation of bed-load transport, Θ → Θc and a 1[(Θ/Θc) – 1] ≈ 0. Hence, one can write
The bed-load transport rate equation, Eq. (5.115), becomes
For high intensity bed-load transport rate, Θ ≫ Θc and (Θ − Θc) → ∞. Hence, it is given by
The bed-load transport rate equation, Eq. (5.115), then becomes
5.9 Equal Mobility Concept for Bed-Load Transport
Parker et al. (1982) developed a concept of equal mobility assuming that the bed-load transport of gravels can be accomplished through mobility of the particles exposed to the flow. The participation of the underneath particles in bed-load transport can only be possible up to the extent of the degradations that can result in an exposure of those particles to the flow. They referred coarser surface layer with bed-load transport as pavement; however, it is different from an armor layer. In this concept, the particle size distribution of bed load is approximated by that of underneath particles for all flows capable of mobilizing the majority of available gravel sizes.
Based on the equal mobility concept, Parker et al. (1982) developed a functional relationship between a bed-load transport function \( \Phi _{{{\text{b}}i}}^{ + } \) and a bed shear stress parameter \( \Theta _{i}^{ + } \) for a gravel size of d i . The \( \Phi _{{{\text{b}}i}}^{ + } \) and \( \Theta _{i}^{ + } \) are given by
where g bi is the bed-load transport rate per unit width for the fractional particle size d i , p i is the fraction by weight of size d i , and \( \tau_{0i}^{ + } = \, 0.0 8 7 5\left( {d_{ 50} /d_{i} } \right) \).
Due to equal mobility of all sizes, a specific particle size, termed subpavement size and denoted by d 50, is used to characterize the bed-load transport. Based on the field data of gravel-bed streams with sizes from 18 to 28 mm, Parker et al. (1982) proposed
where \( \Theta _{50}^{ + } \) is the bed shear stress parameter defined in Eq. (5.120b) corresponding to subpavement size d 50.
5.10 Sediment Pickup Function
Pickup rate, defined as volume rate of sediment removal per unit area, was studied by different investigators. Although the mode of bed-load transport according to the concept of pickup is not clear, there are three concepts of sediment pickup. As already discussed, Einstein (1950) hypothesized that after a period of rest, a sediment particle can only be picked up. The period of rest is longer than that of pickup. In his hypothesis, the total distance between two successive periods of rest can be traveled by a particle by performing several brief jumps. A particle covers an average step length of 100d by performing a jump. However, the pickup definition of Yalin (1977) is different from that of Einstein. Yalin hypothesized that a particle can be picked up when it detaches the bed surface to perform a jump. It implies that a jump by a particle involves a pickup and then deposition. According to de Ruiter (1982, 1983), the period of pickup equals the time period required to travel (from rest) by a particle over a distance of its half the diameter.
The approach of Einstein (1950) was stochastic. He assumed that a sediment particle is lifted when the instantaneous lift having a Gaussian distribution exceeds the submerged weight of the particle. His sediment pickup formula is
where α p is the coefficient and p is pickup or removal probability, that is the time fraction during which a sediment particle is picked up by the flow, which has already been discussed in Einstien’s approach.
Fernandez Luque (1974) used experimental data for 0.9 ≤ d ≤ 1.8 mm and proposed
According to Yalin (1977), the period of pickup is proportional to the ratio of the particle diameter to shear velocity. Using a stochastic approach, he obtained a sediment pickup formula as
Based on experimental data (3 ≤ d ≤ 13.5 mm), Nakagawa and Tsujimoto (1980) suggested
They recommended α p = 0.02 for spherical particles.
According to de Ruiter (1982, 1983), the pickup time period was found to be much smaller than that of instantaneous bed shear stress exceeding its threshold value. Based on stochastic approach, he proposed
where p p is the pickup probability function and σ 0 is the standard deviation of instantaneous bed shear stress. The value of coefficient α p, recommended by de Ruiter, is 0.016.
van Rijn (1984b) conducted experiments with different sand sizes (0.13 ≤ d ≤ 1.5 mm) and proposed an empirical equation of pickup function as
Dey and Debnath (2001) performed experiments with various uniform and nonuniform sand sizes (0.24 ≤ d ≤ 1.55 mm). Considering the effects of sediment nonuniformity, they proposed
where σ g is the geometric standard deviation of particle size distribution.
5.11 Saltation
5.11.1 Characteristics of Saltation
When the bed shear stress just exceeds the threshold value for the initiation of particle motion, the particles roll and/or slide in contact with the bed. As the bed shear stress increases further, the particles move along the bed by a series of short jumps with approximately same step lengths. This phenomenon is called saltation. The saltation of a particle is governed by the hydrodynamic drag and lift forces and also the bed roughness. Due to the gravity, the particle begins to descend and returns to the bed when it is lifted by the hydrodynamic force to a certain height. In this way, the particle undergoes a saltation process as shown in Fig. 5.13. Subsequently, a new step of saltation may begin as a result of an impact against the bed and the lift force. According to the laboratory experimental observations by Francis (1973) and Abbott and Francis (1977), the characteristics of a saltating particle are described as follows:
Schematic of a particle saltation
The particle transport in saltation mode is limited to a maximum height of about ten times the particle diameter. The particle motion is dominated by the gravitational force, although it can be set off by the impulses of velocity fluctuations (near-bed turbulence agitations) during bursting events or by the effects of wall shear flow that a particle experiences a shear lift due to the velocity gradient in the vicinity of the bed. The hydrodynamic pressure and the viscous skin friction can also be the sources to provide momentum to the particles. In the rising stage of particle trajectory, the vertical component of the drag force and the gravitational force are together directed downwards; while in the falling stage of particle trajectory, the vertical component of drag force being directed upwards opposes the gravitational force. The lift force is always directed upwards provided the particle velocity to lag behind the fluid velocity at the saltating particle.
It is observed that some particles move in the form of series of saltations. It means that after the particles coming back to the bed performing a saltation, they immediately perform next saltation without any pause on the bed. It is obvious that the lift force is the main cause of lifting up the particles from the bed. However, the effects of the bed impact force by no means can be neglected. As a saltating particle strikes the bed particles, it may either ricochet off the bed particles or impact against them. During the impact of the particles with the bed particles, majority of the momentum, that they possess, is transferred to the bed particles in a succession of horizontal impulses. It may cause to initiate a rolling motion of the surface particles, termed surface creep. However, a saltating particle may cease motion, if it falls within one of the local depressions on the bed surface.
5.11.2 Particle Trajectory and Characteristic Parameters (van Rijn’s Approach)
The forces acting on a saltating particle, as shown in Fig. 5.13, were analyzed by van Rijn (1984a). In fact, he analyzed the problem deterministically in the context of estimation of bed-load transport rate. In Fig. 5.13, the forces are the submerged weight of the particle F G acting downwards and the hydrodynamic force components in the form of drag and lift. The direction of drag force F D is opposite to the direction of the particle velocity v r relative to the fluid flow; whereas the lift force is in the normal direction.
Equations of motion: The trajectory of a saltating particle can be determined by solving the equations of motion. Assuming a spherical saltating particle and the forces due to fluid accelerations to be of a second order, the equations of motion, according to White and Schultz (1977), can be written as
where m a is the particle mass plus added fluid mass, v r is the particle velocity relative to the fluid flow, that is \( [(\bar{u} - \dot{x})^{2} + \dot{z}^{2} ]^{0.5} \hbox{,}\,\bar{u} \) is the local time-averaged flow velocity in x-direction, \( \dot{x} \) and \( \dot{z} \) are the streamwise and vertical velocities of particle, respectively, and \( {\ddot{\it{x}}} \) and \( {\ddot{\it{z}}} \) are the streamwise and vertical accelerations of particle, respectively.
The added fluid mass or virtual mass is the inertia added to a system. An accelerating or decelerating particle is to move some volume of surrounding fluid, as it moves through it, since the particle and fluid cannot occupy the same physical space simultaneously. For simplicity, this can be assumed as some volume of fluid moving with the particle, though in reality all the fluid is accelerated to various degrees. Therefore, the total mass of a spherical particle can be given by
where α m is the added mass coefficient. Assuming a potential flow, the added mass of a sphere is obtained as the half of the fluid mass displaced by the sphere. However, in real fluid flow, the flow is separated from the sphere and α m may be different from that for a potential flow. The value α m = 0.5 was considered by van Rijn.
The drag force F D, which is resulted from the pressure and the viscous skin frictional effects, can be expressed as
The drag coefficient C D can be determined from the empirical expressions given by Morsi and Alexander (1972).
The lift on a particle in the wall shear layer of flow is induced by two ways. They are due to (1) velocity gradient in the shear layer and (2) spinning motion of the particle as a Magnus effect. For a sphere moving in a viscous fluid flow, Saffman (1968) determined the lift F Ls due to shear as
The Magnus lift F Lm due to spinning motion in a viscous fluid flow obtained by Rubinow and Keller (1961) is given by
where ω is the angular velocity of the particle. The total lift force F L is therefore
The submerged weight of the spherical particle is given by Eq. (4.9). The velocity distribution in the wall shear layer is assumed to follow the logarithmic law given by Eq. (4.27), where the zero-velocity level can be considered as z 0 = 0.11(υ/u *) + 0.03k s.
Boundary conditions and solution scheme: The virtual bed level is assumed to be at 0.25d below the top of the bed particles, as shown in Fig. 5.13. The initial position of the particle lying on closely packed bed particles is 0.6d above the virtual bed level. Here, d is the representative particle size, assumed to be d 50. According to the experimental observations by Francis (1973) and Abbott and Francis (1977), \( \dot{x} = \dot{z} = 2u_{ * } \). Equations (5.129a, b) were first transformedFootnote 4 by van Rijn to a system of ordinary simultaneous differential equations of the first order. Then, he solved them by a numerical method known as automatic step-change differential equation solver. The characteristic parameters of saltating particles were computed for the range u * = 0.04–0.14 m s−1 and d 50 = 0.1–2 mm. He assumed k s = 2d 50 and calibrated C L as C L(R *d ≤ 5) = 1.6, C L(5 < R *d < 70) = 1.6–20 varying linearly, and C L(R *d ≥ 70) = 20.
Characteristic parameters of saltating particles: The saltation length λ b and height h s were first computed. Then, they are empirically correlated with the nondimensional particle parameter D * [= d(Δg/υ 2)1/3] and the nondimensional excess bed shear stress (Θ/Θc) – 1 as follows (van Rijn 1984a):
The above equations suggest that the saltation length and height increase with an increase in particle parameter and excess bed shear stress, but independent of flow depth. Experimental observations by Poreh et al. (1970) on saltation length and Williams (1970) on saltation height confirmed that λ b ≈ 8d 50 for d 50 = 1.35 mm and h s = 5–40d 50 for d 50 = 1.9 mm. The results obtained from Eqs. (5.135a, b) are more or less in conformity with these experimental results. Besides van Rijn’s Eqs. (5.135a, b), Table 5.1 furnishes the formulas of saltation length λ b and height h s proposed by different investigators. It is obvious that their results are quite varying from one another.
For a saltating particle, van Rijn (1984a) computed the mean velocity \( \bar{u}_{\text{b}} \) as a function of nondimensional particle parameter and nondimensional bed shear stress as
Further, van Rijn (1984a) approximated Eq. (5.136) in a simpler form as
Besides Eqs. (5.136) and (5.137), Table 5.2 furnishes the formulas of mean velocity \( \bar{u}_{\text{b}} \) of a saltating particle given by different investigators.
Bed-load transport rate: van Rijn (1984a) defined the bed-load transport rate q b as a product of the particle velocity \( \bar{u}_{\text{b}} \), the volumetric concentration C of transported particles, and the saltation height h s. It is therefore given by
Note that if the saltation height h s is replaced by the thickness δ b of bed-load transport layer, then Eq. (5.138) becomes Eq. (5.3). Analysis of the experimental data by van Rijn (1981) showed that the bed-load concentration C (by volume) can be represented by
where C 0 is the maximum bed-load concentration. He determined C 0 = 0.65. It is interesting to note that the bed-load concentration C is inversely proportional to the nondimensional particle parameter and directly proportional to the nondimensional excess bed shear stress. Using Eqs. (5.135b), (5.137) and (5.139) into Eq. (5.138), van Rijn (1984a) obtained a bed-load transport equation, which has already been given as Eq. (5.26) as a du Boys type equation.
5.12 Fractional Bed Load of Nonuniform Sediments
Natural streams are typically made up of nonuniform sediment mixtures, whose transport phenomenon is therefore of immense importance. Unlike the transport of uniform sediment, the problems related to fractional nonuniform sediment transport are rather complex, especially when the consideration is given to the shelter–exposure interactions of bed particles of different sizes. Einstein (1950) was the pioneer to develop fractional transport rate of nonuniform sediments. Since then, Ashida and Michiue (1972), Parker et al. (1982), Patel and Ranga Raju (1996), Wu et al. (2000), and some other investigators put forward different methods to calculate the fractional bed-load transport rate of nonuniform sediments. Besides, Hsu and Holly (1992) proposed a method to determine the size fractional composition of nonuniform bed load aided by probability and availability of mobile sediments. The probabilistic approach by Einstein (1950) and the equal mobility approach by Parker et al. (1982) taking into the fractional transport rate have already been discussed. Here, some other important methods are introduced.
Ashida and Michiue’s (1972) bed-load transport formula for uniform sediment is given by Eq. (5.22). This formula was found to overestimate the individual size fractions of bed-load transport rate when compared with the experimental data for nonuniform sediment mixtures. They recommended the values of Shields parameters to be corrected for the fractional size of sediment. Thus, the equation of fractional bed-load transport intensity is given by
where Θ i and Θci are the Shields parameter and threshold Shields parameter corresponding to a fraction p i of size d i , respectively, and i b is the fraction of bed-load transport. Equation (5.140) thus can be used to compute total bed-load transport for the entire range of particle size distribution of the bed sediment.Footnote 5
Hsu and Holly’s (1992) method begins with the determination of the size distribution of transported sediment and then ends with the estimation of bed-load transport rate. The each fractional size d i in the transported sediment is hypothesized to be proportional to the joint probability of its mobility under the prevailing hydraulic condition and its availability on the bed surface within the active layer. If the flow velocity fluctuations follow the Gaussian probability distribution, the probability p ri of removal of size d i is derived as
where \( \tilde{u}_{\text{d}} = u^{\prime } /\bar{u}_{\text{d}} \hbox{,}\,u^{\prime } \) is the fluctuations of instantaneous streamwise velocity at the bed particle level, \( \bar{u}_{\text{d}} \) is the time-averaged streamwise velocity at the bed particle level, \( \sigma_{{\tilde{u}_{\text{d}} }} \) is the standard deviation of \( \tilde{u}_{\text{d}} \), and u cri is the near-bed threshold velocity for the initiation of motion of sediment size d i . They employed Qin’s (1980) formula given by Eq. (4.160) for the computation of u cri for a given size d i , and used \( \sigma_{{\tilde{u}_{\text{d}} }} = \, 0. 2 \), as suggested by Yen et al. (1988).
The availability of fractional size d i is equivalent to its fractional representation β bi on the bed surface within the active layer. Therefore, the fraction p i of size d i in the transported sediment is
In this way, the particle size distribution of the transported sediment in bed load is determined. Then, the weighted mean sediment size d m and mean near-bed threshold velocity u crm are estimated. For the estimation of bed-load transport rate for all fractional sizes, Hsu and Holly used Shamov’s formula (Zhang et al. 1989):
Hsu and Holly originally expressed g b in mass of bed-load transport rate per unit width. The right-hand side of Eq. (5.143) is multiplied by g (acceleration due to gravity) to convert the unit to N m−1 s−1.
Patel and Ranga Raju (1996) expressed the fractional bed-load transport intensity Φbi as a function of Θci ξ b. In fact, they corrected the threshold Shields parameter Θci corresponding to fractional size d i by a factor ξ b, termed hiding–exposure correction factor. The estimation of ξ b is as follows:
where M is the Kramer’s uniformity parameter, \( \tau^{\prime}_{0i} = \rho gR^{\prime}_{\text{b}} S_{0} \hbox{,}\,R^{\prime}_{\text{b}} = \, (Un^{\prime } /S_{0}^{0.5} )^{1.5} \hbox{,}n^{\prime } = d_{65}^{1/6} / 2 4\hbox{,} \;{\text{and}}\;\tau_{{0{\text{cg}}}} \) is the threshold bed shear stress for the geometric mean size d g [≈ (d 84.1 d 15.9)0.5] of the nonuniform sediment mixture as per Shields. The variation of fractional bed-load transport intensity Φbi with Θci ξ b obtained by Patel and Ranga Raju (1996) is illustrated in Fig. 5.14.
Curve to estimate fractional bed-load transport rate (Patel and Ranga Raju 1996)
Wu et al. (2000) presented a relationship of fractional bed-load transport intensity Φbi as a function of nondimensional excess particle bed shear stress. It is
where \( n^{\prime } = d_{50}^{1/6} /20\hbox{,}\,\tau_{{0{\text{c}}i}} \) is the threshold bed shear stress corresponding to size d i , τ 0 = ρgR b S 0, R b is the total hydraulic radius, and n is the Manning roughness coefficient \( ( = R_{\text{b}}^{2/3} S_{0}^{0.5} /U) \).
Note that one can use Meyer-Peter and Müller (1948) formula for the estimation of fractional bed-load transport intensity Φbi (van Rijn 1993). The threshold bed shear stress is corrected to account for the nonuniformity effect as ξ i Θc. Then,
In the above, the correction factor ξ i given by Egiazaroff (1965) is
5.13 Sediment Sorting and Streambed Armoring
The time-dependent transport rate of nonuniform sediment mixture is a complicated process due to sorting of a sediment bed in addition to sediment diffusion as suspension. In a sediment mixture, the resistance to an individual particle motion depends upon particle size and shape, as well as sheltering and exposure to the flow. Sediment sorting is defined as a selective transport of different fractional sizes of sediment particles. When the sediment transport rate of a bed exceeds the rate of sediment supply by the approaching flow, the sediment bed starts to degrade. Active layer refers to the surface layer of the sediment bed from which the sediment can be entrained to the flow. Because of the nonuniformity of the sediment, typically, exposed finer particles are transported easily at a faster rate than the coarser ones, and the remaining bed particles are being coarsened. Thus, the size of particles’ sorting takes place. The weakly entrained or unentrained coarse particles tend to accumulate in the surface layer, forming a band of coarser particles. Gradually, this coarsening process stops until a layer of coarse particles is completely developed to cover the streambed protecting the underneath finer sediment particles from being transported. Once this process is completed, the streambed is called armored and the layer of coarsest particles is called the armor layer (Fig. 5.15).
Definition sketch of streambed armoring
Due to variable nature of flow condition of a natural streambed, typically one or more than one layers of armor particles are required to protect the underneath finer sediment particles (Fig. 5.16). Borah (1989) and Froehlich (1995) reported that the natural armor-layer thickness is one to three times the armor-particle sizes. However, the thickness, porosity, and particle size distribution of an armor layer vary with flow and bed evolution. Importantly, fine sediment can still be winnowed at a very feeble rate through the interstices of armored particles.
Natural streambed armoring. Photograph by the author
Borah et al. (1982) considered an active layer to be homogeneous within itself at any given time and proposed estimation for the thickness of the active layer t a from the volumetric consideration as
where M is the fraction of the size d M or larger than d M , that cannot be transported by the flow, d M is the size for the fraction M, and ρ 0M is the porosity for the fraction M. Thus, the fractional size d M and larger sizes contribute to an armor layer.
In an active layer, the rate of transport from the bed surface decreases with time but does not truly go to zero even after a long time (several days). It implies that the development of an armor layer is an asymptotic process. When the bed shear stress increases, the finer particles are transported and coarser ones stay in place. Eventually, an upper limiting condition of the streambed is reached, which is called the threshold armoring condition. The corresponding bed shear stress is used to define the threshold bed shear stress for armoring τ 0ca. Hence, a sediment mixture has a range of bed shear stress over which its bed surface can be armored. The armor layer is thus now formed by the near-coarsest particles d 90 or even coarser than d 90, which are found in a particle size distribution curve of nonuniform sediment. However, for a higher flow rate, when τ 0 > τ 0ca, the armor layer becomes unstable and subsequently is destroyed. Correia and Graf (1988) suggested the median size of armor particles: d 50a ≈ 1.4d 50 and d 50a ≤ 0.6d 90. Raudkivi (1990) gave an empirical relationship for the estimation of stability of particles in an armor layer as
where Θca is the threshold Shields parameter for the armoring particles and d 50a|max is the maximum size of the armoring particles, being determined by extrapolating the particle size distribution curve on the basis of last two or three points. Importantly, no armoring takes place for uniform sediments.
In case of a nonuniform sediment sample with a mixture of fine and large particles (for example, sand and gravel), Chin et al. (1994) observed that the stability of individual large particles and their number in the bed govern the process of formation of a stable armor layer. The removal of finer particles from the bed surface exposes individual large particles. As a large particle is exposed considerably to the flow, it leads to the formation of an erosion pit in the front and a deposition of finer particles at the rear. The large particle may then slide into the erosion pit, reducing its exposure to the flow and becoming more stable. Medium and relatively coarse particles may also be accumulated within the scour pit and finer particles may hide behind and in between the larger ones. Gradually, this rearrangement of the surface particles leads to the formation of clusters of particles of various sizes. A cluster may slowly collapse with an erosion of the bed at its periphery. The anchor large particle may then be moved downstream to another stable position; and the process of cluster formation may be repeated. Thus, the formation of an armor layer in a nonuniform sediment mixture with fine and large particles is a continuous process involving formation and collapse of clusters. In Sect. 8.6, formation of cluster is further discussed.
5.14 Sediment Entrainment Probability to Bed Load
Determination of sediment entrainment probability to bed load is an essential prerequisite in developing a probabilistic theory of bed-load transport. Einstein (1942) laid the foundation of the probabilistic concept to study the bed-load transport, in which the probability of sediment removal was introduced. The most innovative development was due to Einstein (1950) to introduce a formula for the probability of sediment transport [see Eq. (5.70)]. It was based on the probability of hydrodynamic lift induced by the fluctuating velocity to exceed submerged particle weight, using the Gaussian distribution, as already discussed. The probability function p that is given by Eq. (5.69) can be written in a simplified form as
Subsequent investigations by other researchers viewed the probability of sediment removal in different ways and put forward formulation for probability in terms of entrainment or pickup probability function. The entrainment probability function is a function of Shields parameter Θ. Engelund and Fredsøe (1976) gave an empirical formula for the entrainment probability function [see Eq. (5.87)] by using experimental data of Guy et al. (1966) and Fernandez Luque (1974). The formula was subsequently modified by Fredsøe and Deigaard (1992) in the form
However, following Einstein’s concept of bed-load transport, Cheng and Chiew (1998) obtained an expression for the entrainment probability function, based on the assumption of the Gaussian distribution for the streamwise velocity fluctuations. They expressed the sediment entrainment probability in hydraulically rough flow regime as
where u d is the instantaneous near-bed velocity and B = [4Δgd/(3C L)]0.5. They estimated the time-averaged near-bed velocity \( \bar{u}_{\text{d}} \), using the logarithmic law and fixing the zero-plane displacement level at 0.25d and the zero-velocity level z 0 at k s/30 below the top of the closely packed bed particles. Here, k s was considered as 2d. They assumed that a particle placed in an interstice between two bed particles is about to move. In this way, they estimated \( \bar{u}_{\text{d}} = { 5}. 5 2u_{*} \) acting on the particle in an initial position at z = 0.6d. Quoting Kironoto and Graf (1994), Cheng and Chiew (1998) assumed \( \sqrt {\overline{{u^{\prime 2} }} } = { 2}u_{*} \) and finally obtained the entrainment probability as
Cheng and Chiew (1998) selected a value of C L = 0.25 to fit Eq. (5.153) with the previous experimental data, as shown in Fig. 5.17.
Variation of Shields parameter Θ with probability p of sediment entrainment (Bose and Dey 2013). The Θ(p) curves obtained from the approaches given by Einstein (1950), Fredsøe and Deigaard (1992) and Cheng and Chiew (1998), and the experimental data of Guy et al. (1966) and Fernandez Luque (1974) are shown for the comparison
Later, Wu and Lin (2002) noted that since only positive fluctuations in the streamwise velocity could cause entrainment of bed particles, a log-normal distribution could be better suited to derive an expression for the entrainment probability. They therefore modified the concept of entrainment probability as
Wu and Lin (2002) finally expressed the entrainment probability as
Bose and Dey (2013) argued that the Gaussian and the log-normal distributions primarily occur when there is additive and multiplicative accumulation of errors. This is, however, not the case of turbulent velocity fluctuations in open channel flow. Bose–Dey universal probability theory (see Sect. 3.17.1), on the other hand, gave the Gram–Charlier series expansion of the probability densities based on the two-sided exponential or Laplace distribution. They explained that the probability density function (henceforth pdf) \( p_{{\hat{u}}} (\hat{u}) \) for the nondimensional streamwise velocity fluctuations \( \hat{u} \) can be given by
Dey et al. (2012) obtained that the coefficients C 10 and C 30 are of the order of 0.001; while C 20 ≈ –0.5 and C 40 ≈ 0.6. Thus, it was assumed that C 20 ≈ –0.5 and the rest of the coefficients are effectively negligible due to their smallness or division by a large number, such as 384. Then, Eq. (5.156) reduces to
The instantaneous near-bed streamwise velocity u d, which can be decomposed as \( u_{\text{d}} = \bar{u}_{\text{d}} + u^{\prime } \), is the cause of an entrainment of particles lying on the bed. Wu and Lin (2002), following Nelson et al. (1995), argued that the entrainment is only possible when the velocity fluctuations u′ > 0, for which the pdf according to Eq. (5.157) becomes the one-sided exponential based Gram–Charlier series. Therefore,
where p u′(u′) is the pdf for u′. It satisfies the condition
Following Einstein (1950), a particle placed on the bed is likely to be lifted by the flowing fluid provided F L > F G. Importantly, the instantaneous lift force F L acting on a particle fluctuates in accordance with the velocity fluctuations u′ of the near-bed velocity u d; while the submerged weight F G of a particle is a constant for a given particle size. Therefore, F L > F G implies that u d > B or \( u^{\prime } > B - \bar{u}_{\text{d}} \), where B = [4Δgd/(3C L)]0.5. Thus, using Eq. (5.158), the total entrainment probability p is
It is pertinent to mention that Dey et al. (2012) found that when the bed particles move, the von Kármán constant κ diminishes from its universal value 0.41, and the zero-plane displacement level and the zero-velocity level move up as compared to their values in immobile beds [also available in Dey and Raikar (2007), Gaudio et al. (2010), Dey et al. (2011), and Gaudio and Dey (2012)]. These modify the estimation of near-bed velocity from the logarithmic law as \( \bar{u}_{\text{d}} = { 6}. 4u_{*} \), which was used by Bose and Dey. Quoting Kironoto and Graf (1994), Cheng and Chiew (1998) estimated \( \sqrt {\overline{{u^{\prime 2} }} } = { 2}u_{*} \). Using these results, the a can be expressed as
Figure 5.17 depicts the theoretical Θ(p) curve for C L = 0.15 obtained by solving Eq. (5.159) using Eq. (5.160). The theoretical curve matches well with the experimental data of Guy et al. (1966) and Fernandez Luque (1974). The data of Guy et al. (1966) that correspond to dunes have less agreement, because the analysis by Bose and Dey (2013) did not include the flow resistance due to bedforms. However, the curve obtained by Bose and Dey (2013) corresponds closely with the curves of Fredsøe and Deigaard (1992) and Cheng and Chiew (1998) for p < 0.2. The Shields parameter Θ for rough flow regime (R * > 70, where R * is the shear Reynolds number, u * k s/υ) according to Yalin and Karahan’s (1979) diagram is 0.046, for which the probability of entrainment is 0.1 % as obtained from Fig. 5.17. It implies that 0.1 % of all the particles on a given bed area are in motion under the threshold of sediment entrainment.
5.15 Effects of Bed Load on Velocity Distribution
Dey et al. (2012) conducted experiments to measure the velocity distributions and turbulence parameters in mobile-bed flow with bed-load transport and to compare them with those in a clear-water (immobile bed) flow. The experimental data for clear-water flow were used as a reference. For each sediment sample, an experimental set comprised of two different runs, such as clear-water and mobile-bed flow conditions. Fixed-bed roughness for a clear-water flow was prepared by gluing sediment on the flume bottom. The mobile-bed experiments were conducted to simulate the bed-load transport at a certain rate corresponding to the same flow condition as that of the clear-water flow. In mobile-bed experiments, the same sediment that was glued on the flume bottom was fed in the flow at a uniform rate to achieve a dynamic equilibrium condition of the mobile bed. A continuous weak sediment transport (as bed load) was established by the flow in the form of a thin sediment layer disallowing any bedforms to develop.
Figure 5.18 shows the vertical distributions of nondimensional time-averaged streamwise velocity u + for clear-water and mobile-bed flows. In order to fit the data points in the inner layer (z ≤ 0.2h) to the universal logarithmic law of wall, the time-averaged streamwise velocity \( \bar{u} \) and the vertical distance z are scaled by u * and d 50, such that \( u^{ + } = \bar{u}/u_{*} \) and z + = z/d 50. For the convenience, the origin of z-axis is set at the top of the bed particles (that is the bed surface). As the flow regime was the rough-turbulent flow, it is customary to use d 50 to scale z. Dey et al. used the values of u * that were obtained from the Reynolds shear stress plots by extrapolating a linear curve fitting onto the bed surface. To plot the experimental data, they consider the logarithmic law expressed in nondimensional form. It is
where Δ z + = Δ z/d 50, Δ z is the depth of the virtual bed level from the bed surface, ζ + = z 0/d 50, and z 0 is the zero-velocity level. Figure 5.18 describes the logarithmic law showing the variations of u + with z + + Δ z + for the experimental datasets. It is clear that a prior estimation of Δ z + was an essential prerequisite to plot the data, and subsequent determination of κ and ζ + was required to fit the data to the logarithmic law given by Eq. (5.161). The determination of these parameters was done independently, as described below:
Vertical distributions of nondimensional time-averaged streamwise velocity u + for clear-water and mobile-bed flows (Dey et al. 2012)
-
Step 1:
Having obtained u * from the Reynolds shear stress plots by projecting straight line on the bed surface [see Eq. (3.20) and Fig. 3.11], such that \( u_{*} = \left. {( - \overline{{u^{\prime}w^{\prime}}} )^{0.5} } \right|_{z = 0} \), the dataset u +(z +) for the range z ≤ 0.2h were prepared for the data analysis.
-
Step 2:
As an initial trial, considering Δ z + = 0, the values of κ and ζ + were determined from Eq. (5.161) by the regression analysis, and then, the regression coefficient RC was evaluated.
-
Step 3:
The values Δ z + were increased at a regular interval by a small magnitude (say 0.001), and the values of κ and ζ + were determined in the same way as in step 2. The values of RC for each value of Δ z + were checked, till RC became the maximum. Then, the corresponding values of Δ z +, κ and ζ + were considered as parameters for Eq. (5.161).
The average values of Δ z + = 0.39, κ = 0.413, and ζ + = 0.034 obtained for clear-water flow are in agreement with those for the traditional logarithmic law over rough boundary. Typically, the customary values of these parameters for the rough beds are Δ z + = 0.25, κ = 0.41, and ζ + = 0.033 (van Rijn 1984a). Thus, for clear-water flow, the data collapse well on the average logarithmic law curve shown by a solid line in Fig. 5.18. On the other hand, the average values of Δ z + = 0.21, κ = 0.37, and ζ + = 0.04 obtained for mobile-bed flow suggest the modified values of the parameters for the logarithmic law over a rough mobile bed. It is obvious that for mobile-bed flow, the data exhibit some degree of scatter about the average logarithmic law curve. A comparison of the values of Δ z + and ζ + for clear-water and mobile-bed flows reveals that the virtual bed and zero-velocity levels move up in the presence of bed-load transport. Although the data analysis related to the logarithmic law was done considering the data range z ≤ 0.2h, Fig. 5.18 displays all the data plots for z ≤ 0.2h and z > 0.2h. Thus, the data plots depart from the logarithmic law in the outer layer to some extent. Additionally, the values of Nikuradse's equivalent sand roughness k s can be determined from the relationship of zero-velocity level as k s = 30ζ + d 50. Finally, it can be concluded that for mobile-bed flow, (1) the von Kármán constant decreases and (2) the virtual bed and the zero-velocity levels move up.
5.16 Effects of Bed Load on Length Scales of Turbulence
According to Prandtl, the mixing length l, which defines a distance that a fluid parcel (eddy) keeps its original characteristics before dispersing into the surrounding fluid, is given by
To calculate the mixing length l from Eq. (5.162), Dey et al. (2012) used the measured velocity profiles to determine the velocity gradients d\(\bar{u}\)/dz by a smooth curve fitting to the data plots. They obtained the values of \( - \overline{{u^{\prime}w^{\prime}}} \) directly from the measured Reynolds shear stress distributions. The variations of nondimensional mixing length \( \tilde{l} \left( { = l/h} \right) \) with \( \tilde{z} \left( { = z/h} \right) \) for clear-water and mobile-bed flows as obtained by Dey et al. are shown in Fig. 5.19. Within the wall shear layer (z ≤ 0.2h), \( \tilde{l} \) varies linearly with \( \tilde{z} \). All the experimental data points for clear-water and mobile-bed flows collapse reasonably on a single band, which is in conformity with Prandtl’s mixing length hypothesis. Also, the data points collapse satisfactorily on the curves obtained from the theoretical equation of \( \tilde{l} = \kappa \tilde{z}( 1 { }{-}\tilde{z})^{0.5} \) given by Nezu and Nakagawa (1993). The slope of the linear portion defining von Kármán constant \( \kappa ( = \tilde{l}/\tilde{z} = l/z) \) for mobile-bed flow is smaller than that for clear-water flow. It suggests that the traversing length of an eddy decreases with bed-load transport and increases more rapidly with z in a clear-water flow. A detailed discussion on nonuniversality of von Kármán constant κ is given in next section.
Nondimensional mixing length \( \tilde{l} \) as a function of nondimensional vertical distance \( \tilde{z} \) for clear-water and mobile-bed flows (Dey et al. 2012)
Studies by Gore and Crowe (1991), Hetsroni (1993), Crowe (1993), Best et al. (1997) argued that in flow with transported particles, the ratio of the size of transported particles to the length scale of turbulence is involved in influencing the enhancement or attenuation of the streamwise turbulence intensity. Taylor microscale λ T, which defines the eddy size in the inertial subrange, is the relevant length scale of turbulence and is given by
where ε is the turbulent kinetic energy dissipation rate. The estimation of ε is done by using Kolmogorov second hypothesis that predicts the following equality describing the true inertial subrange (Pope 2001):
where k w is the wave number, S uu is the spectral density function for u′, and C is the constant approximately equaling 0.5 (Monin and Yaglom 2007).
In Fig. 5.20a, the spectra \( S_{uu} [ = \, (0. 5\bar{u}/\pi )F_{uu} \left( f \right)] \) as a function of \( k_{\text{w}} [ = \, ( 2\pi/\bar{u})f] \) are drawn using the despiked instantaneous velocity data at z = 2 mm (near-bed point) from the bed surface having d 50 = 2.6 mm for clear-water and mobile-bed flows. For both flow conditions, the depth-averaged flow velocity was 0.851 m s−1 and the flow depth 0.15 m. The bed-load transport rate in mobile-bed experiment was 7 × 10−3 kg s−1 m−1. The inertial subranges in clear-water and mobile-bed flows are characterized by Kolmogorov’s –5/3-th power law. It corresponds to a subrange of k w, where the average value of \( k_{\text{w}}^{5/3} \) S uu is relatively constant (that is independent of k w), as shown in Fig. 5.20b. Then, the ε was estimated from Eq. (5.164) and λ T from Eq. (5.163).
a Velocity power spectra S uu (k w). b Estimation of turbulent dissipation rate ε for clear-water and mobile-bed flows (Dey et al. 2012)
Figure 5.21 shows the variations of the ratio of sediment size to Taylor microscale, that is \( \hat{\lambda }_{\text{d}} = d_{ 50} /\lambda_{\text{T}} \), with \( \tilde{z} \) obtained by Dey et al. (2012) for the same flow condition mentioned above (clear-water and mobile-bed cases). Near the bed (z ≤ 0.1 h), \( \hat{\lambda }_{\text{d}} \) for mobile-bed flow is smaller than that for a clear-water flow. In the outer layer, \( \hat{\lambda }_{\text{d}} \) for both the cases, the variation being almost same decreases away from the bed. The values of λ T near the bed are 2 and 2.44 mm in clear-water and mobile-bed flows, respectively. Hence, the eddy size close to the bed increases in the presence of bed-load transport. Other studies on two-phase flows reported that the range \( \hat{\lambda }_{\text{d}} \approx \, 0. 2{-} 1. 2 \) corresponds to the turbulence enhancement; while the range \( \hat{\lambda }_{\text{d}} \approx \, 0. 2 { }{-}0.0 6 5 \) corresponds to the turbulence attenuation (Gore and Crowe 1991; Hetsroni 1993; Best et al. 1997).
Ratio of particle size to Taylor microscale \( \hat{\lambda }_{\text{d}} \) as a function of nondimensional vertical distance \( \tilde{z} \) for clear-water and mobile-bed flows (Dey et al. 2012)
Figure 5.22 presents the data plots of the ratio of particle size to Taylor microscale, \( \hat{\lambda }_{\text{d}} \), for mobile-bed flow as a function of relative difference of streamwise turbulence intensities \( \varDelta \sigma_{uu} [ = \, (\overline{{u^{\prime 2} }} )^{0. 5} |_{\text{mb}} /(\overline{{u^{\prime 2} }} )^{0. 5} |_{\text{cw}} {-}{ 1}] \). Here, subscripts “cw” and “mb” refer to clear-water and mobile-bed flows, respectively. The positive values of \( \varDelta \sigma_{uu} (\tilde{z} < 0. 2) \) indicate that the streamwise turbulence intensity in mobile bed is greater than that in clear-water flow. This is in conformity to the findings of Sumer et al. (2003), who studied the role of externally induced turbulence fields on bed-load transport and argued that the sediment transport rate increases considerably with an increase in streamwise turbulence intensity \( (\overline{{u^{\prime 2} }} )^{0.5} \).
Ratio of particle size to Taylor microscale \( \hat{\lambda }_{\text{d}} \) for mobile-bed flow as a function of relative difference of streamwise turbulence intensities ∆ σ uu between clear-water and mobile-bed flows (Dey et al. 2012)
5.17 Effects of Bed Load on von Kármán Constant κ
During bed-load transport, the sediment motion (by rolling, sliding, and saltation) produces an expansion of the roughness layer. Recking et al. (2008) reported that the Nikuradse's equivalent sand roughness k s increases from the particle size d 50 for immobile-bed condition to 2.6d 50 for intense bed-load transport condition. The expansion of the roughness layer modifies the logarithmic wall shear layer, resulting in the variation of von Kármán constant κ from its universal value 0.41. Gaudio et al. (2010) and Gaudio and Dey (2012) reviewed the studies on the effects of sediment transport on κ, which is discussed below:
Gust and Southard (1983) analyzed the velocity data in the wall shear layer (z/h ≤ 0.2) measured by a hot-wire anemometer. They observed a decrease in κ from its universal value with an increase in bed-load transport rate. After a transitional regime corresponding to the entrainment threshold of sediment, κ adjusted to a constant value of 0.32 ± 0.04 for all the experiments with bed-load transport, in which the transport rate varied by a factor 10. Best et al. (1997) used a phase Doppler anemometer to differentiate the characteristics of the fluid from those of the sediment particles and to quantify the influence of the sediments on the carrier fluid turbulence. They observed that the average value of κ was 0.385 in the presence of bed-load transport. Nikora and Goring (2000) reported a study on the characteristics of turbulent structure of high Reynolds number in quasi-two-dimensional flow with fixed and weakly mobile gravel-beds. The flow measurements by an acoustic Doppler velocimeter in an irrigation field canal were carried out for two bed conditions: fixed and weakly mobile beds. Measurements were first taken with a weakly mobile-bed flow (WMBF) and then repeated for a fixed-bed flow (FBF). They obtained κ ≈ 0.29 for the WMBF being significantly smaller than κ ≈ 0.4 for the FBF. They argued that the value κ ≈ 0.4 with the WMBF would have been achieved with an adjustment of the virtual bed level if the bed level was shifted by 30 mm upwards. Since such a shift is physically unjustifiable, it corroborates that the difference of κ values between the WMBF and the FBF is possible due to the effects of bed-load transport. Bennett and Bridge (1995), Nikora and Goring (1999), and Gallagher et al. (1999) also revealed an appreciable decrease in κ under bed-load transport. Nikora and Goring (1999) anticipated that the reduction in κ might reflect the special turbulence characteristics within a narrow range of the Shields parameter when the bed shear stress is approximately equal to the threshold bed shear stress. In Nikora and Goring (2000), the drag reduction effects were expressed as decreased values of κ. The general concept is that the drag reduction prevails when the spacing between turbulent bursting events increases in comparison as the spacing in flow with no sediment (Tiederman et al. 1985). However, it is revealed that the κ reduces when spanwise (lateral) spacing between bursting events increases; while streamwise spacing remains unchanged (Hetsroni et al. 1997). Nikora and Goring (2000) found that the streamwise spacing between bursting events was approximately the same for both the WMBF and the FBF, referring to an increase in spanwise spacing for the WMBF. Dey and Raikar (2007) reported the laboratory experimental results on the turbulent flow characteristics measured by an acoustic Doppler velocimeter. The primary endeavor was to investigate the response of the turbulent flow field, having zero-pressure gradient, to the uniform gravel-beds at the near-threshold of sediment of motion. They observed that the variation of mixing length is considerably linear with the distance from the bed within the wall shear layer, whose thickness was obtained as 0.23 times the boundary layer thickness; and von Kármán κ was estimated as 0.35.
Gaudio et al. (2011) performed laboratory tests in a narrow flume with sediment feeding to simulate bed load on a rough bed and measured the velocity within the wall shear layer (z/h ≤ 0.2) by using a Pitot-Prandtl tube. They obtained a decrease in κ, that κ varied from 0.3 with bed load (0.0334 ≤ g b ≤ 0.0649 kg s−1 m−1) to 0.4 with clear-water flow condition. Further, Dey et al. (2012) fitted a logarithmic law for mobile-bed flow to obtain κ = 0.37 for bed-load transport rates (2 × 10−3 ≤ g b ≤ 7 × 10−3 kg s−1 m−1), as already discussed. Table 5.3 furnishes a summary of the results on κ in flow with bed-load transport. The available experimental data are so limited that the variation of κ with bed-load transport rate (q b or g b) is not so specific, although it has been well-recognized that the κ values with bed-load transport are less than its universal value 0.41.
5.18 Examples
Example 5.1
The flow velocity in a wide river is 1.65 m s−1, flow depth 3.2 m, and energy slope 5 × 10−4. The flow is uniform within the measuring reach. The bed sediment has a median size d 50 = 1.5 mm, d 65 = 1.8 mm, and d 90 = 3 mm, a static angle of repose of 32°, a dynamic angle of repose of 20°, and a relative density of 2.65. Consider coefficient of kinematic viscosity of water υ = 10−6 m2 s−1 and mass density of water ρ = 103 kg m−3.
Compute the bed-load transport rate (in volume per unit time and width) by using formulas/methodologies proposed by du Boys, Shields, Schoklitsch, Meyer-Peter and Müller, Einstein, Brown/Julien (empirical form of Einstein’s method), Bagnold, Engelund and Fredsøe, Yalin, and van Rijn.
Also, compute the saltation characteristics of the particle.
Solution
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Given data are as follows:
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Flow velocity, U = 1.65 m s−1; flow depth, h = 3.2 m; energy slope, S f = 5 × 10−4; sediment size, d 50 = 1.5 mm, d 65 = 1.8 mm, and d 90 = 3 mm; static angle of repose, ϕ = 32°; dynamic angle of repose, ϕ d = 20°; relative density, s = 2.65; kinematic viscosity of water, υ = 10−6 m2 s−1; and mass density of water, ρ = 103 kg m−3
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For uniform flow, the energy slope equals the streamwise bed slope. Thus, S f = S 0 = 5 × 10−4
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Applied bed shear stress, τ 0 = ρghS 0 = 103 × 9.81 × 3.2 × 5 × 10−4 = 15.7 Pa
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Shear velocity, u * = (τ 0/ρ)0.5 = (15.7/103)0.5 = 0.125 m s−1
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Shields parameter, Θ = τ 0/(Δρgd 50) = 15.7/(1.65 × 103 × 9.81 × 1.5 × 10−3) = 0.647
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Use van Rijn’s empirical formula for the determination of threshold bed shear stress and threshold shear velocity (see Table 4.1):
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Particle parameter, D * = d 50(Δg/υ 2)1/3 = 1.5 × 10−3[1.65 × 9.81/(10−6)2]1/3 = 37.94
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Threshold Shields parameter, \( \Theta _{\text{c}} ( 20 \, < D_{*} \le 1 50) \, = \, 0.0 1 3D_{*}^{0.29} = \, 0.0 1 3\times 3 7. 9 4^{0. 2 9} = \, 0.0 3 7 \)
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Threshold bed shear stress, τ 0c = ΘcΔρgd 50 = 0.037 × 1.65 × 103 × 9.81 × 1.5 × 10−3 = 0.898 Pa
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Threshold shear velocity, u *c = (τ 0c/ρ)0.5 = (0.898/103)0.5 = 0.03 m s−1
Computation of bed load by du Boys formula
Computation of bed load by Shields formula
Computation of bed load by Schoklitsch formula
Computation of bed load by Meyer-Peter and Müller formula
Meyer-Peter and Müller recommended Θc = 0.047 and corresponding τ 0c = ΘcΔρgd 50 = 0.047 × 1.65 × 103 × 9.81 × 1.5 × 10−3 = 1.14 Pa
Computation of bed load by Einstein’s method Assume \( R^{\prime}_{\text{b}} = R_{\text{b}} = h = { 3}. 2 {\text{ m}} \) (for the wide channel)
From Fig. 5.10, Φb (Ψb = 1.86) = 4
Computation of bed load by empirical form of Einstein’s method Ψb = 1.86 ≤ 1.92; thus, Julien formula, given by Eq. (5.72), is applicable.
Computation of bed load by Bagnold formula Assume e b = 0.1
Computation of bed load by Engelund and Fredsøe formula Dynamic coefficient of friction, μ d = tan 20°
The formula of Engelund and Fredsøe seems to produce a higher estimation. Computation of bed load by Yalin formula
Computation of bed load by van Rijn formula
Computation of saltation characteristics
By van Rijn formulas:
Further, estimates of saltation length λ b, height h s, and particle velocity \( \bar{u}_{\text{b}} \) by using the formulas (see Tables 5.1 and 5.2) proposed by different investigators are given in Table 5.4.
Example 5.2
Water flows with a depth-averaged velocity of 1.5 m s−1 through a wide channel having a uniform flow depth of 3 m. The channel has a streamwise bed slope of 8 × 10−4. The size classes of nonuniform sediment obtained from the sieve analysis are 35 % between 0.1 and 0.5 mm, 30 % between 0.5 and 1 mm, 20 % between 1 and 2 mm, 10 % between 2 and 3 mm, and 5 % between 3 and 4 mm. Relative density of sediment is 2.65; and sediment size, d 50 = 0.75 mm and d 90 = 3 mm.
Find the bed-load transport rate by using the methods of (1) Meyer-Peter and Müller and (2) Ashida and Michiue.
Solution
Given data are as follows:
Flow velocity, U = 1.5 m s−1; flow depth, h = 3 m; bed slope, S 0 = 8 × 10−4; bed sediment size, d 50 = 0.75 mm and d 90 = 3 mm; and relative density, s = 2.65
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1.
Meyer-Peter and Müller’s method
Weighted mean size, d m = ∑p i d i = 1.06 mm (see Table 5.5)
Applied bed shear stress, τ 0 = ρghS 0 = 103 × 9.81 × 3 × 8 × 10−4 = 23.54 Pa
Threshold Shields parameter, Θc = 0.034 that is obtained from van Rijn’s empirical formula (Table 4.1) for the sediment size d m = 1.06 mm. Bed-load transport rate for fractional size \( d_{i}\; \text{is}\; q_{{{\text{b}}i}} = { 8}(\Delta g)^{0. 5} p_{i} d_{i}^{1.5} (\eta_{\text{C}}\Theta _{i} {-}\xi_{i}\Theta _{\text{c}} )^{ 1. 5} \)[see Eq. ( 5.146)]
Therefore, the total bed-load transport rate for all size fractions, q b = 2.146 × 10−4 m2 s−1 (Table 5.5)
However, one can check the difference in estimation of bed-load transport rate obtained using the weighted mean size d m.
which is almost equaling the estimate of total bed-load transport rate for all size fractions.
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2.
Ashida and Michiue’s method
Bed-load transport rate for fractional size \( d_{i}\,\text{is}\,q_{{{\text{b}}i}} = { 17}(\Delta g)^{0. 5} p_{i} d_{i}^{1.5} (\Theta _{i} {-}\Theta _{{{\text{c}}i}} )\times(\Theta _{i}^{0.5} {-}\Theta _{{\text c}i}^{0.5} ) \) [see Eq. (5.140)]
Threshold Shields parameter \( \Theta _{{{\text{c}}i}} \) for the fractional size d i can be obtained by using van Rijn’s empirical formula (Table 4.1) for the size d i (Table 5.6).
Therefore, the total bed-load transport rate for all size fractions, \( q_{\text{b}} = { 3}. 1 4\times 10^{{{-} 3}}\,{\text{m}}^{ 2}\, {\text{s}}^{{{-} 1}} \) (Table 5.6)
However, one can check the difference in estimation of bed-load transport rate obtained using the weighted mean size d m.
which is very close to the estimate of total bed-load transport rate for all size fractions.
Notes
- 1.
The probability of a particle performing (n + 1) number of jumps is (1 − p)p n(n + 1). Then,
$$ \sum\nolimits_{n = 0}^{\infty } {(1 - p)p^{n} } (n + 1) = 1 + p +p^2+p^3++p^n=(1 - p)^{-1}.$$ - 2.
To minimize the errors, the standard deviation of lift force fluctuations given by Eq. (5.68) should be small, then B * → ∞ as η 0 → ∞. Hence,
$$ - B_{ * }\Psi _{\text{b*}} - \frac{1}{{\eta_{0} }} = - \infty \;{\text{and}}\;B_{ * }\Psi _{\text{b*}} - \frac{1}{{\eta_{0} }} \ne 0 $$ - 3.
The use of Einstein’s Ψb*(Φb*) curve as shown in Fig. 5.10 is as follows:
Step 1: From the given bed sediment and flow conditions, compute Ψb* from Eq. (5.67). Then, the correction factors ξ and Y can be obtained from Figs. 5.8 and 5.9, respectively. The other parameters required to be computed are B * from Eq. (5.67), Δ k = k s/x k, x k from Fig. 5.7, \( u_{*}^{{\prime }} = (gR_{\text b}^{{\prime }} S_{0} )^{0.5} \hbox{,}\,\delta^{{\prime }} = { 11}. 6\upsilon /u_{*}^{{\prime }} \), and Ψb and B from Eq. (5.66).
Step 2: From Fig. 5.10, determine Φb* for the computed Ψb*. Thus, q b or g b can be obtained from Eq. (5.2).
- 4.
The procedure of transformation of second order differential equation to first order and the numerical solution methodology of a system of ordinary simultaneous differential equations can be found in Bose (2009).
- 5.
The procedure of computation of total bed-load transport for the entire range of particle size distribution of the bed sediment is as follows:
Step 1: Compute Φbi for the fraction p i of sediment size d i by using Einstein’s approach or Ashida and Michiue’s bed-load transport formula or by any other standard method given in this chapter.
Step 2: Compute i b q b by using Eq. (5.140) as
$$ \Phi _{{{\text{b}}i}} = \frac{{q_{{{\text{b}}i}} }}{{p_{i} (\Delta gd_{i}^{3} )^{0.5} }} \quad \wedge\quad q_{{{\text{b}}i}} = i_{\text{b}} q_{\text{b}} \; \Rightarrow\; i_{\text{b}} q_{\text{b}} =\Phi _{{{\text{b}}i}} \times p_{i} (\Delta gd_{i}^{3} )^{0.5} $$Step 3: For each size fraction, the i b q b can be computed in this way. The total bed-load transport can therefore be obtained by summing up the results over the entire range of particle size distribution.
Note: In case of a mixture of small size of sediment spread, the size d 35 can be approximated as an effective sediment size for the approximate estimation of total bed-load.
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Dey, S. (2014). Bed-Load Transport. In: Fluvial Hydrodynamics. GeoPlanet: Earth and Planetary Sciences. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19062-9_5
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