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Bed-Load Transport

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Fluvial Hydrodynamics

Part of the book series: GeoPlanet: Earth and Planetary Sciences ((GEPS))

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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Notes

  1. 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. 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. 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. 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. 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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    Subhasish Dey

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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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