A general bedload transport equation for homogeneous grains

Bed load is one of two types of sediment load (the other one suspended load) transported in natural rivers (Fig. 1) and is originated from channel bed or banks. Although it only takes about 5% to 20% of the total sediment load, bedload transport is important as it controls bedforms and channel morphology, influences channel stability, and affects flow resistance. In natural gravel-bed rivers, bed load typically consists of grains of mixed sizes, while in sand-bed rivers and in laboratory flumes, bed load may be comprised of identical particles. Under the steady, uniform condition, under which no net erosion or deposition would occur on channel or flume beds, a flow transporting bed load of homogenous grains can create bed forms such as ripples, dunes, and antidunes. Consequently, the flow strength, which may be defined as the bed shear stress τ = ρghs where ρ is flow density, g is acceleration of gravity, h is mean flow depth, and s is energy slope, which is often practically represented by water surface slope or bed slope, is partitioned into two components: τ', the shear stress directly causing the transport of bed materials (also referred to as shear stress due to grain roughness), and τ'', the remaining shear stress for overcoming the flow resistance due to the bed forms (also referred to as shear stress due to bed form). The amount of transported bed load is directly controlled by τ'. Therefore, most shear stress based bedload equations use τ'rather than τ. For simplicity, τ' is referred to as τ hereafter.

Bed load is transported in different modes with different values of τ. At low to medium shear stress, bed load may be transported through sliding, rolling, or saltating (i.e. jumping) along the bed. The flow conditions associated with these transport modes is termed as the saltation regime. At high shear stress, bed load is mainly transported in a loosely defined granular sheets or laminations wherein high grain concentration constrains the vertical motion of grains and promotes grains moving in layers. The associated flow condition is termed as the sheet-flow regime (Gao, 2008). In the past, bedload transport equations were developed either in the saltation or the sheet-flow regimes. This study established a general bedload equation that can be used in both the saltation and sheet-flow regimes.

From an energy perspective, a bedload-laden flow can be regarded as a transporting machine whose mechanic performance can be described by the energy equation (Bagnold, 1973)

(1) i_b tan α = e_bω

where i_b = q_bg(ρ_s - ρ) is the immersed bedload transport rate (J s^-1 m^-2), q_b is the volumetric bedload transport rate (m² s^-1), tan α is the dynamic friction coefficient, e_b is the efficiency of bedload transport, ω = uτis the flow power per unit bed area (W m^-2), and u is the mean flow velocity (m s^-1). The left side of the equation represents the rate of doing work transporting bed load per unit area of the bed (J s^-1 m^-2), whereas the right side indicates the power expended on bedload transport over the same area (W m^-2). Converting equation (1) into a general bedload transport equation requires the determination of both tanα and e_b. The definition of tanα (Bagnold, 1973) assumes that bed load is supported entirely by a normal stress arising from grain-to-grain and grain-to-bed collisions, which is commonly regarded as a constant of 0.6 and is only valid in the sheet-flow regime. Extending this concept into the saltation regime leads to a new concept, termed as stress coefficient s_b, which is defined as (Abrahams and Gao, 2006):

(2) s_b = T/W' = (T_g + T_f)/W'

In eq. (2), T is the tangential bed shear stress (kg m^-1 s^-2) that is required to keep the bed load in motion and is defined as τ = T + τ_c where τ_c is the critical fluid shear stress (kg m^-1 s^-2) below which bedload movement ceases. T may be divided into two parts: that part transmitted to the bed by grain collisions and termed grain shear stress T_g (kg m^-1 s^-2) and that part transmitted to the bed by fluid drag and termed fluid shear stress T_f (kg m^-1 s^-2). W' is the immersed weight of the bed load.

To develop a functional equation for s_b, data from a variety of flume experiments on bedload transport with homogeneous grains were compiled to represent a wide range of hydraulic and sedimentological conditions that are available in both the saltation and sheet-flow regimes. The 322 flume data representing rough turbulent two-dimensional open-channel flows moving well sorted non-cohesive sediments over plane mobile beds were selected for the analysis. Nonlinear regression analysis gave rise to the following functional equation for s_b (Fig.2):

(3) s_b = 0.6G^–2

where G is a normalized measure of sediment transport stage and is defined as 1-(Θ_c/Θ) in which Θ, the dimensionless bed shear stress, defined as Θ=τ/gD(ρ_s-ρ) where D is the median grain diameter (m), and Θ_c is the critical value of Θ at which bed sediments begin to move. Combining eqs. (1) and (3) and employing the known values of i_b and ω from the compiled data resulted in the functional equation for e_b (Fig. 3):

(4) e_b = 0.6G^1.4

Replacing sb and eb by their functional equations (e.g. eqs.(3) and (4)) and rearranging eq. (1) produces a general bedload transport equation for both the saltatation and sheet-flow regimes (Fig.4):

(5) i_b = ωG^3.4

or its dimensionless form (Fig. 5):

(6) Φ_b = Θ^1.5G^3.4(u/u_*)

where Φ_b = q_b/((g((ρ_s-ρ)/ρ)D)^0.5D) and ρ_s is the density of bedload grains. Eq. (5) is a simple and parsimonious bedload transport equation that can accurately predict bedload transport rates in both the salutation and sheet-flow regimes. This equation provides a standard against which measured bed-load transport rates in flows subject to influences such as bed forms, bed armoring, and limited sediment availability may be compared to assess the impact of these influences on bed-load transport rates.

• Abrahams A. and Gao P. 2006. A bed-load transport model for rough turbulent open-channel flows on plane beds. Earth Surface Processes and Landforms, 31: 910-928.
• Bagnold RA. 1973. The nature of saltation and of 'bed-load' transport in water. Proceedings of the Royal Society of London, Series A 332: 473-504.
• Gao P. 2008. Transition between Two Bed-Load Transport Regimes: Saltation and Sheet Flow. Journal of Hydraulic Engineering 134: 340-349.