The bed-load velocity is the average particle velocity during transport. It is not the average velocity of the bed-load material, which is what is typically measured in a tracer studies or mobile bed laboratory experiments. The bed-load velocity different formulations available are plotted in the figure below and described in the following sub sections. In general, the formulations produce similar results except for the Julien and Bounvilay (2013) formulation which does not have a critical threshold for bed-load transport.
Figure 1. Comparison of bed-load velocity functions.
Phillips and Sutherland
Phillips and Sutherland (1989) proposed the following formula for the bed-load velocity
| 1) |
u_{bk}=8.5\left\{\frac{\tau '_{b}}{\rho _{w}}\max \left[1-\left(\frac{\tau _{crk}}{\tau '_{b}}\right)^{1/2},0\right]\right\}^{1/2} |
where
τ'_b = grain-related bed shear stress [M/L/T2]
τ_{crk} = critical bed shear stress for the kth size class [M/L/T2]
ρ_{w} = water density [M/L3]
n' = grain-related Manning’s roughness coefficient [T/L1/3]
van Rijn
The van Rijn (1984a) formula for computing the bed load velocity (ubk) is given by
| 2) |
\frac{u_{bk}}{\sqrt{R_{k}gd_{k}}}=1.5\max \left(\frac{\tau '_{b}}{\tau _{crk}}-1,0\right)^{0.6} |
where
R_{k} = ρ_{sk}/ρ_w−1 = submerged specific gravity for the kth grain class [-]
ρ_{sk} = particle density for the kth grain class [M/L3]
ρ_{w} = water density [M/L3]
g = gravitational constant (~9.81 m/s2) [L/T2]
d_{k} = characteristic grain diameter for the kth size class [L]
τ'_b = grain-related bed shear stress [M/L/T2]
τ_{crk} = critical bed shear stress for the kth size class [M/L/T2].
The van Rijn (1984a) formula is based on laboratory experiments of Fernandez Luque (1974; 1976).
van Rijn-Wu
Wu et al. (2006) recalibrated the coefficients of the van Rijn (1984a) formula as
| 3) |
\frac{u_{bk}}{\sqrt{R_{k}gd_{k}}}=1.64\max \left(\frac{\tau '_{b}}{\tau _{crk}}-1,0\right)^{0.5} |
where
R_{k} = ρ_{sk}/ρ_w−1 = submerged specific gravity for the kth grain class [-]
ρ_{sk} = particle density for the kth grain class [M/L3]
ρ_{w} = water density [M/L3]
g = gravitational constant (~9.81 m/s2) [L/T2]
d_{k} = characteristic grain diameter for the kth size class [L]
τ′_b = grain-related bed shear stress [M/L/T2]
τ_{crk} = critical bed shear stress for the kth size class [M/L/T2]
The Wu et al. (2006) formula produces similar bed-load velocities as the van Rijn (1984a) formula for low shear stresses but significantly lower bed-load velocities for high shear stresses where the van Rijn (1984a) departs from the other bed-load velocity formulations.