Total-load Horizontal Diffusion Coefficient
The total-load horizontal mixing/diffusion coefficient is determined as
| 1) |
ε_{tk} = r_{sk}ε_{sk} + (1−r_{sk})ε_{bk} \quad \text{for } k = 1,\hdots, N |
where
r_{sk} = fraction of suspended-load [-]
ε_{sk} = suspended-load mixing coefficient [L2/T]
ε_{bk} = bed-load mixing coefficient [L2/T]
The calculation of the suspended- and bed-load mixing coefficients is described below.
Suspended-load Horizontal Diffusion Coefficient
The suspended-load horizontal mixing coefficient (εsk) represents the effects of turbulent diffusion. The horizontal sediment mixing coefficient is assumed to be related to the turbulent eddy viscosity as
| 2) |
\varepsilon _{sk}=\frac{\nu _{t}}{\sigma _{sk}}\,\,\,\,\,\mathrm{for}\,\,k=1,\ldots ,N |
where
σ_{sk} =Schmidt number for kth grain class [-]
ν_{t} = turbulent eddy viscosity [L2/T]
If a turbulent eddy viscosity is not available either because the flow model solves a Diffusion Wave equation or simply because it was ignored in the flow model, then it may be calculated as
where
c_{M} = empirical coefficient (cM ≈ 0.5 − 6) [-]
u_{*} = bed shear velocity [L/T]
h = water depth [L]
It is noted that the horizontal and vertical sediment mixing coefficients have similar formulations but generally the horizontal coefficient will be much larger.
Bed-load Horizontal Diffusion Coefficient
The bed-load horizontal diffusion coefficient (εbk) represents the sediment mixing due to spatially and temporally varying bed-load velocities. The following formula is proposed for the bed-load horizontal diffusion coefficient.
where
c_{B } = empirical coefficient (c_{B} \approx 5) [-]
u_{*} = bed shear velocity [L/T]
d_{k} = grain size diameter [L]
The bed-load horizontal mixing coefficient is much smaller than the suspended-load mixing coefficient. In addition, the dispersion produced by the mixing in the bed is usually much more significant than horizontal bed-load diffusion. Therefore, the bed-load diffusion coefficient can usually be ignored.