Sediment Transport 1

Sediment Transport

Overview

For sand transport, the wash-load (i.e. sediment transport which does not contribute to the bed-material) can be assumed to be zero, and therefore, the total-load transport is equal to the sum of the bed- and the suspended-load transports: $q_{tk*}=q_{sk*}+q_{bk*}$.

There are currently three sediment transport models available in CMS:

(1) Equilibrium total load

(2) Equilibrium bed load plus non-equilibrium suspended load, and

(3) Non-equilibrium total-load.

The first two models are single-size sediment transport models and are only available with the explicit time-stepping schemes. The third is multiple-sized sediment transport model and is available with both the explicit and implicit time-stepping schemes.

Equilibrium Total-load Transport Model

In this model, both the bed load and suspended load are assumed to be in equilibrium. The bed change is solved using a simple mass balance equation known as the Exner equation.

${\frac {\rho _{s}(1-\rho _{m}^{'})}{f_{morph}}}{\frac {\partial z_{b}}{\partial t}}={\frac {\partial q_{t*j}}{\partial x_{j}}}+{\frac {\partial }{\partial x_{j}}}\left(D_{s}\mid q_{t*}\mid {\frac {\partial z_{b}}{\partial x_{j}}}\right)$ (2-42)

for $j=1,2;k=1,2,.....,N$ , where N is the number of sediment size classes and

t = time [s]

h = total water depth [m]

$x_{j}$ =Cartesian coordinate in the j^th direction [m]

$q_{t*}$ = equilibrium total-load transport rate [kg/m/s]

$z_{b}$ = bed elevation with respect to the vertical datum [m]

$p_{m}^{'}$ = bed porosity [-]

$f_{morph}$ = morphologic acceleration factor [-]

$\rho _{s}$ = sediment density [~2650 kg/m³ for quartz sediment]

$D_{s}$ = empirical bed-slope coefficient (constant) [-]

Because the model assumes that both the sediment transport is equilibri-um, it only recommended for coarse grids with resolutions larger than 50-100 m where the assumption of equilibrium sediment transport is more appropriate. As mentioned above the equilibrium total-load sediment transport model is a single-size sediment transport model and is only available with the explicit time-stepping scheme. For more information on the equilibrium sediment transport model, the reader is referred to Buttolph et al. (2007).

Equilibrium Bed-load plus Nonequilibrium Suspended Load Transport Model

Calculations of suspended load and bed load are conducted separately. The bed load is assumed to be in equilibrium and is included in the bed change equation while the suspended load is solved through the solution of an advection-diffusion equation. Actually the advection diffusion equation is a non-equilibrium formulation, but because the bed load is assumed to be in equilibrium, this model is referred to the "Equilibrium A-D" model.

Suspended-load Transport Equation

The transport equation for the suspended load is given by

${\frac {\partial (hC)}{\partial t}}+{\frac {\partial (hV_{j}C)}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}\left(v_{s}h{\frac {\partial C}{\partial x_{j}}}\right)+E_{b}-D_{b}$ (2-43)

where

t = time[s]

h = water depth [m]

$x_{j}$ = Cartesian coordinate in the

$E_{b}$ entrainment or pick-up function [kg/m²/s]

$D_{b}$ deposition or settling function [kg/m²/s]

The entrainment and deposition functions are calculated as

$E_{b}=-\varepsilon _{s}{\frac {\partial c}{\partial z}}\left|_{z=a}\right.=\omega _{s}c_{a*}$ (2-44)

$E_{b}=\omega _{s}c_{a}(2-45)$

where

z = vertical coordinate from the bed [m]

$\varepsilon _{s}$ = vertical sediment mixing coefficient [m²/s]

c = local sediment concentration [kg/m³]

$\omega _{s}$ = sediment fall velocity [m/s]

$c_{a}$ = calculated sediment concentration at an elevation a above the bed [kg/m³]

$c_{a*}$ = equilibrium (capacity) sediment concentration at an elevation a above the bed [kg/m³]

Bed Change Equation

The bed change is calculated as

${\frac {\rho _{s}(1-p_{m}^{'})}{f_{morph}}}{\frac {\partial z_{b}}{\partial t}}={\frac {\partial q_{b*j}}{\partial x_{j}}}+{\frac {\partial }{\partial x_{j}}}\left(D_{s}\mid q_{t}*\mid {\frac {\partial z_{b}}{\partial x_{j}}}\right)+E_{b}-D_{b}$ (2-46)

where

$z_{b}$ = bed elevation with respect to the vertical datum [m]

$p_{m}^{'}$ = bed porosity [-]

$f_{morph}$ = morphologic acceleration factor [-]

$\rho _{s}$ = sediment density [~2650 kg/m³ for quartz sediment]

$D_{s}$ = empirical bed-slope coefficient (constant) [-]

$q_{bk}=hUC_{tk}(1-r_{sk})$ is the bed load mass transport rate [kg/m/s]

As mentioned above the equilibrium total-load sediment transport model is a single-size sediment transport model and is only available with the explicit time-stepping scheme. For more information on the equilibrium sediment transport model, the reader is referred to Buttolph et al. (2007).

Nonequilibrium Total-Load Transport Model

Total-load Transport Equation

The single-sized sediment transport model described in Sánchez and Wu (2011a) was extended to multiple-sized sediments within CMS by Sánchez and Wu (2011b). In this model, the sediment transport is separated into current- and wave-related transports. The transport due to currents includes the stirring effect of waves; and the wave-related transport includes the transport due to asymmetric oscillatory wave motion as well as steady contributions by Stokes drift, surface roller, and undertow. The current-related bed and suspended transports are combined into a single total-load transport equation, thus reducing the computational costs and simplifying the bed change computation. The two-dimensional horizontal (2DH) transport equation for the current-related total load is

${\frac {\partial }{\partial t}}\left({\frac {hC_{tk}}{\beta _{tk}}}\right)+{\frac {\partial (hV_{j}C_{tk})}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}\left[v_{s}h{\frac {\partial (r_{sk}C_{tk})}{\partial x_{j}}}\right]+\alpha _{t}\omega _{sk}(C_{t*k}-C_{tk})(2-47)$