# CMS-Flow:Non-equilibrium Sediment Transport

UNDER CONSTRUCTION

written by Alejandro Sanchez Last date modified: December 8, 2010

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 2DH transport equation for the current-related total load is

 ${\displaystyle {\frac {\partial }{\partial t}}{\biggl (}{\frac {hC_{tk}}{\beta _{tk}}}{\biggr )}+{\frac {\partial (U_{j}hC_{tk})}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}{\biggl [}\nu _{s}h{\frac {\partial (r_{sk}C_{tk})}{\partial x_{j}}}{\biggr ]}+\alpha _{t}\omega _{sk}(C_{t*k}-C_{tk})}$ (1)

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

${\displaystyle C_{tk}}$ = actual depth-averaged total-load sediment concentration [kg/m3] for size class k defined as ${\displaystyle C_{tk}=q_{tk}/(Uh)}$ in which ${\displaystyle q_{tk}}$ is the total-load mass transport
${\displaystyle C_{tk*}}$ = equilibrium depth-averaged total-load sediment concentration [kg/m3] for size class k and described in the equilibrium concentration and transport rates section
${\displaystyle \beta _{tk}}$ = total-load correction factor described in the Total-Load Correction Factor section [-]
${\displaystyle r_{sk}}$ = fraction of suspended load in total load for size class k described in fraction of suspended sediments section [-]
${\displaystyle v_{s}}$ = horizontal sediment mixing coefficient described in the horizontal sediment mixing coefficient section [m2/s]
${\displaystyle \alpha _{t}}$ = total-load adaptation coefficient described in the adaptation coefficient section [-]
${\displaystyle \omega _{sk}}$ = sediment fall velocity [m/s].

In the above equation, the first term represents the temporal variation of ${\displaystyle C_{tk}}$; the second term represents the horizontal advection; the third term represents the horizontal diffusion and dispersion of suspended sediments; and the last term represents the erosion and deposition. The equation may be applied to single-sized sediment transport by using a single sediment size class (i.e., N = 1). The bed composition, however, does not vary when using a single sediment size class. The units of sediment concentration used here are kg/m3 rather than dimensionless volume concentrations in order to avoid numerical precision errors at low concentrations.

In the above equations, it is assumed that the wave flux velocity is not included in the momentum equations. If the wave flux velocity is included, then the total flux velocity (V) should be used instead of the depth-averaged current velocity (U). The reason for this is because without a wave-induced sediment transport to counter the offshore directed transport due to the undertow, the model would predict excessive movement of sediment offshore.

# Fraction of Suspended Sediment

In order to solve the system of equations for sediment transport implicitly, the fraction of suspended sediment must be determined explicitly. This is done by assuming

 ${\displaystyle r_{sk}={\frac {q_{sk}}{q_{tk}}}\simeq {\frac {q_{sk*}}{q_{tk*}}}}$ (2)

where ${\displaystyle q_{sk}{\text{and }}q_{tk}}$ are the actual suspended- and total-load transport rates and ${\displaystyle q_{sk*}{\text{and }}q_{tk*}}$ are the equilibrium suspended- and total-load transport rates.

The total-load adaptation coefficient ${\displaystyle (\alpha _{t})}$ is an important parameter in the sediment transport model. There are many variations of this parameter in literature (Lin 1984; Gallappatti and Vreugdenhil 1985; and Armanini and di Silvio 1986). CMS uses a total-load adaptation coefficient ${\displaystyle (\alpha _{t})}$ that is related to the total-load adaptation length (Lt ) and time (Tt) by

 ${\displaystyle L_{t}={\frac {Uh}{\alpha _{t}\omega _{s}}}=UT_{t}}$ (3)

where:

${\displaystyle \omega _{s}}$ = sediment fall velocity corresponding to the transport grain size for single-sized sediment transport or the median grain size for multiple-sized sediment transport [m/s]
U = depth-averaged current velocity [m/s]
h = water depth [m].

The adaptation length or time is a characteristic distance or time for sediment to adjust from non-equilibrium to equilibrium transport. Because the total load is a combination of the bed and suspended loads, the associated adaptation length may be calculated as ${\displaystyle L_{t}=r_{s}L_{s}+(1-r_{s})L_{b}{\text{ or }}L_{t}={\text{ max }}(L_{s},L_{b}),{\text{where }}L_{s}{\text{and }}L_{b}}$ are the suspended- and bed-load adaptation lengths. The symbol ${\displaystyle L_{s}}$ is defined as

 ${\displaystyle L_{s}={\frac {Uh}{\alpha \omega _{s}}}=UT_{s}}$ (4)

in which ${\displaystyle \alpha {\text{ and }}T_{s}}$ are the adaptation coefficient lengths for suspended load. The adaptation coefficient ${\displaystyle (\alpha )}$ can be calculated either empirically or based on analytical solutions to the pure vertical convection-diffusion equation of suspended sediment. One example of an empirical formula is that proposed by Lin (1984):

 ${\displaystyle \alpha =3.25+0.55\ ln{\big (}{\frac {\omega _{s}}{\kappa u_{*}}}{\big )}}$ (5)

where u* is the bed shear stress, and ${\displaystyle \kappa }$ is the von Karman constant. Armanini and di Silvio (1986) proposed an analytical equation:

 ${\displaystyle {\frac {1}{a}}={\frac {\delta }{h}}+\left(1-{\frac {\delta }{h}}\right)\exp \left[-1.5\left({\frac {\delta }{h}}\right)^{-1/6}{\frac {\omega _{s}}{u_{*}}}\right]}$ (6)

where ${\displaystyle \delta }$ is the thickness of the bottom layer defined by ${\displaystyle \delta =33z_{0}{\text{ and }}z_{0}}$ is the zero-velocity distance from the bed. Gallappatti (1983) proposed the following equation to determine the suspended-load adaptation time:

 {\displaystyle T_{s}={\frac {h}{u_{*}}}exp\left[{\begin{aligned}&(1.57-20.12u_{r})\omega _{*}^{3}+(326.832u_{r}^{2.2047}-0.2)\omega _{*}^{2}\\&+(0.1385\ ln\ u_{r}-6.4061)\omega _{*}+(0.5467u_{r}-2.1963)\end{aligned}}\right]} (7)

where u* is the current related bottom shear velocity, ${\displaystyle u_{r}=u_{*}/U,{\text{ and }}\omega _{*}=\omega _{s}/u_{*}}$.

The bed-load adaptation length (Lb) is generally related to the dimension of bed forms such as sand dunes. Large bed forms are generally proportional to the water depth, and, therefore, the bed-load adaptation length can be estimated as Lb = abh in which ab is an empirical coefficient on the order of 5-10. Although limited guidance exists on methods to estimate Lb, the determination of Lb is still empirical and in the developmental stage. For a detailed discussion of the adaptation length, the reader is referred to Wu (2007). In general, it is recommended that the adaptation length be calibrated with field data in order to achieve the best and most reliable results.

The total-load correction factor (βtk) accounts for the vertical distribution of the suspended sediment concentration and velocity profiles as well as the fact that bed load travels at a slower velocity than the depth-averaged current velocity (Figure 1) (Wu 2007). By definition, βtk is the ratio of the depth-averaged total-load and flow velocities.

Figure 1. Schematic of sediment and current vertical profiles.

In a combined bed-load and suspended-load model, the correction factor is given by

 ${\displaystyle \beta _{tk}={\frac {1}{r_{sk}/\beta _{sk}+(1-r_{sk})U/u_{bk}}}}$ (8)

where ubk is the bed-load velocity, and βsk is the suspended-load correction factor and is defined as the ratio of the depth-averaged suspended sediment and flow velocities. Since most sediment is transported near the bed, both the total- and suspended-load correction factors (βtk and βsk) are usually less than 1 and typically in the range of 0.3 and 0.7, respectively. By assuming logarithmic current velocity and exponential suspended sediment concentration profiles, an explicit expression for the suspended-load correction factor (βsk) may be obtained as (Sánchez and Wu 2011b)

 ${\displaystyle \beta _{sk}={\frac {\int _{z_{b}+a}^{\overline {\eta }}uc_{k}dz}{U\int _{z_{b}+a}^{\overline {\eta }}c_{k}dz}}={\frac {E_{1}(\phi _{k}A)-E_{1}(\phi _{k})+ln(A/Z)e^{-\phi _{k}A}-ln(1/Z)e^{-\phi _{k}}}{e^{-\phi _{k}A}[ln(1/Z)-1][1-e^{-\phi _{k}(1-A)}]}}}$ (9)

where:

${\displaystyle \phi _{k}=\omega _{sk}h/\epsilon _{k}[-]}$

A = a/h [-]

Z = za/h [-]

${\displaystyle \epsilon _{k}}$ = vertical mixing coefficient [m2/s]

a = reference height near the bed for the suspended load [m]

za = apparent roughness length [m]

${\displaystyle E_{1}(x)=\int _{x}^{\infty }{\frac {e^{-t}}{t}}dt}$ (exponential integral).

The equation can be further simplified by assuming that the reference height is proportional to the apparent roughness length (e.g., a = 30zz) so that ${\displaystyle \beta _{sk}=\beta _{sk}(Z,\phi _{k})}$. Figure 2 shows a comparison of the suspended-load correction factor based on the logarithmic velocity with exponential and Rouse suspended sediment concentration profiles. Both cases show similar behavior for the suspended-load correction factor (${\displaystyle \beta _{sk}}$). For fine sediments (small fall velocity), ${\displaystyle \beta _{sk}}$ is close to 1.0 and experiences smaller influences by the bottom roughness, while for course sediments, ${\displaystyle \beta _{sk}}$can be as low as 0.5 and is largely influenced by the bottom roughness. This is to be expected since course sediments are transported more closely to the bottom, compared to fine sediments.

Figure 2. Suspended load correction factors based on the logarithmic velocity profile and (a) exponential and (b) Rouse suspended sediment profile. The Rouse number is ${\displaystyle r=\omega _{s}/(\kappa u_{*})}$

The bed-load velocity (ubk) in Equation 8 is calculated using the van Rijn (1984a) formula with re-calibrated coefficients from Wu et al. (2006):

 ${\displaystyle u_{bk}=1.64\left({\frac {\tau _{b}^{'}}{\tau _{crk}}}-1\right)^{0.5}{\sqrt {(s-1)gd_{k}}}}$ (10)

where:

s = specific gravity [-]

g = gravitational constant (~9.81 m/s2)

dk = characteristic grain diameter for the kth size class [m]

${\displaystyle \tau _{b}^{'}=(n^{'}/n)^{3/2}\tau _{b}}$ = grain-related bed shear stress [Pa]

${\displaystyle n^{'}=d_{50}^{1/6}/20\ }$ = grain-related Manning’s roughness coefficient [s/m1/3]

${\displaystyle \tau _{crk}}$ = critical bed shear stress for the kth size class [Pa].

# Bed Change Equation

The change in the water depth is calculated by

 ${\displaystyle \rho _{s}(1-p'_{m}){\biggl (}{\frac {\partial z_{b}}{\partial t}}{\biggr )}_{k}=\alpha _{t}\omega _{sk}(C_{tk}-C_{tk*})+{\frac {\partial }{\partial x_{j}}}{\biggl [}D_{s}q_{bk}{\frac {\partial z_{b}}{\partial x_{j}}}{\biggr ]}}$ (11)

where:

zb = bed elevation with respect to the vertical datum [m]

pm' = bed porosity [-]

${\displaystyle \rho _{s}}$ = sediment (material) density (~2650 kg/m3 for quartz sediment)

Ds = empirical bed-slope coefficient (constant) [-]

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

The first term on the right-hand side of the above equation represents the bed change due to sediment exchange near the bed. The last term accounts for the effect of the bed slope on bed-load transport. The bed-slope coefficient (Ds) is usually about 0.1 to 3.0. For a detailed derivation of the above equation, the reader is referred to Sanchez and Wu (2011a). The total bed change is calculated as the sum of the bed changes for all size classes:

 ${\displaystyle {\frac {\partial z_{b}}{\partial t}}=\sum _{k}\left({\frac {\partial z_{b}}{\partial t}}\right)_{k}}$ (12)

# Bed Sorting and Gradation (nonuniform sediments)

When simulating nonuniform sediments it is necessary to keep track of the vertical variation in bed composition. In order to do this, the bed is divided into discrete layers. The top layer is the mixing or active layer and is the layer of sediment which is actively being exchanged with the bed and suspended loads. The temporal variation of the bed-material in the mixing and second layers is calculated as (Wu 2004)

 ${\displaystyle {\frac {\partial (\delta _{m}p_{bk})}{\partial t}}+{\biggl (}{\frac {\partial \zeta }{\partial t}}{\biggr )}_{k}=-{\frac {\partial (\delta _{m}p_{bk})}{\partial t}}=p_{bk}^{*}{\biggl (}{\frac {\partial \delta _{m}}{\partial t}}+{\frac {\partial \zeta }{\partial t}}{\biggr )}}$ (11)

where ${\displaystyle \delta _{m}}$ and ${\displaystyle \delta _{s}}$ are the thicknesses of the mixing and second layers respectively. ${\displaystyle p_{bk}^{*}=p_{bk}}$ for ${\displaystyle \partial \zeta /\partial t+\partial \delta _{m}/\partial t\leq 0}$ or ${\displaystyle p_{bk}^{*}=p_{sk}}$ otherwise. Figure 3 shows an example of the bed layer evolution over time for varying bed deposition and erosion.

Figure 3. Example of bed layer variation for varying deposition and erosion.

# Boundary Conditions

There are three types of boundary conditions in the sediment transport: Wet-dry, Outflow and Inflow.

1. Wet-dry interface.

The interface between wet and dry cells has a zero-flux boundary condition. Both the advective and diffusive fluxes are set to zero at the wet-dry interfaces. Note that avalanching may still occur between wet-dry cells.

2. Outflow Boundary Condition

Outflow boundaries are assigned a zero-gradient boundary condition and sediments are allowed to be transported freely out of the domain.

3. Inflow Boundary Condition

When flow is entering the domain, it is necessary to specify the sediment concentration. In CMS-Flow, the inflow sediment concentration is set to the equilibrium sediment concentation. For some cases, it is desired to reduce the amount of sediment entering from the boundary such as in locations where the sediment source is limited (i.e. coral reefs). The inflow equilibrium sediment concentration may be adjusted by multiplying by a loading scaling factor and is specified by the Advanced Card NET_LOADING_FACTOR or SEDIMENT_INFLOW_LOADING_FACTOR.

# Numerical Methods

The governing equations are discretized using the Finite Volume Method on a staggered, non-uniform Cartesian grid. Time integration is calculated with a simple explicit forward Euler scheme. Diffusion terms are discretized with the standard central difference scheme. Advection terms are discretized with either the first order upwind scheme or the second order Hybrid Linear/Parabolic Approximation (HLPA) scheme of Zhu (1991). The default advection scheme is HLPA but may be changed with the Advanced Card ADVECTION_SCHEME.

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