CMS-Flow:Non-equilibrium Sediment Transport

Non-equilibrium Sediment Transport in CMS - UNDER CONSTRUCTION

Introduction

The Coastal Modeling System (CMS) was developed with the objective of providing an operational numerical simulation system for coastal hydrodynamics, sediment transport, and morphology change for operating and managing federal coastal navigation projects (typically, coastal inlet and ports). Typical applications involve estimation of navigation channel infilling, wave conditions in the presence of jetties and breakwaters, and sand bypassing. The CMS consists of three components that are coupled together: (1) a depth-averaged hydrodynamic model, (2) a steady-state spectral wave model, and (3) a depth-averaged sediment transport and morphology change model. This wiki section describes a Non-equilibrium Sediment Transport (NET) model which has been added in the CMS Release v3.75 as one of several sediment transport option and has been implemented in the Surface Water Modeling System (SMS). The NET simulates non-cohesive, single size sediment transport and bed change using a Finite Volume method and includes advection, diffusion, hiding and exposure, and avalanching.

Transport Equation

Non-cohesive sediment transport is calculated with a non-equilibrium bed-material (total load) formulation. In this approach, the suspended- and bed-load transport equations are combined into a single equation and thus there is one less empirical parameter to estimate (adaptation length). The transport equation is derived by adding the suspended- and bed-load transport equations to obtain the general sediment mass balance equation and then substituting a non-equilibrium expression for the bed elevation change as suggested by Wu (2004):

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

where $h$ is the total water depth ($h=\zeta +\eta$), $C_{t}$ is the total load concentration, $C_{t*}$ is the sediment transport capacity, $\beta _{t}$ is the total load correction factor, $\nu _{s}$ is the diffusion coefficient, $r_{s}$ is the fraction of suspended sediments, $\alpha _{t}$ is the total load adaptation coefficient, and $\omega _{s}$ is the sediment fall velocity.

The concentration capacity may be calculated with either the Lund-CIRP (Carmenen and Larson 2007), the van Rijn (2007), or the Watanabe (1987) transport equations. The calculated sediment concentration capacities from these formula are multiplied by transport scaling factors which typically vary from 0.5-2.0 and have default value of 1.0.

In CMS, the sediment fall velocity is calculated with the Soulsby (1997) equation. If the sediment fall velocity has been measured in the laboratory, then it may be specified with the Advanced Card

    SEDIMENT_FALL_VELOCITY <white space> #

where # is the sediment fall velocity in m/s. It is not recommended to use the sediment fall velocity as a calibration parameter.

The total-load correction factor accounts for the time lag (hysteresis) between flow and sediment transport and is given by

   $\beta _{t}={\frac {1}{r_{s}\beta _{s}+(1-r_{s})U/u_{b}}}$

where $\beta _{s}$ is the suspended-load correction factor defined by

   $\beta _{s}={\frac {\int _{a}^{h}u_{s}cdz}{U\int _{a}^{h}cdz}}$

Bed Change Equation

If the advection-diffusion (A-D) equation is selected to simulate the sediment transport and mixing, the change in the water depth is calculated by the sediment continuity equation

       $(1-p'_{m}){\frac {\partial \zeta }{\partial t}}=\alpha _{t}\omega _{s}(C_{t*}-C_{t})+{\frac {\partial }{\partial x_{j}}}{\biggl [}D_{s}|U|h(1-r_{s})C_{t}{\frac {\partial \zeta }{\partial x_{j}}}{\biggr ]}$

where $p'_{m}$ is the sediment porosity, and $D_{s}$ is a bedslope coefficient.

Hiding and Exposure

At many sites, the bed material can be characterized by a single sediment size, with other sizes or materials (shell hash) which do not contribute significantly to morphology change, but do modify the sediment transport through hiding and exposure. By assuming that the spatial distribution of the bed material composition is constant in time, a hiding and exposure correction function can be introduced to correct the critical shields parameter $\Theta _{ck}^{he}=\xi _{k}\Theta _{ck}$ where $\xi _{k}$ is the dimensionless hiding and exposure function and $\Theta _{ck}$ is the critical shear stress of the transport grain size. In CMS, a formula similar to that of Parker et al. (1995) and others is implemented $\xi _{k}=(d_{k}/d_{50})^{-m}$ where is the grain size corresponding to the 50th percentile, and $m$ is an empirical coefficient between 0.5-1.0 (default is 0.7).

The transport grain size is specified in the Advanced Cards section as:

  TRANSPORT_GRAIN_SIZE <white space> #

where # is the sediment grain size in mm. The transport grain size should be the dominant grain size in the area of interest.

To change the value of $m$ another Advanced Card is used:

  HIDING_EXPOSURE_COEFFICIENT <white space> #

If it is desired to test the model with a constant grain size and ignore the information in the D50_dataset, the Advanced Card can be used

  CONSTANT_GRAIN_SIZE <white space> #

Avalanching

The process of avalanching is simulated by enforcing the angle of repose while maintaining mass continuity between adjacent cells. The presented approach adopts a relaxation method between adjacent cells and is stable and efficient. The equation for bed change due to avalanching is obtained by combining the equation of angle of repose and the continuity equation to obtain

     $\Delta \zeta _{p}^{a}=R\sum _{i}{\frac {A_{i}\delta x_{i}}{A_{p}+A_{i}}}({\textrm {tan}}\phi _{i}-{\textrm {sgn}}\phi _{i}{\textrm {tan}}\phi _{r}){\textrm {H}}(|\phi _{i}|-\phi _{r})$

where again the summation convention is applied for repeated indices. The subscripts p and i indicate the center and neighboring cells respectively, $\delta x$ is the cell center distance between p and i, $\Delta \zeta _{p}$ is the bed change due to avalanching and A is the cell area, $\phi$ is the bed slope, $\phi _{r}$ is the sediment repose angle, R is an under-relaxation factor (approximately 0.25-0.5), and ${\textrm {H}}$ and ${\textrm {sgn}}$ are the Heaviside and sign functions.

Boundary Conditions

The sediment flux on cell faces between dry and wet cells is assigned to zero. Outflow boundaries are assigned a zero-gradient boundary conditions (BC). Inflow boundaries may be assigned a specific concentration, the concentration capacity or a zero-gradient BC. The inflow boundary condition is specified with Advanced Card:

   NET_INFLOW_BC_CONC <white space> <type>

where <type> may be either EQUILIBRIUM OR ZERO-GRADIENT. 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 <white space> #

where # is the loading factor in dimensionless units.

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:

  NET_ADVECTION_SCHEME <white space> <name>

where <name> may be either HLPA, UPWIND, or OFF.

References

Buttolph, A. M., C. W. Reed, N. C. Kraus, N. Ono, M. Larson, B. Camenen, H. Hanson, T. Wamsley, and A. K. Zundel. (2006). “Two-dimensional depth-averaged circulation model CMS-M2D: Version 3.0, Report 2: Sediment transport and morphology change.” Coastal and Hydraulics Laboratory Technical Report ERDC/CHL TR-06-9. Vicksburg, MS: U.S. Army Engineer Research and Development Center, U.S.A.

Camenen, B., and Larson, M. (2007). “A unified sediment transport formulation for coastal inlet application”. Technical Report ERDC-CHL CR-07-01. Vicksburg, MS: U.S. Army Engineer Research and Development Center, U.S.A

Parker, G., Kilingeman, P. C., and McLean, D. G. (1982). “Bed load and size distribution in paved gravel-bed streams.” J. Hydr. Div., ASCE, 108(4), 544-571.

Soulsby, R. L. (1997). "Dynamics of marine sands, a manual for practical applications". H. R. Wallingford, UK: Thomas Telford.

Watanabe, A. (1987). “3-dimensional numerical model of beach evolution”. Proc. Coastal Sediments ’87, ASCE, 802-817.

Wu, W. (2004).“Depth-averaged 2-D numerical modeling of unsteady flow and nonuniform sediment transport in open channels”. J. Hydraulic Eng., ASCE, 135(10), 1013–1024.

van Rijn, L. C. (1985). “Flume experiments of sedimentation in channels by currents and waves.” Report S 347-II, Delft Hydraulics laboratory, Deflt, Netherlands.

Zhu, J. (1991). “A low diffusive and oscillation-free convection scheme”. Com. App. Num. Meth., 7, 225-232.

Zundel, A. K. (2000). “Surface-water modeling system reference manual”. Brigham Young University, Environmental Modeling Research Laboratory, Provo, UT.

External Links

• Aug 2006 Two-Dimensional Depth-Averaged Circulation Model CMS-M2D: Version 3.0, Report 2, Sediment Transport and Morphology Change [1]

• Aug 2008 CMS-Wave: A Nearshore Spectral Wave Processes Model for Coastal Inlets and Navigation Projects [2]

Powerpoint presentation on NET

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CMS-Flow