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 using a bed-material formulation. The equation is obtained by adding the suspended- and bed-load transport equations to obtain the general sediment balance equation and then substituting a non-equilibrium expression for the bed elevation change as suggested by Wu (2004). The resulting transport equation is

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

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 (1985), or the Watanabe (1987) transport equations. The advantage of a bed-material approach is that the suspended- and bed-load transport equations are combined into a single equation and there is one less empirical parameter to estimate.

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) that 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 introducedto correct the critical shields parameter such as where is the dimensionless hiding and exposure function, is the critical shear stress of the transport grain size and is the corrected shields parameter. The CMS adapts the formula of Parker et al. (1995) where is the grain size corresponding to the 50th percentile, and is an empirical coefficient set to 0.7.

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

     Failed to parse (unknown function "\tt"): {\displaystyle  \Delta \zeta ^a _p = R \frac{ A_i \delta x_i }{A_p + A_i} H( | \phi _i | - \phi _r ) \biggl[tan \phi _i - {\tt sgn }( \phi _i) tan \phi _r \biggr] }

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 and R is an under-relaxation factor (approximately 0.25-0.5). The coefficient $\gamma$ = 1 for $\zeta _{p}<\zeta _{i}$ (upslope) and $\gamma$ = -1 for $\zeta _{p}>\zeta _{i}$ (downslope).

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.

Numerical Methods

The governing equations are discretized using the Finite Volume Method on a staggered, non-uniform Cartesian grid. Advection terms are discretized with the near second order Hybrid Linear/Parabolic Approximation (HLPA) scheme of Zhu (1991). Time integration is calculated with a simple explicit forward Euler scheme. Diffusion terms are discretized with the standard central difference scheme.

CMS Cards Related to NET

To turn on the non-equilibrium sediment transport (NET) option in the CMS, simply check the box that reads “Non-equilibrium sediment transport” within the “Transport” tab in the “CMS Model Control Window”. This can also be done by setting the CMS card SED_TRAN_FORMULATION in the CMS cards file (*.cmcards) to NET. The sediment fall velocity can be set with the advanced card SEDIMENT_FALL_VELOCITY followed the fall velocity in m/s. The advection scheme is specified in CMS with the card NET_ADVECTION_SCHEME as OFF, UPWIND or HLPA. The default scheme is HLPA. The total-load adaptation length is set with the card TOTAL_LOAD_ADAPTATION_LENGTH.

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.

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]