CMS-Flow:Non-equilibrium Sediment Transport

Non-equilibrium Sediment Transport in CMS - UNDER CONSTRUCTION

Introduction

This wiki report describes the Non-equilibrium Sediment Transport (NET) in the Coastal Modeling System (CMS) developed by the U.S. Army Corps of Engineers (USACE) with the objective of understanding the coastal hydrodynamics, and sediment transport in order to improve the usage Operation and Maintenance funds. The modeling system consists of three main components which are coupled together: (1) a depth-averaged hydrodynamic, (2) a steady-state spectral wave model, and (3) a depth-averaged sediment transport model. This paper focuses on the description of a new non-equilibrium sediment transport formulation for calculation of morphology change. The NET adopts a total load approach as described in the report by Wu (2007) in the Coastal Modeling System (CMS) circulation model CMS-Flow.

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

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

where $h$ is the total water depth, $C_{t}$ is the total load concentration, $C_{t*}$ is the sediment transport capacity, $\beta _{t}$ is the total load correction factor, $\mu$ 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.

Bed Change Equation

In the case where the advection-diffusion equation is used to simulate the sediment transport and mixing, the change in the water depth is calculated using the sediment continuity equation

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

Hiding and Exposure

In order to simulate long periods of time, it is usually not feasible or desirable to calculate multiple sediment sizes since this requires solving a separate A-D equation for sediment size and also calculating the bed layering and sorting. In many cases the bed material is dominated 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 effects. By assuming that the spatial distribution of the bed material composition is constant in time, a hiding and exposure correction function can be used to 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. In this study the formula of Parker et al. (1995) is used and is given by where is the grain size corresponding to the 50th percentile, and is an empirical coefficient set here 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

     $\Delta \zeta _{p}=R\sum _{i}{\frac {A_{i}\delta x(tan\phi -\gamma tan\phi _{r})}{A_{p}+A_{i}}}$

where 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 $\zeta _{p}<\zeta _{p}$ (upslope) and $\gamma$ = -1 for (downslope). Note that for Cartesian grids equation (6) may be simplified to where is the cell width at cell .

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

Discretization of Governing Equations The governing equations are discretized using the Finite Volume Method on a staggered, non-uniform Cartesian grid. Advection terms in the flow equations are discretized using the upwind scheme, and the advection terms of the sediment transport equation are discretized using a near second order Hybrid Linear/Parabolic Approximation (HLPA) scheme of Zhu (1991). Time integration is done using the simple explicit forward Euler scheme. Diffusion terms in the flow and sediment transport are discretized using 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 by the user using the card SEDIMENT_FALL_VELOCITY followed the fall velocity in m/s. The advection scheme is specified in CMS using the card NET_ADVECTION_SCHEME as OFF, UPWIND or HLPA. The default scheme is HLPA.