CMS-Flow:Non-equilibrium Sediment Transport

1.0 Introduction

1.0 Introduction

This report describes the numerical implementation of the non-equilibrium sediment transport using a total load approach as described in the report by Wu (2007) in the Coastal Modeling System (CMS) circulation model CMS-Flow. 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.

2.0 Cartesian Grid

The governing equations are discretized on a rectilinear grid as shown in Figure 1. Vector values such as velocities and sediment transport rates are referenced with respect to their corresponding cell faces, while scalar values such as water depths, water surface elevations and sediment concentrations are referenced with respect to the cell centers.

3.0 Advection-Diffusion Equation

The total load sediment concentration is solved using an Advection-Diffusion (AD) equation.

where is the total water depth, is the total load concentration, is the sediment transport capacity, is the total load correction factor, is the current velocity in the x-direction, is the current velocity in the y-direction, is the diffusion coefficient, is the fraction of suspended sediments, is the total load adaptation coefficient, is the sediment fall velocity and is the bed slope term. The sediment fall velocity can be set by the user using the card SEDIMENT_FALL_VELOCITY followed the fall velocity in m/s. The concentration change is discretized using a first order Euler scheme. The advection terms are discretized using an upwind or Hybrid Linear/Parabolic Approximation (HLPA) scheme. The advection scheme is specified in CMS using the card NET_ADVECTION_SCHEME as OFF, UPWIND or HLPA. The default scheme is HLPA.

4.0 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

5.0 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.

6.0 Avalanching

The process of avalanching is simulated by enforcing , where is 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 where the subscripts and indicate the center and neighboring cells respectively, is the cell center distance between and , is the bed change due to avalanching and is the cell area, is the bed slope, is the sediment repose angle and R is an under-relaxation factor (approximately 0.25-0.5). The coefficient = 1 for (upslope) and = -1 for (downslope). Note that for Cartesian grids equation (6) may be simplified to where is the cell width at cell .

7.0 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.

8.0 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.