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):

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial}{\partial t} \biggl( \frac{ h C_t }{\beta _t} \biggr) + \frac{\partial (U_j h C_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) }

(1)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h} is the total water depth (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h = \zeta + \eta } ), Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_t} is the total load concentration, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{t*} } is the sediment transport capacity, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta _t} is the total load correction factor, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu _s } is the diffusion coefficient, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r_s} is the fraction of suspended sediments, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha_t} is the total load adaptation coefficient, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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. 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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_t = \frac{1}{r_s\beta_s + (1-r_s)U_c/u_b}}

(2)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_c} is the current depth-averaged current magnitude, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_b} is the bed-load transport velocity, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_s} is the suspended-load correction factor defined by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_s = \frac{\int_{a}^{h} u_s c dz } { U \int_{a}^{h} c dz} }

(3)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c} is the local sediment concentration, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a} is the thickness of the bed-load layer, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_s} is the stream-wise local current velocity. Assuming a logarithmic velocity distribution in non-dimensional form (shape function)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u'_s(z) = \frac{u_s(z)}{U_c} = \frac{\ln{(z'/z'_0)}}{ \ln{(1/z'_0)} - 1} }

(4)

and an exponential concentration profile also in non-dimensional form

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c'(z') = \frac{c(z)}{c_a} = \exp{ (-\phi'(z'-a')) } }

(5)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c} is the local concentration, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_a} is a near-bottom concentration specified at a reference height Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi = w_s/ \epsilon_s } , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi'=\phi h} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z'=z/h} , and $a'=a/h$. Using (4) and (5) he analytical solution to (3) is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_s = \frac{e^{\phi' a'}}{(\ln(1/z'_0)-1)(1-e^{-\phi'(1-a')})} \biggl[ E_1(\phi' a') - E_1(\phi') + \frac{\ln{(a'/z'_0)}}{e^{\phi' a'}} - \frac{\ln{(1/z'_0)}}{e^{\phi'}} \biggr] }

(6)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_1(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}\, dt } is the exponential integral. Because the exponential integral requires and series expansion to solve it is more efficient to create a look up table and interpolate the values during the simulation which is what is done in CMS. A look up table is also created for when using the Rouse sediment concentration profile.

In CMS, the bed load velocity is determined using the Van Rijn formula (1984)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{u_b}{\sqrt{(\rho_s/\rho-1)gd_k}} = aT^b }

(4)

where and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T} is the non-dimensional transport stage number given by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T=\tau'_b/\tau_cr -1} , and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a,b} are empirical coefficients equal to 1.64 and 0.5 (Wu et al. 2006).

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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (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] }

(4)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p'_m } is the sediment porosity, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta_{ck}^{he} = \xi_k \Theta_{ck}} where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi_k} is the dimensionless hiding and exposure function and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi_k = (d_k/d_{50})^{-m} } where is the grain size corresponding to the 50th percentile, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m } is an empirical coefficient between 0.5-1.0 (default is 0.7).

The transport grain size is specified in the Advanced Card TRANSPORT_GRAIN_SIZE. The transport grain size should be the dominant grain size in the area of interest. To change the value of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m } another Advanced Card HIDING_EXPOSURE_COEFFICIENT. If it is desired to test the model with a constant grain size and ignore the information in the D50_dataset, the Advanced Card CONSTANT_GRAIN_SIZE.

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 (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta \zeta ^a _p = 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 ) }

(5)

where the subscripts p and i indicate the center and neighboring cells respectively, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta x} is the cell center distance between cells p and i, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta \zeta ^a _p } is the bed change due to avalanching, A is the cell area, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} is the bed slope, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _r} is the sediment repose angle, R is an under-relaxation factor (approximately 0.25-0.5), and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textrm{H}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textrm{ sgn}} are the Heaviside and sign functions. The equation is applied every morphologic time step by sweeping through all of the computational cells to calculate Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta \zeta } and then modifying the bathymetry as $\zeta ^{n+1}=\zeta ^{n}+\Delta \zeta ^{a}$.

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 <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