CMS-Flow:Non-equilibrium Sediment Transport: Difference between revisions

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__NOTOC__
''' UNDER CONSTRUCTION'''
<big>
'''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 ==
''written by Alejandro Sanchez''
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):
Last date modified: December 8, 2010
        {{Equation| <math> \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) </math>|2=1}}


where <math>h</math> is the total water depth (<math> h = \zeta + \eta </math>), <math>C_t</math> is the total load concentration, <math>C_{t*} </math> is the sediment transport capacity, <math>\beta _t</math> is the total load correction factor, <math> \nu _s </math> is the diffusion coefficient, <math>r_s</math> is the fraction of suspended sediments, <math>\alpha_t</math> is the total load adaptation coefficient, and <math>\omega_s</math> is the sediment fall velocity.
= Total-load Transport Equation =
The single-sized sediment transport model described in Sánchez and Wu (2011a) was extended to multiple-sized sediments within CMS by Sánchez and Wu (2011b). In this model, the sediment transport is separated into current- and wave-related transports. The transport due to currents includes the stirring effect of waves, and the wave-related transport includes the transport due to asymmetric oscillatory wave motion as well as steady contributions by Stokes drift, surface roller, and undertow. The current-related bed and suspended transports are combined into a single total-load transport equation, thus reducing the computational costs and simplifying the bed change computation. The 2DH transport equation for the current-related total load is


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.
{{Equation|
<math>
\frac{\partial}{\partial t} \biggl( \frac{ h C_{tk} }{\beta _{tk}} \biggr) + \frac{\partial (U_j h C_{tk})}{\partial x_j} = \frac{\partial }{\partial x_j} \biggl[ \nu _s h \frac{\partial (r_{sk} C_{tk})}{\partial x_j} \biggr] + \alpha _t \omega _{sk} (C_{t*k} - C_{tk})
</math>|1}}


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.
for ''j''= 1,2; k=1,2...,N, where ''N'' is the number of sediment size classes and


The total-load correction factor accounts for the time lag (hysteresis) between flow and sediment transport and is given by
:<math>C_{tk}</math> = actual depth-averaged total-load sediment concentration [kg/m<sup>3</sup>] for size class ''k'' defined as <math>C_{tk} = q_{tk}/(Uh)</math> in which <math>q_{tk}</math> is the total-load mass transport
    {{Equation| <math>\beta_t = \frac{1}{r_s\beta_s + (1-r_s)U_c/u_b}</math>|2=2}}


where <math>U_c</math> is the current depth-averaged current magnitude, <math>u_b</math> is the bed-load transport velocity,  and <math>\beta_s</math> is the suspended-load correction factor defined by
:<math>C_{tk*}</math> = equilibrium depth-averaged total-load sediment concentration [kg/m<sup>3</sup>] for size class ''k'' and described in the equilibrium concentration and transport rates section
    {{Equation|<math>\beta_s = \frac{\int_{a}^{h} u_s c dz } { U \int_{a}^{h} c dz} </math>|2=3}}


where <math>c</math> is the local sediment concentration, <math>a</math> is the thickness of the bed-load layer, and <math>u_s</math> is the stream-wise local current velocity. Assuming a logarithmic velocity distribution in non-dimensional form (shape function)
:<math>\beta_{tk}</math> = total-load correction factor described in the Total-Load Correction Factor section [-]
    {{Equation|<math> u'_s(z) = \frac{u_s(z)}{U_c} = \frac{\ln{(z'/z'_0)}}{ \ln{(1/z'_0)} - 1} </math>|2=4}}


and an exponential concentration profile also in non-dimensional form
:<math>r_{sk}</math> = fraction of suspended load in total load for size class k described in fraction of suspended sediments section [-]
    {{Equation|<math> c'(z') = \frac{c(z)}{c_a} = \exp{ (-\phi'(z'-a')) } </math>|2=5}}


where <math>c</math> is the local concentration,  <math>c_a</math> is a near-bottom concentration specified at a reference height <math>a</math>, <math> \phi = w_s/ \epsilon_s </math>, <math>\phi'=\phi h</math>, <math>z'=z/h</math>, and <math>a'=a/h</math>. Using (4) and (5) he analytical solution to (3) is
:<math>v_s</math> = horizontal sediment mixing coefficient described in the horizontal sediment mixing coefficient section [m<sup>2</sup>/s]
    {{Equation|<math> \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] </math>|2=6}}


where <math>E_1(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}\, dt </math> 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. Figure 1 and 2 compare the suspended load correction factor for exponential and Rouse sediment concentration profiles
:<math>\alpha_t</math> = total-load adaptation coefficient described in the adaptation coefficient section [-]


[[File:Betas_LogExp.jpg|thumb|left|400px|Figure 1. Suspended load correction factor based on logarithmic velocity and exponential concentration profiles.]]
:<math>\omega_{sk}</math> = sediment fall velocity [m/s].


[[File:Betas_LogRouse.jpg|thumb|none|400px|Figure 2. Suspended load correction factor based on logarithmic velocity and Rouse concentration profiles.]]
In the above equation, the first term represents the temporal variation of <math>C_{tk}</math>; the second term represents the horizontal advection; the third term represents the horizontal diffusion and dispersion of suspended sediments; and the last term represents the erosion and deposition. The equation may be applied to single-sized sediment transport by using a single sediment size class (i.e., N = 1). The bed composition, however, does not vary when using a single sediment size class. The units of sediment concentration used here are kg/m<sup>3</sup> rather than dimensionless volume concentrations in order to avoid numerical precision errors at low concentrations.


<br style="clear:both" />
In the above equations, it is assumed that the wave flux velocity is not included in the momentum equations. If the wave flux velocity is included, then the total flux velocity ''(V)'' should be used instead of the depth-averaged current velocity ''(U)''. The reason for this is because without a wave-induced sediment transport to counter the offshore directed transport due to the undertow, the model would predict excessive movement of sediment offshore.
 
= Fraction of Suspended Sediment =
 
In order to solve the system of equations for sediment transport implicitly, the fraction of suspended sediment must be determined explicitly. This is done by assuming
 
{{Equation|
<math>r_{sk} = \frac{q_{sk}}{q_{tk}} \simeq \frac{q_{sk*}}{q_{tk*}}
</math>|2}}
 
where <math>q_{sk} \text{and } q_{tk}</math> are the actual suspended- and total-load transport rates and <math>q_{sk*} \text{and } q_{tk*}</math> are the equilibrium suspended- and total-load transport rates.
 
=Adaptation Coefficient=
 
The total-load adaptation coefficient <math>(\alpha_t )</math> is an important parameter in the sediment transport model. There are many variations of this parameter in literature (Lin 1984; Gallappatti and Vreugdenhil 1985; and Armanini and di Silvio 1986). CMS uses a total-load adaptation coefficient <math>(\alpha_t )</math> that is related to the total-load adaptation length (L<sub>t</sub> ) and time (T<sub>t</sub>) by
 
{{Equation|
<math>L_t = \frac{Uh}{\alpha_t \omega_s} = UT_t</math>
|3}}
 
where:
:<math>\omega_s</math> = sediment fall velocity corresponding to the transport grain size for single-sized sediment transport or the median grain size for multiple-sized sediment transport [m/s]
 
:U = depth-averaged current velocity [m/s]
 
:h = water depth [m].
 
The adaptation length or time is a characteristic distance or time for sediment to adjust from non-equilibrium to equilibrium transport. Because the total load is a combination of the bed and suspended loads, the associated adaptation length may be calculated as <math> L_t = r_s L_s + (1 - r_s)L_b \text{ or } L_t = \text{ max }(L_s, L_b ), \text{where } L_s \text{and } L_b</math> are the suspended- and bed-load adaptation lengths. The symbol <math>L_s</math> is defined as
 
{{Equation|
<math>L_s = \frac {Uh}{\alpha \omega_s} = UT_s
</math>|4}}
 
in which <math>\alpha \text{ and } T_s</math> are the adaptation coefficient lengths for suspended load. The adaptation coefficient ''<math>(\alpha )</math>'' can be calculated either empirically or based on analytical solutions to the pure vertical convection-diffusion equation of suspended sediment. One example of an empirical formula is that proposed by Lin (1984):
 
{{Equation|
<math>\alpha = 3.25 + 0.55 \ ln \big(\frac{\omega_s}{\kappa u_*}\big) </math>|5}}
 
where u<sub>*</sub> is the bed shear stress, and <math>\kappa</math> is the von Karman constant. Armanini and di Silvio (1986) proposed an analytical equation:
 
{{Equation|<math>\frac{1}{a} = \frac{\delta}{h} + \left(1 - \frac{\delta}{h}\right)\exp \left[-1.5\left(\frac{\delta}{h}\right)^{-1/6} \frac{\omega_s}{u_*} \right]</math>|6}}
 
where <math>\delta</math> is the thickness of the bottom layer defined by <math>\delta = 33 z_0 \text { and } z_0</math> is the zero-velocity distance from the bed. Gallappatti (1983) proposed the following equation to determine the suspended-load adaptation time:
 
{{Equation|<math>T_s = \frac{h}{u_*}exp
\left[\begin{align}&(1.57 - 20.12u_r)\omega_* ^3 + (326.832u_r ^{2.2047} - 0.2)\omega_* ^2\\& +(0.1385 \ ln \ u_r - 6.4061)\omega_* + (0.5467u_r - 2.1963)\end{align}\right]</math>|7}}
 
where u<sub>*</sub> is the current related bottom shear velocity, <math>u_r = u_*/U, \text{ and } \omega_* = \omega_s /u_*</math>.
 
The bed-load adaptation length (L<sub>b</sub>) is generally related to the dimension of bed forms such as sand dunes. Large bed forms are generally proportional to the water depth, and, therefore, the bed-load adaptation length can be estimated as L<sub>b</sub> = a<sub>b</sub>h in which a<sub>b</sub> is an empirical coefficient on the order of 5-10. Although limited guidance exists on methods to estimate L<sub>b</sub>, the determination of L<sub>b</sub> is still empirical and in the developmental stage. For a detailed discussion of the adaptation length, the reader is referred to Wu (2007). In general, it is recommended that the adaptation length be calibrated with field data in order to achieve the best and most reliable results.
 
= Total-load correction factor =
The total-load correction factor (β<sub>tk</sub>) accounts for the vertical distribution of the suspended sediment concentration and velocity profiles as well as the fact that bed load travels at a slower velocity than the depth-averaged current velocity (Figure 1) (Wu 2007). By definition, β<sub>tk</sub> is the ratio of the depth-averaged total-load and flow velocities.
 
[[File:fig_2_3.png]]<br>
'''Figure 1. Schematic of sediment and current vertical profiles.'''
 
In a combined bed-load and suspended-load model, the correction factor is given by
 
{{Equation|<math>\beta_{tk} = \frac{1}{r_{sk} / \beta_{sk} + (1-r_{sk})U/u_{bk}} </math>|8}}
 
where u<sub>bk</sub> is the bed-load velocity, and β<sub>sk</sub> is the suspended-load correction factor and is defined as the ratio of the depth-averaged suspended sediment and flow velocities. Since most sediment is transported near the bed, both the total- and suspended-load correction factors (β<sub>tk</sub> and β<sub>sk</sub>) are usually less than 1 and typically in the range of 0.3 and 0.7, respectively. By assuming logarithmic current velocity and exponential suspended sediment concentration profiles, an explicit expression for the suspended-load correction factor (β<sub>sk</sub>) may be obtained as (Sánchez and Wu 2011b)
 
{{Equation|<math>
\beta_{sk} = \frac{\int_{z_b + a}^{\overline{\eta}}uc_k dz}{U\int_{z_b + a}^{\overline{\eta}}c_k dz} =
\frac{E_1(\phi_k A) - E_1 (\phi_k) + ln(A/Z)e^{-\phi_k A} - ln(1/Z)e^{-\phi_k}}{e^{-\phi_k A}[ln(1/Z)-1][1-e^{-\phi_k(1-A)}]}</math>|9}}
 
where:
 
<math>\phi_k = \omega_{sk}h/\epsilon_k [-]</math>
 
A = a/h [-]
 
Z = z<sub>a</sub>/h [-]
 
<math>\epsilon_k</math> = vertical mixing coefficient [m<sup>2</sup>/s]
 
a = reference height near the bed for the suspended load [m]
 
z<sub>a</sub> = apparent roughness length [m]
 
<math>E_1 (x) = \int_x ^\infty \frac{e^{-t}}{t} dt</math> (exponential integral).
 
The equation can be further simplified by assuming that the reference height is proportional to the apparent roughness length (e.g., a = 30z<sub>z</sub>) so that <math>\beta_{sk} = \beta_{sk}(Z,\phi_k)</math>. Figure 2 shows a comparison of the suspended-load correction factor based on the logarithmic velocity with exponential and Rouse suspended sediment concentration profiles. Both cases show similar behavior for the suspended-load correction factor (<math>\beta_{sk}</math>). For fine sediments (small fall velocity), <math>\beta_{sk}</math> is close to 1.0 and experiences smaller influences by the bottom roughness, while for course sediments, <math>\beta_{sk}</math>can be as low as 0.5 and is largely influenced by the bottom roughness. This is to be expected since course sediments are transported more closely to the bottom, compared to fine sediments.
 
[[File:fig_2_4.png|400px]]
<br>
'''Figure 2. Suspended load correction factors based on the logarithmic velocity profile and (a) exponential and (b) Rouse suspended sediment profile. The Rouse number is <math>r = \omega_s /(\kappa u_* )</math>'''
 
The bed-load velocity (u<sub>bk</sub>) in Equation 8 is calculated using the van Rijn (1984a) formula with re-calibrated coefficients from Wu et al. (2006):
 
{{Equation|<math>
u_{bk} = 1.64 \left(\frac{\tau_b ^'}{\tau_{crk}} - 1 \right)^{0.5} \sqrt{(s-1)gd_k}
</math>|10}}
 
where:
 
s = specific gravity [-]
 
g = gravitational constant (~9.81 m/s<sup>2</sup>)
 
d<sub>k</sub> = characteristic grain diameter for the k<sup>th</sup> size class [m]
 
<math>\tau_b ^' = (n^'/n)^{3/2}\tau_b</math> = grain-related bed shear stress [Pa]


In CMS, the bed load velocity is determined using the Van Rijn formula (1984)
<math>n^' = d_{50}^{1/6}/20 \ </math> = grain-related Manning’s roughness coefficient [s/m<sup>1/3</sup>]
    {{Equation|<math> \frac{u_b}{\sqrt{(\rho_s/\rho-1)gd_k}} = aT^b </math>|2=4}}


where and <math>T</math> is the non-dimensional transport stage number given by <math>T=\tau'_b/\tau_cr -1</math>, and <math>a,b</math> are empirical coefficients equal to 1.64 and 0.5 (Wu et al. 2006).
<math>\tau_{crk}</math> = critical bed shear stress for the k<sup>th</sup> size class [Pa].


== Bed Change Equation ==
The change in the water depth is calculated by
      {{Equation| <math> (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] </math>|2=4}}


where <math> p'_m </math> is the sediment porosity, and <math> D_s </math> 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 <math>\theta_{ck}^{he} = \xi_k \theta_{ck}</math> where <math>\xi_k</math> is the dimensionless hiding and exposure function and <math>\theta_{ck}</math> 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 <math>\xi_k = (d_k/d_{50})^{-m} </math> where is the grain size corresponding to the 50th percentile, and <math> m </math> 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.
= Bed Change Equation =
To change the value of <math> m </math> 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 change in the water depth is calculated by  
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
{{Equation|<math>
      {{Equation|<math> \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 ) </math>|2=5}}
\rho_s (1 - p'_m) \biggl( \frac{\partial z_b}{\partial t} \biggr)_k = \alpha _t \omega _{sk} (C_{tk} - C_{tk*}) + \frac{\partial }{\partial x_j} \biggl[ D_s q_{bk} \frac{\partial z_b}{\partial x_j} \biggr]</math>|11}}


where the subscripts p and i indicate the center and neighboring cells respectively, <math>\delta x</math>  is the cell center distance between cells p and i, <math>  \Delta \zeta ^a _p </math> is the bed change due to avalanching, A is the cell area, <math>\phi</math> is the bed slope, <math>\phi _r</math> is the sediment repose angle, R is an under-relaxation factor (approximately 0.25-0.5), and <math>\textrm{H}</math> and <math>\textrm{ sgn}</math> are the Heaviside and sign functions. The equation is applied every morphologic time step by sweeping through all of the computational cells to calculate <math> \Delta \zeta </math> and then modifying the bathymetry as <math>\zeta^{n+1}=\zeta^n + \Delta \zeta^a  </math>.
where:


== Boundary Conditions ==
z<sub>b</sub> = bed elevation with respect to the vertical datum [m]
There are three types of boundary conditions in the sediment transport: Wet-dry, Outflow and Inflow.


1. Wet-dry interface.
p<sub>m</sub><sup>'</sup> = bed porosity [-]
: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
<math>\rho_s</math> = sediment (material) density (~2650 kg/m<sup>3</sup> for quartz sediment)
: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
D<sub>s</sub> = empirical bed-slope coefficient (constant) [-]
: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 or SEDIMENT_INFLOW_LOADING_FACTOR.


== Numerical Methods ==
<math>q_{bk} = hUC_{tk}(1-r_{sk})</math> is the bed-load mass transport rate magnitude [kg/m/s].
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 ADVECTION_SCHEME.


== References ==
The first term on the right-hand side of the above equation represents the bed change due to sediment exchange near the bed. The last term accounts for the effect of the bed slope on bed-load transport. The bed-slope coefficient (D<sub>s</sub>) is usually about 0.1 to 3.0. For a detailed derivation of the above equation, the reader is referred to Sanchez and Wu (2011a). The total bed change is calculated as the sum of the bed changes for all size classes:


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.
{{Equation|<math>
\frac{\partial z_b}{\partial t} = \sum_k \left(\frac{\partial z_b}{\partial t}  \right)_k
</math>|12}}


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
= Bed Sorting and Gradation (nonuniform sediments) =
When simulating nonuniform sediments it is necessary to keep track of the vertical variation in bed composition. In order to do this, the bed is divided into discrete layers. The top layer is the mixing or active layer and is the layer of sediment which is actively being exchanged with the bed and suspended loads. The temporal variation of the bed-material in the mixing and second layers is calculated as (Wu 2004)
{{Equation|
<math>
\frac{\partial (\delta_m p_{bk}) }{\partial t} + \biggl( \frac{\partial \zeta}{\partial t}  \biggr)_k = -\frac{\partial (\delta_m p_{bk}) }{\partial t} = p^*_{bk} \biggl(\frac{\partial \delta_m }{\partial t} + \frac{\partial \zeta }{\partial t} \biggr)
</math>|11}}


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.
where <math> \delta_m </math> and <math> \delta_s </math> are the thicknesses of the mixing and second layers respectively. <math> p^*_{bk} = p_{bk} </math> for <math> \partial \zeta / \partial t + \partial \delta_m / \partial t \le 0 </math> or <math> p^*_{bk} = p_{sk} </math> otherwise. Figure 3 shows an example of the bed layer evolution over time for varying bed deposition and erosion.


Soulsby, R. L. (1997). "Dynamics of marine sands, a manual for practical applications". H. R. Wallingford, UK: Thomas Telford.
[[File:Bed_layering_V2.jpg|thumb|left|500px|Figure 3. Example of bed layer variation for varying deposition and erosion.]]


Watanabe, A. (1987). “3-dimensional numerical model of beach evolution”. Proc. Coastal Sediments ’87, ASCE, 802-817.  
<br style="clear:both" />
= Boundary Conditions =
There are three types of boundary conditions in the sediment transport: Wet-dry, Outflow and Inflow.  


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.
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.  


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


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


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


== External Links ==
= 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 [http://libweb.erdc.usace.army.mil/Archimages/705.PDF]
* Aug 2006 Two-Dimensional Depth-Averaged Circulation Model CMS-M2D: Version 3.0, Report 2, Sediment Transport and Morphology Change [http://libweb.erdc.usace.army.mil/Archimages/705.PDF]


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Latest revision as of 16:33, 6 November 2014

UNDER CONSTRUCTION

written by Alejandro Sanchez Last date modified: December 8, 2010

Total-load Transport Equation

The single-sized sediment transport model described in Sánchez and Wu (2011a) was extended to multiple-sized sediments within CMS by Sánchez and Wu (2011b). In this model, the sediment transport is separated into current- and wave-related transports. The transport due to currents includes the stirring effect of waves, and the wave-related transport includes the transport due to asymmetric oscillatory wave motion as well as steady contributions by Stokes drift, surface roller, and undertow. The current-related bed and suspended transports are combined into a single total-load transport equation, thus reducing the computational costs and simplifying the bed change computation. The 2DH transport equation for the current-related total load 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 \frac{\partial}{\partial t} \biggl( \frac{ h C_{tk} }{\beta _{tk}} \biggr) + \frac{\partial (U_j h C_{tk})}{\partial x_j} = \frac{\partial }{\partial x_j} \biggl[ \nu _s h \frac{\partial (r_{sk} C_{tk})}{\partial x_j} \biggr] + \alpha _t \omega _{sk} (C_{t*k} - C_{tk}) }

(1)

for j= 1,2; k=1,2...,N, where N is the number of sediment size classes 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 C_{tk}} = actual depth-averaged total-load sediment concentration [kg/m3] for size class k defined as 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_{tk} = q_{tk}/(Uh)} in which 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 q_{tk}} is the total-load mass transport
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_{tk*}} = equilibrium depth-averaged total-load sediment concentration [kg/m3] for size class k and described in the equilibrium concentration and transport rates section
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_{tk}} = total-load correction factor described in the Total-Load Correction Factor section [-]
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_{sk}} = fraction of suspended load in total load for size class k described in fraction of suspended sediments section [-]
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 v_s} = horizontal sediment mixing coefficient described in the horizontal sediment mixing coefficient section [m2/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 \alpha_t} = total-load adaptation coefficient described in the adaptation coefficient section [-]
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_{sk}} = sediment fall velocity [m/s].

In the above equation, the first term represents the temporal variation 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 C_{tk}} ; the second term represents the horizontal advection; the third term represents the horizontal diffusion and dispersion of suspended sediments; and the last term represents the erosion and deposition. The equation may be applied to single-sized sediment transport by using a single sediment size class (i.e., N = 1). The bed composition, however, does not vary when using a single sediment size class. The units of sediment concentration used here are kg/m3 rather than dimensionless volume concentrations in order to avoid numerical precision errors at low concentrations.

In the above equations, it is assumed that the wave flux velocity is not included in the momentum equations. If the wave flux velocity is included, then the total flux velocity (V) should be used instead of the depth-averaged current velocity (U). The reason for this is because without a wave-induced sediment transport to counter the offshore directed transport due to the undertow, the model would predict excessive movement of sediment offshore.

Fraction of Suspended Sediment

In order to solve the system of equations for sediment transport implicitly, the fraction of suspended sediment must be determined explicitly. This is done by assuming

 

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_{sk} = \frac{q_{sk}}{q_{tk}} \simeq \frac{q_{sk*}}{q_{tk*}} }

(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 q_{sk} \text{and } q_{tk}} are the actual suspended- and total-load transport rates 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 q_{sk*} \text{and } q_{tk*}} are the equilibrium suspended- and total-load transport rates.

Adaptation Coefficient

The total-load adaptation 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 (\alpha_t )} is an important parameter in the sediment transport model. There are many variations of this parameter in literature (Lin 1984; Gallappatti and Vreugdenhil 1985; and Armanini and di Silvio 1986). CMS uses a total-load adaptation 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 (\alpha_t )} that is related to the total-load adaptation length (Lt ) and time (Tt) 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 L_t = \frac{Uh}{\alpha_t \omega_s} = UT_t}

(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 \omega_s} = sediment fall velocity corresponding to the transport grain size for single-sized sediment transport or the median grain size for multiple-sized sediment transport [m/s]
U = depth-averaged current velocity [m/s]
h = water depth [m].

The adaptation length or time is a characteristic distance or time for sediment to adjust from non-equilibrium to equilibrium transport. Because the total load is a combination of the bed and suspended loads, the associated adaptation length may be calculated as 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 L_t = r_s L_s + (1 - r_s)L_b \text{ or } L_t = \text{ max }(L_s, L_b ), \text{where } L_s \text{and } L_b} are the suspended- and bed-load adaptation lengths. The symbol 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 L_s} is defined as

 

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 L_s = \frac {Uh}{\alpha \omega_s} = UT_s }

(4)

in which 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 \text{ and } T_s} are the adaptation coefficient lengths for suspended load. The adaptation 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 (\alpha )} can be calculated either empirically or based on analytical solutions to the pure vertical convection-diffusion equation of suspended sediment. One example of an empirical formula is that proposed by Lin (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 \alpha = 3.25 + 0.55 \ ln \big(\frac{\omega_s}{\kappa u_*}\big) }

(5)

where u* is the bed shear stress, 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 \kappa} is the von Karman constant. Armanini and di Silvio (1986) proposed an analytical 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 \frac{1}{a} = \frac{\delta}{h} + \left(1 - \frac{\delta}{h}\right)\exp \left[-1.5\left(\frac{\delta}{h}\right)^{-1/6} \frac{\omega_s}{u_*} \right]} (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 \delta} is the thickness of the bottom layer 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 \delta = 33 z_0 \text { and } z_0} is the zero-velocity distance from the bed. Gallappatti (1983) proposed the following equation to determine the suspended-load adaptation time:

  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_s = \frac{h}{u_*}exp \left[\begin{align}&(1.57 - 20.12u_r)\omega_* ^3 + (326.832u_r ^{2.2047} - 0.2)\omega_* ^2\\& +(0.1385 \ ln \ u_r - 6.4061)\omega_* + (0.5467u_r - 2.1963)\end{align}\right]} (7)

where u* is the current related bottom shear velocity, 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_r = u_*/U, \text{ and } \omega_* = \omega_s /u_*} .

The bed-load adaptation length (Lb) is generally related to the dimension of bed forms such as sand dunes. Large bed forms are generally proportional to the water depth, and, therefore, the bed-load adaptation length can be estimated as Lb = abh in which ab is an empirical coefficient on the order of 5-10. Although limited guidance exists on methods to estimate Lb, the determination of Lb is still empirical and in the developmental stage. For a detailed discussion of the adaptation length, the reader is referred to Wu (2007). In general, it is recommended that the adaptation length be calibrated with field data in order to achieve the best and most reliable results.

Total-load correction factor

The total-load correction factor (βtk) accounts for the vertical distribution of the suspended sediment concentration and velocity profiles as well as the fact that bed load travels at a slower velocity than the depth-averaged current velocity (Figure 1) (Wu 2007). By definition, βtk is the ratio of the depth-averaged total-load and flow velocities.

Fig 2 3.png
Figure 1. Schematic of sediment and current vertical profiles.

In a combined bed-load and suspended-load model, the correction factor 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_{tk} = \frac{1}{r_{sk} / \beta_{sk} + (1-r_{sk})U/u_{bk}} } (8)

where ubk is the bed-load velocity, and βsk is the suspended-load correction factor and is defined as the ratio of the depth-averaged suspended sediment and flow velocities. Since most sediment is transported near the bed, both the total- and suspended-load correction factors (βtk and βsk) are usually less than 1 and typically in the range of 0.3 and 0.7, respectively. By assuming logarithmic current velocity and exponential suspended sediment concentration profiles, an explicit expression for the suspended-load correction factor (βsk) may be obtained as (Sánchez and Wu 2011b)

  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_{sk} = \frac{\int_{z_b + a}^{\overline{\eta}}uc_k dz}{U\int_{z_b + a}^{\overline{\eta}}c_k dz} = \frac{E_1(\phi_k A) - E_1 (\phi_k) + ln(A/Z)e^{-\phi_k A} - ln(1/Z)e^{-\phi_k}}{e^{-\phi_k A}[ln(1/Z)-1][1-e^{-\phi_k(1-A)}]}} (9)

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 \phi_k = \omega_{sk}h/\epsilon_k [-]}

A = a/h [-]

Z = za/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 \epsilon_k} = vertical mixing coefficient [m2/s]

a = reference height near the bed for the suspended load [m]

za = apparent roughness length [m]

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} (exponential integral).

The equation can be further simplified by assuming that the reference height is proportional to the apparent roughness length (e.g., a = 30zz) so that 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_{sk} = \beta_{sk}(Z,\phi_k)} . Figure 2 shows a comparison of the suspended-load correction factor based on the logarithmic velocity with exponential and Rouse suspended sediment concentration profiles. Both cases show similar behavior for the suspended-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 \beta_{sk}} ). For fine sediments (small fall velocity), 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_{sk}} is close to 1.0 and experiences smaller influences by the bottom roughness, while for course 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 \beta_{sk}} can be as low as 0.5 and is largely influenced by the bottom roughness. This is to be expected since course sediments are transported more closely to the bottom, compared to fine sediments.

Fig 2 4.png
Figure 2. Suspended load correction factors based on the logarithmic velocity profile and (a) exponential and (b) Rouse suspended sediment profile. The Rouse number 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 r = \omega_s /(\kappa u_* )}

The bed-load velocity (ubk) in Equation 8 is calculated using the van Rijn (1984a) formula with re-calibrated coefficients from Wu et al. (2006):

  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_{bk} = 1.64 \left(\frac{\tau_b ^'}{\tau_{crk}} - 1 \right)^{0.5} \sqrt{(s-1)gd_k} } (10)

where:

s = specific gravity [-]

g = gravitational constant (~9.81 m/s2)

dk = characteristic grain diameter for the kth size class [m]

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 \tau_b ^' = (n^'/n)^{3/2}\tau_b} = grain-related bed shear stress [Pa]

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 n^' = d_{50}^{1/6}/20 \ } = grain-related Manning’s roughness coefficient [s/m1/3]

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 \tau_{crk}} = critical bed shear stress for the kth size class [Pa].



Bed Change Equation

The change in the water depth is calculated 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 \rho_s (1 - p'_m) \biggl( \frac{\partial z_b}{\partial t} \biggr)_k = \alpha _t \omega _{sk} (C_{tk} - C_{tk*}) + \frac{\partial }{\partial x_j} \biggl[ D_s q_{bk} \frac{\partial z_b}{\partial x_j} \biggr]} (11)

where:

zb = bed elevation with respect to the vertical datum [m]

pm' = bed porosity [-]

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 \rho_s} = sediment (material) density (~2650 kg/m3 for quartz sediment)

Ds = empirical bed-slope coefficient (constant) [-]

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 q_{bk} = hUC_{tk}(1-r_{sk})} is the bed-load mass transport rate magnitude [kg/m/s].

The first term on the right-hand side of the above equation represents the bed change due to sediment exchange near the bed. The last term accounts for the effect of the bed slope on bed-load transport. The bed-slope coefficient (Ds) is usually about 0.1 to 3.0. For a detailed derivation of the above equation, the reader is referred to Sanchez and Wu (2011a). The total bed change is calculated as the sum of the bed changes for all size classes:

  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 z_b}{\partial t} = \sum_k \left(\frac{\partial z_b}{\partial t} \right)_k } (12)

Bed Sorting and Gradation (nonuniform sediments)

When simulating nonuniform sediments it is necessary to keep track of the vertical variation in bed composition. In order to do this, the bed is divided into discrete layers. The top layer is the mixing or active layer and is the layer of sediment which is actively being exchanged with the bed and suspended loads. The temporal variation of the bed-material in the mixing and second layers is calculated as (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 (\delta_m p_{bk}) }{\partial t} + \biggl( \frac{\partial \zeta}{\partial t} \biggr)_k = -\frac{\partial (\delta_m p_{bk}) }{\partial t} = p^*_{bk} \biggl(\frac{\partial \delta_m }{\partial t} + \frac{\partial \zeta }{\partial t} \biggr) }

(11)

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 \delta_m } and are the thicknesses of the mixing and second layers respectively. for or otherwise. Figure 3 shows an example of the bed layer evolution over time for varying bed deposition and erosion.

Figure 3. Example of bed layer variation for varying deposition and erosion.


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 or SEDIMENT_INFLOW_LOADING_FACTOR.

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 ADVECTION_SCHEME.

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]

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