Model help:SuspSedDensityStrat
SuspSedDensityStrat
This model is used for calculating the effect of density stratification on the vertical profiles of velocity and suspended sediment.
Model introduction
The model is the calculation of Density Stratification Effects Associated with Suspended Sediment in Open Channels.
This program calculates the effect of sediment self-stratification on the streamwise velocity and suspended sediment concentration profiles in open-channel flow. Two options are given. Either the near-bed reference concentration Cr can be specified by the user, or the user can specify a shear velocity due to skin friction u*s and compute Cr from the Garcia-Parker sediment entrainment relation.
Model parameters
Uses ports
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Provides ports
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Main equations
[math]\displaystyle{ k_{c} = {\frac{11 H}{e^ \left ( \kappa C z \right ) }} }[/math] (1)
[math]\displaystyle{ Cz = C_{f} ^ \left ( {\frac{-1}{2}} \right ) ={\frac{U}{u_{*}}} }[/math] (2)
[math]\displaystyle{ R = {\frac{\rho _{s}}{\rho}} - 1 }[/math] (3)
[math]\displaystyle{ Re_{p} = {\frac{sqrt \left ( R g D \right ) D}{\nu}} }[/math] (4)
[math]\displaystyle{ R_{f} = f \left ( Re_{p} \right ) }[/math] (5)
[math]\displaystyle{ R_{f} = {\frac{v_{s}}{sqrt \left ( R g D \right )}} }[/math] (6)
[math]\displaystyle{ \nu _{t} {\frac{du^*}{dz}} = u_{*} ^2 \left ( 1 - {\frac{z}{H}} \right ) }[/math] (7)
[math]\displaystyle{ v_{s} c^* + \nu _{t} {\frac {dc^*}{dz}} = 0 }[/math] (8)
[math]\displaystyle{ \nu_{t} = \kappa u_{*} H F_{1} \left ( \zeta \right ) F_{2} \left ( Ri \right ) }[/math] (9)
[math]\displaystyle{ \zeta = {\frac{z}{H}} }[/math] (10)
[math]\displaystyle{ Ri = -Rg {\frac{{\frac{dc^*}{dz}}}{\left ( {\frac{du^*}{dz}} \right ) ^2}} }[/math] (11)
[math]\displaystyle{ {\frac{u^* |_{\zeta_{t}}}{u_{*}}} = {\frac{1}{\kappa}} ln \left ( 30 \zeta _{r} {\frac{H}{k_{c}}} \right ) }[/math] (12)
- Dimensionless forms
[math]\displaystyle{ u={\frac{u^*}{u_{*}}} }[/math] (13)
[math]\displaystyle{ c = {\frac{c^*}{c_{r} ^*}} }[/math] (14)
[math]\displaystyle{ {\frac{du}{d\zeta}} = {\frac{\left ( 1 - \zeta \right )}{\kappa F_{1} \left ( \zeta \right ) F_{2} \left ( Ri \right )}} }[/math] (15)
[math]\displaystyle{ {\frac{dc}{d \zeta}} = {\frac{1}{\kappa u_{*r}}} {\frac{1}{F{1} \left ( \zeta \right ) F_{2} \left ( Ri \right ) }} c }[/math] (16)
[math]\displaystyle{ Ri = - Ri_{*} {\frac{{\frac{dc}{d \zeta}}}{\left ( {\frac{du}{d \zeta}} \right ) ^2 }} }[/math] (17)
[math]\displaystyle{ u | _{*r} = {\frac{u_{*}}{v_{s}}} }[/math] (18)
[math]\displaystyle{ Ri_{*} = {\frac{R g H c_{r} ^*}{u_{*} ^2}} }[/math] (19)
- Forms for the functions F_{1} and F{2}
[math]\displaystyle{ F{1} = \zeta \left ( 1 - \zeta \right ) }[/math] (20)
- Simth and McLean (1977)
ζ_{r} <= ζ < 0.3
[math]\displaystyle{ F_{1} = \zeta + 1.32892 \zeta ^2 - 16.8632 \zeta ^3 + 25.22663 \zeta ^4 }[/math] (21)
0.3 <= ζ <= 1
[math]\displaystyle{ F{1} = 0.160552 +0.075605 \zeta -0.1305618 \zeta ^2 - 0.1055945 \zeta ^3 }[/math] (22)
Gelfenbaum and Smith (1986)
[math]\displaystyle{ F_{1} = \zeta exp \left ( - \zeta - 3.2 \zeta ^2 + {\frac{2}{3}} \zeta ^2 \right ) }[/math] (23)
Smith and McLean (1977)
[math]\displaystyle{ F{2} = 1 - 4.7 Ri }[/math] (24)
Gelfenbaum and Smith (1986)
[math]\displaystyle{ F_{2} = {\frac{1}{1 + 10.0 X}} }[/math] (25)
[math]\displaystyle{ X = {\frac{1.35 Ri}{1 + 1.35 Ri}} }[/math] (26)
- Form for near-bed concentration
[math]\displaystyle{ c_{r} ^* = {\frac{A X_{e} ^*}{1 + {\frac{A}{0.3} X_{e} ^5}}} }[/math] (27)
[math]\displaystyle{ X_{e} = {\frac{u_{*s}}{v_{s}}} Re_{p} ^ \left ( 0.6 \right ) }[/math] (28)
- Solution equation
[math]\displaystyle{ u = {\frac{1}{\kappa}} ln \left ( 30 {\frac{H}{k_{s}}} \right ) }[/math] (29)
[math]\displaystyle{ c = \left ( {\frac{\left ( 1 - \zeta \right ) / \zeta }{\left ( 1 - \zeta _{r} \right ) \zeta _{r}}}\right ) ^ \left ( {\frac{1}{\kappa u_{*r}}} \right ) }[/math] (30)
[math]\displaystyle{ Ri =Ri_{*} {\frac{\kappa \zeta F_{2}}{u_{*r} \left ( 1 - \zeta \right ) }} c }[/math] (31)
| Symbol | Description | Unit |
|---|---|---|
| u* | streamwise velocity | m / s |
| c* | volume suspended sediment concentration | - |
| H | depth | m |
| u* | shear velocity | m / s |
| u*s | shear velocity due to skin friction | m / s |
| kc | composite roughness | - |
| U | depth-averaged flow velocity | m / s |
| ks | grain roughness | - |
| ρ | water density | kg / m3 |
| ν | kinematic viscosity | - |
| ρs | sediment density | kg / m3 |
| vs | fall velocity | m / s |
| Rep | Reynolds number | - |
| g | gravitational acceleration | m / s2 |
| νt | kinematic eddy viscosity | - |
| κ | Karman constant, equals to 0.4 | - |
| Ri | gradient Richardson number | - |
| A | coefficient, equals to 1.3 * 10-7 | - |
| Dsx | size in the surface material, such that x percentage of the material is finer | mm |
Notes
Any notes, comments, you want to share with the user
Numerical scheme
Examples
An example run with input parameters, BLD files, as well as a figure / movie of the output
Follow the next steps to include images / movies of simulations:
- Upload file: http://csdms.colorado.edu/wiki/Special:Upload
- Create link to the file on your page: [[Image:<file name>]].
See also: Help:Images or Help:Movies
Developer(s)
References
Dietrich, W. E. 1982 Settling velocity of natural particles. Water Resources Research, 18(6), 1615-1626.
Garcia, M. and Parker, G. 1991 Entrainment of bed sediment into suspension. J. Hydraul. Engrg., ASCE, 117(4), 414-435.
Gelfenbaum, G. and Smith, J. D. 1986 Experimental evaluation of a generalized suspended-sediment transport theory. In Shelf and Sandstones, Canadian Society of Petroleum Geologists Memoir II, Knight, R. J. and McLean, J. R., eds., 133 – 144.
Smith, J. D. and McLean, S. R. 1977 Spatially averaged flow over a wavy surface. J. Geophys. Res., 82(2), 1735-1746.
