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
This will be something that the CSDMS facility will add
Provides ports
This will be something that the CSDMS facility will add
Main equations
- definitions
1) Composite bed roughness
[math]\displaystyle{ k_{c} = {\frac{11 H}{e^ \left ( \kappa C z \right ) }} }[/math] (1)
2) Dimensionless Chezy resistance coefficient
[math]\displaystyle{ Cz = C_{f} ^ \left ( {\frac{-1}{2}} \right ) ={\frac{U}{u_{*}}} }[/math] (2)
3) Submerged specific gravity of the sediment
[math]\displaystyle{ R = {\frac{\rho _{s}}{\rho}} - 1 }[/math] (3)
4) Explicit particle Reynolds number
[math]\displaystyle{ Re_{p} = {\frac{\sqrt {R g D } D}{\nu}} }[/math] (4)
5) Fall number
[math]\displaystyle{ R_{f} = f \left ( Re_{p} \right ) = {\frac{v_{s}}{\sqrt { R g D }}} }[/math] (5)
- Basic forms
1) Momentum conservation equation for the flow
[math]\displaystyle{ \nu _{t} {\frac{d \bar{u}}{dz}} = u_{*} ^2 \left ( 1 - {\frac{z}{H}} \right ) }[/math] (6)
2) Conservation equation for the suspended sediment
[math]\displaystyle{ v_{s} \bar{c} + \nu _{t} {\frac {d \bar{c}}{dz}} = 0 }[/math] (7)
3) Eddy viscosity
[math]\displaystyle{ \nu_{t} = \kappa u_{*} H F_{1} \left ( \zeta \right ) F_{2} \left ( Ri \right ) }[/math] (8)
4) Dimensionless upward normal coordinate
[math]\displaystyle{ \zeta = {\frac{z}{H}} }[/math] (9)
5) Gradient Richardson number
[math]\displaystyle{ Ri = -Rg {\frac{{\frac{dc^*}{dz}}}{\left ( {\frac{du^*}{dz}} \right ) ^2}} }[/math] (10)
6) Bottom boundary condition of velocity (using the rough logarithmic law)
[math]\displaystyle{ {\frac{\bar{u} |_{\zeta_{t}}}{u_{*}}} = {\frac{1}{\kappa}} ln \left ( 30 \zeta _{r} {\frac{H}{k_{c}}} \right ) }[/math] (11)
- Dimensionless forms
[math]\displaystyle{ u={\frac{\bar{u}}{u_{*}}} }[/math] (12)
[math]\displaystyle{ c = {\frac{\bar{c}}{\bar{c} _{r} ^*}} }[/math] (13)
[math]\displaystyle{ {\frac{du}{d\zeta}} = {\frac{\left ( 1 - \zeta \right )}{\kappa F_{1} \left ( \zeta \right ) F_{2} \left ( Ri \right )}} }[/math] (14)
[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] (15)
[math]\displaystyle{ Ri = - Ri_{*} {\frac{{\frac{dc}{d \zeta}}}{\left ( {\frac{du}{d \zeta}} \right ) ^2 }} }[/math] (16)
[math]\displaystyle{ u | _{*r} = {\frac{u_{*}}{v_{s}}} }[/math] (17)
[math]\displaystyle{ Ri_{*} = {\frac{R g H \bar{c} _{r} ^*}{u_{*} ^2}} }[/math] (18)
- Forms for the functions F_{1} and F{2}
1) Standard form for the function F1
[math]\displaystyle{ F{1} = \zeta \left ( 1 - \zeta \right ) }[/math] (19)
2)Alternative form for the function F1 (Simth and McLean (1977)) ζ_{r} <= ζ < 0.3
[math]\displaystyle{ F_{1} = \left\{\begin{matrix} \zeta + 1.32892 \zeta ^2 - 16.8632 \zeta ^3 + 25.22663 \zeta ^4 & \zeta _{r} \lt = \zeta \lt 0.3 \\ 0.160552 +0.075605 \zeta -0.1305618 \zeta ^2 - 0.1055945 \zeta ^3 & 0.3 \lt = \zeta \lt = 1 }[/math] (20)
3) Alternative form for the function F1 (Gelfenbaum and Smith (1986))
[math]\displaystyle{ F_{1} = \zeta exp \left ( - \zeta - 3.2 \zeta ^2 + {\frac{2}{3}} \zeta ^2 \right ) }[/math] (21)
4) Form for function F2 (Smith and McLean (1977))
[math]\displaystyle{ F{2} = 1 - 4.7 Ri }[/math] (22)
5) Alternative form for the function F2 (Gelfenbaum and Smith (1986))
[math]\displaystyle{ F_{2} = {\frac{1}{1 + 10.0 X}} }[/math] (23)
[math]\displaystyle{ X = {\frac{1.35 Ri}{1 + 1.35 Ri}} }[/math] (24)
- Form for near-bed concentration
specification for reference sediment concentration
[math]\displaystyle{ \bar{c} _{r} ^* = {\frac{A X_{e} ^*}{1 + {\frac{A}{0.3} X_{e} ^5}}} }[/math] (25)
[math]\displaystyle{ X_{e} = {\frac{u_{*s}}{v_{s}}} Re_{p} ^ \left ( 0.6 \right ) }[/math] (26)
- Solution equation
1) Calculation for velocity
[math]\displaystyle{ u = {\frac{1}{\kappa}} ln \left ( 30 {\frac{H}{k_{s}}} \right ) }[/math] (27)
2) Calculation for suspended sediment concentration
[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] (28)
3) Gradient Richardson number
[math]\displaystyle{ Ri =Ri_{*} {\frac{\kappa \zeta F_{2}}{u_{*r} \left ( 1 - \zeta \right ) }} c }[/math] (29)
Symbol | Description | Unit |
---|---|---|
u* | streamwise velocity | m / s |
c* | volume suspended sediment concentration | - |
H | water 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 |
R | submerged specific gravity | - |
D | grain diameter | mm |
k | composite roughness height (includes bedform effects) | mm |
C | shear velocity due to skin friction | cm / s |
η | height above bed surface relative to the water surface height | - |
uno | normalized velocity without stratification | - |
cno | normalized concentration without stratification | - |
un | normalized velocity with stratification | - |
cn | normalized concentration with stratification | - |
Ri | gradient Richardson number | - |
Notes
The C value in the inputs can be one of two things: either it is the Shear Velocity due to Skin Friction, OR it is the Reference Volume Concentration; the user should input the value that they want to use, and the program will prompt the user, what they want this value to be (either use Garcia-Parker to find the Reference Concentration, or user-inputted).
The program will run through however many iterations it takes (up to 200) for the error on all the cn and un values to be less than 0.001 to account for the stratification effects.
The reference height is set at 0.05H, the number of intervals is set at 50, the constant of Von Korman, κ, is given a value of 0.4.
There is no GetData function for this program, because there is no time loop for which values may need to be retrieved.
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.