# 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

Parameter Description Unit
Input directory path to input files
Site prefix Site prefix for Input/Output files
Case prefix Case prefix for Input/Output files
Parameter Description Unit
Specific gravity of sediment -
Sediment grain size mm
Flow depth m
Composite Roughness height (including bedform effects) mm
Shear velocity shear velocity cm / s
Kinematic Viscosity of Water cm2 / s
Shear velocity due to Skin Friction cm / s
Parameter Description Unit
Model name name of the model -
Author name name of the model author -

## 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

 $\displaystyle{ k_{c} = {\frac{11 H}{e^ \left ( \kappa C z \right ) }} }$ (1)

2) Dimensionless Chezy resistance coefficient

 $\displaystyle{ Cz = C_{f} ^ \left ( {\frac{-1}{2}} \right ) ={\frac{U}{u_{*}}} }$ (2)

3) Submerged specific gravity of the sediment

 $\displaystyle{ R = {\frac{\rho _{s}}{\rho}} - 1 }$ (3)

4) Explicit particle Reynolds number

 $\displaystyle{ Re_{p} = {\frac{\sqrt {R g D } D}{\nu}} }$ (4)

5) Fall number

 $\displaystyle{ R_{f} = f \left ( Re_{p} \right ) = {\frac{v_{s}}{\sqrt { R g D }}} }$ (5)
• Basic forms

1) Momentum conservation equation for the flow

 $\displaystyle{ \nu _{t} {\frac{d \bar{u}}{dz}} = u_{*} ^2 \left ( 1 - {\frac{z}{H}} \right ) }$ (6)

2) Conservation equation for the suspended sediment

 $\displaystyle{ v_{s} \bar{c} + \nu _{t} {\frac {d \bar{c}}{dz}} = 0 }$ (7)

3) Eddy viscosity

 $\displaystyle{ \nu_{t} = \kappa u_{*} H F_{1} \left ( \zeta \right ) F_{2} \left ( Ri \right ) }$ (8)

4) Dimensionless upward normal coordinate

 $\displaystyle{ \zeta = {\frac{z}{H}} }$ (9)

 $\displaystyle{ Ri = -Rg {\frac{{\frac{d \bar{c}}{dz}}}{\left ( {\frac{d \bar{u}}{dz}} \right ) ^2}} }$ (10)

6) Bottom boundary condition of velocity (using the rough logarithmic law)

 $\displaystyle{ {\frac{\bar{u} |_{\zeta_{r}}}{u_{*}}} = {\frac{1}{\kappa}} ln \left ( 30 \zeta _{r} {\frac{H}{k_{c}}} \right ) }$ (11)
• Dimensionless forms
 $\displaystyle{ u={\frac{\bar{u}}{u_{*}}} }$ (12)
 $\displaystyle{ c = {\frac{\bar{c}}{\bar{c} _{r}}} }$ (13)
 $\displaystyle{ {\frac{du}{d\zeta}} = {\frac{\left ( 1 - \zeta \right )}{\kappa F_{1} \left ( \zeta \right ) F_{2} \left ( Ri \right )}} }$ (14)
 $\displaystyle{ {\frac{dc}{d \zeta}} = {\frac{1}{\kappa u_{*r}}} {\frac{1}{F_{1} \left ( \zeta \right ) F_{2} \left ( Ri \right ) }} c }$ (15)
 $\displaystyle{ Ri = - Ri_{*} {\frac{{\frac{dc}{d \zeta}}}{\left ( {\frac{du}{d \zeta}} \right ) ^2 }} }$ (16)
 $\displaystyle{ u_{*r} = {\frac{u_{*}}{v_{s}}} }$ (17)
 $\displaystyle{ Ri_{*} = {\frac{R g H \bar{c} _{r}}{u_{*} ^2}} }$ (18)
• Forms for the functions F_{1} and F{2}

1) Standard form for the function F1

 $\displaystyle{ F{1} = \zeta \left ( 1 - \zeta \right ) }$ (19)

2)Alternative form for the function F1 (Simth and McLean (1977)) ζ_{r} <= ζ < 0.3

 $\displaystyle{ F_{1} = \left\{\begin{matrix} \zeta + 1.32892 \zeta ^2 - 16.8632 \zeta ^3 + 25.22663 \zeta ^4 & \zeta _{r} \lt = \zeta \lt 0.3 \\ 0.160552 +0.075605 \zeta -0.1305618 \zeta ^2 - 0.1055945 \zeta ^3 & 0.3 \lt = \zeta \lt = 1 \end{matrix}\right. }$ (20)

3) Alternative form for the function F1 (Gelfenbaum and Smith (1986))

 $\displaystyle{ F_{1} = \zeta exp \left ( - \zeta - 3.2 \zeta ^2 + {\frac{2}{3}} \zeta ^2 \right ) }$ (21)

4) Form for function F2 (Smith and McLean (1977))

 $\displaystyle{ F{2} = 1 - 4.7 Ri }$ (22)

5) Alternative form for the function F2 (Gelfenbaum and Smith (1986))

 $\displaystyle{ F_{2} = {\frac{1}{1 + 10.0 X}} }$ (23)
 $\displaystyle{ X = {\frac{1.35 Ri}{1 + 1.35 Ri}} }$ (24)
• Form for near-bed concentration

specification for reference sediment concentration

 $\displaystyle{ \bar{c} _{r} = {\frac{A X_{e} ^*}{1 + {\frac{A}{0.3} X_{e} ^5}}} }$ (25)
 $\displaystyle{ X_{e} = {\frac{u_{*s}}{v_{s}}} Re_{p} ^ \left ( 0.6 \right ) }$ (26)
• Solution equation

1) Calculation for velocity

 $\displaystyle{ u = {\frac{1}{\kappa}} ln \left ( 30 {\frac{H}{k_{s}}} \zeta \right ) }$ (27)

2) Calculation for suspended sediment concentration

 $\displaystyle{ c = \left ( {\frac{\left ( 1 - \zeta \right ) / \zeta }{\left ( 1 - \zeta _{r} \right ) \zeta _{r}}}\right ) ^ \left ( {\frac{1}{\kappa u_{*r}}} \right ) }$ (28)

 $\displaystyle{ Ri =Ri_{*} {\frac{\kappa \zeta F_{2}}{u_{*r} \left ( 1 - \zeta \right ) }} c }$ (29)

## Notes

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:

## 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.