# 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)