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
First parameter	Description parameter	[Units]

Parameter	Description	Unit
First parameter	Description parameter	[Units]

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

[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 {R g D } D}{\nu}} }[/math]

(4)

[math]\displaystyle{ R_{f} = f \left ( Re_{p} \right ) }[/math]

(5)

[math]\displaystyle{ R_{f} = {\frac{v_{s}}{\sqrt { R g D }}} }[/math]

(6)

Basic forms

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

(7)

[math]\displaystyle{ v_{s} \bar{c} + \nu _{t} {\frac {d \bar{c}}{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{\bar{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{\bar{u}}{u_{*}}} }[/math]

(13)

[math]\displaystyle{ c = {\frac{\bar{c}}{\bar{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 \bar{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{ \bar{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)

Nomenclature

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
k_c	composite roughness	-
U	depth-averaged flow velocity	m / s
k_s	grain roughness	-
ρ	water density	kg / m³
ν	kinematic viscosity	-
ρ_s	sediment density	kg / m³
v_s	fall velocity	m / s
Re_p	Reynolds number	-
g	gravitational acceleration	m / s²
ν_t	kinematic eddy viscosity	-
κ	Karman constant, equals to 0.4	-
Ri	gradient Richardson number	-
A	coefficient, equals to 1.3 * 10^-7	-
D_sx	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	-
u_no	normalized velocity without stratification	-
c_no	normalized concentration without stratification	-
u_n	normalized velocity with stratification	-
c_n	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 c_n and u_n 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>]].

Developer(s)

Gary Parker

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.

Links

[Model:SuspSedDensityStrat]