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

[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{d \bar{c}}{dz}}}{\left ( {\frac{d \bar{u}}{dz}} \right ) ^2}} }[/math]

(10)

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

[math]\displaystyle{ {\frac{\bar{u} |_{\zeta_{r}}}{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 \end{matrix}\right. }[/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}}} \zeta \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)

Nomenclature

Symbol	Description	Unit
ū	streamwise velocity	L / T
bar{c}	volume suspended sediment concentration	M / L³
z	coordinate upward normal from bed	L
H	water depth	L
u_*	shear velocity	L / T
u_*s	shear velocity due to skin friction	L / T
k_c	composite bed roughness(includes bedform effects)	-
U	depth-averaged flow velocity	L / T
k_s	grain roughness	-
ρ	water density	M / L³
ν	kinematic viscosity	-
C_z	dimensionless Chezy resistance coefficient	-
C_f	bed friction coefficient	-
D	sediment size	L
ρ_s	sediment density	M / L³
v_s	fall velocity	L / T
F₁, F₂	specified functions	-
ζ	dimensionless upward normal coordinate	-
Re_p	explicit particle Reynolds number	-
g	gravitational acceleration	L / T²
R_f	fall number	-
ν_t	kinematic eddy viscosity	-
κ	Von Karman constant, equals to 0.4	-
Ri	gradient Richardson number	-
u	dimensionless velocity	-
c	dimensionless suspended sediment concentration	-
bar{c}_r	near-bed sediment concentration	-
A	coefficient, equals to 1.3 * 10^-7	-
u_*s	shear velocity due to skin friction only	L / T
X_e	similarity variable for uniform sediment	-
D_sx	size in the surface material, such that x percentage of the material is finer	L
R	sediment submerged specific gravity	-
ζ_r	normalized reference bed elevation	L
Ri_*		-

Output

Symbol	Description	Unit
η	height above bed surface relative to the water surface height	L
u_no	normalized velocity without stratification	L / T
c_no	normalized concentration without stratification	M / L³
u_nao	depth-averaged normalized velocity without stratification effect	L / T
c_nao	depth-averaged normalized concentration without stratification effect	M / L³
q_so	depth-aveeraged normalized volume suspended sediment transport per unit width without stratification effect	L² / T
u_na	depth-averaged normalized velocity with stratification effect	L / T
c_na	depth-averaged normalized concentration with stratification effect	M / L³
q_s	depth-averaged normalized volume suspended sediment transport per unit width with stratification effect	M / L³

Notes

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